Reductio ad Absurdum

twins game

2011-06-01T19:46:00.006+05:30

Will and I got to go to a Twins game last night. We saw the LG Twins beat the Kia Tigers at Jamsil Stadium. It rained on us for most of the game, but Will was a good sport and we made it nine innings. A bit different from an American baseball game, but not importantly. We sat in center field, it was a good game with a lively crowd. Mostly I just enjoyed seeing baseball. We got to see some of the fun bits, like a couple of double plays, a guy stealing two bases on an overthrow, an attempt at stealing first on a dropped third strike. While I momentarily attempted to explain some of these events, Will's questions showed that he was still sorting out how many teams there were, what the black team (the umpires) was up to, and how to predict the next exciting hit to our local center fielder.

This last picture is my favorite, I think:

I was appreciating that baseball really has such a fantastic point of interest more specifically than other sports. Everyone in the game and watching the game is a chaos of attentions until the windup and the pitch, and we share a moment of intense expectation and focus. The evening is punctuated by dozens of these moments, and while many of them pass without remark, a big play means that we all share it, and I can look around and know that I saw exactly what the people around me saw. The fact that the experience is about a ball thrown around by people we do not know is immaterial - what matters is perhaps that we see something, anything, that so many other people see with us, we share a thrill, and I wonder if that is part of what makes sports enjoyable.

I remember Coach Vogel showing me that I didn't have to fear the ball, I didn't have to stand out there hoping they didn't hit to me. He taught me to creep in with the windup, to anticipate and to be chomping at the bit, to hope to be the lucky guy that gets to play that ball. I miss it. I miss the wild concentration and that asymptotic excitement of the pitch, when everyone commits at once to their carefully weighed decision. I forgot how much I missed baseball until a couple weeks ago when I got to play catch with a few of the teachers here.

escher waterfall illusion

2011-04-13T08:42:00.001+05:30

This is admirably done.

kajsa is here!

2011-03-02T11:37:00.003+05:30

Our daughter Kajsa was born on February 18. She is very cute.

She is more than a week old, which is why my paternity leave is almost finished. I have been grateful to have someone covering my classes while I have been able to sort out the paperwork and various details surrounding a new baby. Finding Kajsa's birth certificate was not nearly as exciting as it was for Annie, which is good.

Joie's mom was here for a visit and we really appreciated all of her help. She watched our kids, washed our dishes, let us sleep in, made us bullar, and generally took care of us. She walked around Seoul a bit with us as we saw a few more parts of our big city.

a post!

2011-03-02T11:37:00.001+05:30

There is a swelling volume of blog posts out there apologizing for neglected blogs. These insincere notes, when conjured to our screens from the recesses of the internet where they lie etched in indelible patterns of electrical charges that will outlive some of our most intentional achievements, tell of excuses and good intentions. These posts do provide some philosophical substance... to ponder how many such posts occupy space and energy in this internet... to consider how many such posts punctuate our unobserved efforts as we, like Father Mckenzie from Eleanor Rigby, are writing the words of a sermon that no one will hear... But I do not intend to apologize and I have no very good reason for abandoning the blog or for coming back to it. I guess it ends up as such a post anyway.

But I wanted to shout out from South Korea, as I have not yet posted since I moved here several months ago. After a busy summer and a busier semester, I still feel like I have just arrived in Seoul.

leaving

2010-06-16T11:22:00.003+05:30

Well, our last days at Woodstock have been full of packing what we need and getting rid of what we don't. The very real constraints of space and weight and their costs have made us sharpen our ideas of what is important among the piles of our papers and clothes and other junk which has become such a burden in the light of our impending move. It is refreshing and liberating, and a bit melancholy. I feel like Ma Joad sitting next to her wood stove, taking one last look and pushing into the flames those papers that need to not make the trip. Quite literally, as this process has included some actual pushing letters into the wood stove.

That is something special about traveling, is a more frequent need to let go, to evaluate. Sometimes I just want a big attic or a garage with the potential to hold on to any box of papers I simply do not want to decide about. And other times I enjoy the healthy feeling of being rid.

And the leaving is very anticlimactic as we say goodbyes and then see people a few more times, or don't say goodbye because we didn't realize we wouldn't run into somebody during these last few frantic days. C'est la vie.

5-star challenge: hailstone numbers

2010-06-09T14:43:00.005+05:30

Luke asked why I stopped with the math contest questions. As he forms a strong percentage of recent commenters on this blog, I suppose I ought to humour him. The fact is, these questions are tricky to find. I don't want something that just amounts to boring computations, and I don't want something that can be artlessly mined from Google or WolframAlpha.

This is one of the recent 5* challenge problems I ran. It's tied to a set of sequences that could keep me (has kept me) busy for hours. It's not like useful math, it's more the sort that could be warmly amusing and deeply intriguing, the kind of thing I would be thankful for on a desert island.

A hailstone sequence is defined recursively by this equation:

a_n = { 3*(a_n-1)+1 if a_n-1 is odd; (a_n-1)/2 if a_n-1 is even}

For example, if we start with 10, the next term (because 10 is even) is 5, the next term (because 5 is odd) is 16, then 8, then 4, then 2, then 1, then 4, 2, 1, 4, 2, 1… for ever and ever. The shortest hailstone sequence starting with 10 and concluding with 1 is 10, 5, 16, 8, 4, 2, 1 and its length is seven.

Find the sum of all positive integers N such that the length of the shortest hailstone sequence starting with N and concluding with 1 is thirteen.

geogebra applet - taylor polynomial

2010-06-03T10:49:00.005+05:30

I put this on my math blog but I know that some of you don't look at the math blog and boy would I feel bad if you missed out on this opportunity.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Nate Burchell, Created with GeoGebra

brainy legos

2010-06-03T10:33:00.003+05:30

Two Lego projects for your consideration:
A Lego printer
A Lego Rubik's Cube solver
Oh, the people on that internet. My own obsession with Legos was so much more primitive. The internet brings us all so frighteningly close to some properly obscure endeavors.

book: silence

2010-05-25T14:08:00.004+05:30

I just finished reading Silence by Shusaku Endo. It was recommended to me by Bryan, an English teacher at my school who described it as one of the only Christian books he really likes for its literary value. It is a novel about Portuguese missionaries in Japan during the 1600's. Japan had outlawed Christianity and the missionaries who were caught were tortured and killed if they did not apostatize.

The novel explores the concept of inter-cultural missions and a claim that Japan's culture would simply not support Christianity as it was presented by the Europeans. Any Christianity that allegedly did take hold was corrupted and confused in translation. The priest Rodrigues steals into Japan to follow up on the reports that his mentor, an older priest, has defected and denied the faith.

My reaction to this book and my interpretation of the author's message has been so muddled, it was such a distressing story at times, and its value lies in a few bright details, a few words in the book that really seemed to make the story right. For much of the book, I could sense that the ending would determine if I joyed in the story or resented it, and that is how it was. But not just the ending, the few days after the ending, it is the pondering which has made me like this book.

The story painted a murky picture of the confused motives, futile hopes, and deniable victories of the Portuguese missionaries. And all the while, through all the pain, after all the sacrifice, God's silence.

The human events of Silence include as an element the claim that Christianity does not translate and should not be translated into certain cultures. This message of course is especially common in the academic world, but by people who also lean on absolute truth. They appeal to some sort of absolute truth to explain that Christianity absolutely cannot be absolutely true. They claim some sort of universally undeniable logic to say that Christianity cannot be undeniably universal. In the end the most common argument presented against Christianity is the failures of those who promote it, and the story shows the rejection of a Christianity that Christians have failed to make relevant.

In Silence, Endo invites the reader to ask: Can a truth exceed in power our ability to express it? Can God take imperfect human efforts and complete the statement? Can we afford to interpret God's silence as his inaction? Can we presume that our designs are the limit and the fullness of God's actions through us?

moving on

2010-05-17T21:07:00.001+05:30

We are down to our last month in India. After a quick month in the USA, we will be heading off to Seoul, where I will teach at Yongsan International School. Joie and I are very excited about this new direction in our life. We will miss Woodstock and the people here and life in Redwood Cottage, where we have spent most of our last four years, most of our marriage, and most of our time with Will and Annie.

joie's songs

2010-05-10T14:50:00.002+05:30

Joie just finished her CD! You can listen to her 10 songs or download them from her music page. She did a great job with it and I'm so blessed to have such a talented wife.

the doon valley

2010-05-07T23:19:00.001+05:30

the rain

2010-05-07T22:08:00.003+05:30

I had forgotten about Elijah. This morning I woke up to a drumming rain and a grumbling thunder that didn't stop. I was so deeply grateful for the rain, it means much more to me than it used to. In India on this mountain we have seen every year a dry season building up to the monsoon. Every year the rains come, every year the Lord provides. But before the rain, the dryness builds, the people grow strained, the land starts to die. Many years there are wildfires before the rain finally comes. Sometimes there are clouds that disappoint, or rain that stops after a minute or two. When the rain comes, it is a gift, it is life, it cleans and renews a dry land. When I hear these early rains on our tin roof in redwood cottage, I feel excited for every drop and trickle that soaks through between the dusty pebbles and into this thirsty land, I find myself praising God for his provision for this land. I never took the rain so personally.

It reminded me of Elijah, announcing the end of a severe drought, sending the thirsty servant seven times to look toward the sea, to six times stare wearily at a cloudless horizon before spotting a cloud the size of a man's hand. The hope and hopelessness of that poor servant, and the tension of thirst that so much of the world knows year after year as they wait for the rains. I suffer from a terrible memory, and I think that most of the time I spend waiting for refreshment I have indeed a very dry and distant notion of what refreshment must be, and it is generally better than what I thought I was waiting for.

the dark tower

2010-04-09T21:11:00.003+05:30

I just finished reading The Dark Tower by C.S. Lewis. This was a tangent begun serendipitously whilst I was waiting to locate a copy of The Last Battle, to finish my recent trip through the Narnia books. I love his writing, he gives his words so much more meaning than most of us can.

The Dark Tower was gripping from the very first page, promising to be a philosophical story about time travel and souls of men, featuring Dr. Elwin Ransom from the space trilogy. The story centers around a "chronoscope" that reveals to a few professors a sinister semi-parallel Cambridge in some other time. In the end I was sad to find the manuscript come to an abrupt stopping point. Lewis just stopped writing it, apparently not liking it or not knowing how to finish it.

The story (what there was) was a pretty good read, but more along the lines of That Hideous Strength, somewhat disturbing, but with an underlying theme of redemption and clarity. Not as cheery as some of the Narnia books, but rather intriguing. Mostly I was delighted to have found another book with Ransom, it was much like the time I discovered The Club of Queer Trades, featuring another G.K. Chesterton detective that I hadn't known about.

The unfinished story is a perplexing animal of literature, and yet it stands to reason that every great author must leave a few. Some writing is immortalized by publication, but what of that writing that never fully happened? It wasn't really approved and sent out by the writer, does it even count? One could hardly hold the writer responsible if the book did not meet expectations. And yet it was written, and he did hang on to it, perhaps letting it inspire his later books, perhaps waiting to someday return to it. It reminded me of Michelangelo's David statue in Florence, surrounded by half a dozen other huge blocks of marble with partial figures trying to emerge.

biblical mathematics

2010-04-07T16:22:00.002+05:30

Something got me thinking about mathematics and the Christian world view, and about being a Christian math teacher. I have blogged a bit about this before here.

Ecclesiastes 3:11 tells us that God has set eternity in the hearts of men. This reminder is illuminating as we consider the pursuit of mathematics from a Biblical world view. There are evidences of the eternity in our hearts all around us in the way people live their lives, striving to be close to something bigger, something more permanent, something less bound by the temporal cages that define the edges of our existence.

Our God has created us with inquisitive and creative minds that are hard-wired to pursue the infinite, to wonder about our Maker, to ponder eternity. Mathematics has been moved forward throughout history not by practical people but by seekers and worshippers, artists and philosophers, obsessed, committed, awestruck, determined, half-crazed and wild-eyed people who gave their lives to grasp at the fringe of the robes of the only One who can explain the eternity residing in their hearts.

Many students eventually ask, "Why do people care about math?" and we often fail to dignify this, we respond by explaining that math is used to decide how much grass seed you should buy for your lawn, or to divide a box of cookies between you and your n-1 friends, or to calculate a tip at a restaurant. Or to find the volume of a cone or the number of Mondays in the eighteenth century or the roots of a quadratic equation or a way to ferry our foxes and geese safely across the river. Purely practical skills, right? In short, we fail to dignify them. We offer a few empty incentives to learn math, without really addressing their question. We respond almost defensively, as though mathematics would die without our help, or as though we are afraid our students will grow up and be unable to tip at restaurants.

It is in fact a great question deserving of our consideration: why DO people care about math? Instead of reducing the question to something harmless, let students confront it, let it intrigue and inspire them, let it be a context and a backdrop for the math that happens. Let us lead students to questions about knowledge and proof and logic. Let us ask them to think critically when they hear that science and religion should be separate—because they never have been separate. Teaching mathematics is inviting students to enjoy those gems of thought that have no earthly right to be in our minds, whose presence and beauty is only really explained by the eternity that God has set in our hearts.

Why did the Greeks revere a particular number as divine? Why did the Pythagoreans despair at the discovery of irrational numbers? Why did men like Newton give their lives to pursue mathematics and science in tormented seclusion? Why do people gamble? I am not talking merely about a tingly thrill of profundity that drives numerology and superstition, but also the real gripping and consuming rigor of mathematics, the desire to prove and to irrefutably establish the slightest fragment of logically solid ground. Why do we seek such?

If we want our students to think critically about the teaching and learning of mathematics, we might encourage them to look at the people who have been involved through the centuries. They will see that indeed mathematics is the product of earnest attempts to prove or disprove God's existence, to demonstrate the sufficiency of humanity or the greatness of God, to describe the beginning of the universe or predict its end, to map the heavens, to fathom knowledge itself, to experience a moment of the eternity that God has set in the hearts of men.

The Christian world view does offer an explanation for the eternity that we find in our hearts, and it does tell us that people will be obsessed with the study of mathematics and all manner of science. I am not convinced that a God-denying world view offers a satisfactory explanation of these pursuits or the vigor with which humans engage in these pursuits. The motive of math often goes unexplained. Much of Mathematics was built as an altar to an unknown god, and by teaching mathematics I can hope to bring glory to Him who has been worshipped as something unknown.

a novel thought

2010-03-03T16:06:00.003+05:30

I just finished reading the Count of Monte Cristo by Alexandre Dumas. This is probably the longest book I have ever read, but except for a few slow parts it clipped along nicely. I enjoy reading novels and I like to write as well, but just to try to express myself, to formulate a thought that fails to satisfy until it is written. There certainly is a need to articulate. I have never really tried to write people.

You write to express yourself, sure, but it seems that you write a novel to express a person that lives within you whom you are not really excited to share, that you don't know how to share and to own. I have read novels that do not seem thus inspired, but many are. A person needs to write when once they realize that the thoughts and philosophies that define their beliefs are incongruous with one another and would make more sense if presented as the thoughts and philosophies of half a dozen distinct characters. The novel then is a stage for a writer to play out the people within him, the people who tolerate and despise and resent one another, who challenge and sharpen and object to one another. Each character in a novel is an experiment, the writer tries to prove or fears to allow that the person could exist in a life. In this way, the writer gives voice to his vengeance and mercy alike, his acceptance and rejection of life, his assurance of clarity and his confounded despair, dispensed through the voices of a sufficiently large number of people to make such a noise of thoughts believable to the reader, who in turn fancies himself to be more lucid and coherent than he perhaps is.

will quotes

2010-02-28T14:02:00.002+05:30

So I noticed recently that while nobody really seems interested in my posts about abstract computing theories, it is always a hit when I tell funny stories about Will.

Joie has been diligently teaching Will his numbers and his alphabet, but his progress is slow. Every once in a while he casually takes an interest in the games that we try to encourage and one of his favorite questions for us is, "What number starts with 'car'?"

A few days ago on the return trip from the bazaar we found ourselves about to start walking up Mullingar Hill, a long steep winding climb at the near end of our bazaar. The hill is quite steep and we frequently see drivers stop to let traffic pass and then need to roar their engine like a jet to get moving again. Guys on motorcycles sometimes have to make a few of their friends get off and walk up the hill. Will looked up at the coming climb with an air of weary discouragement. "I'm not gonna like this... upping thing," he said, all the more pathetic for being unable to come up with the word "hill".

That's my boy.

valentine candy

2010-02-12T10:58:00.001+05:30

A few days ago we received a package of goodies from Mormor and Morfar. After Will and Annie got a taste of some Valentine's candy we put the box out of reach where they wouldn't be tempted beyond what they can handle, which isn't much. "Pappa, can I please have some more?" "No, you need to wait until later, you've had enough now". Then Will spotted in the package a pack of shiny alphabet stickers, which he recognized as ABCD's but was unable to read. "Pappa, what does this say?" Then Will tried to read it: "Mormor... loves... us... and... we... can... eat... whatever... we... want".

Thanks for the candy, it was a hit.

game of life

2010-02-07T14:38:00.002+05:30

John Conway's Game of Life is not actually a game as there are no players or winners or decisions. Rather, it is a sequence of states that result from a single set of initial conditions by employing a set of a few simple rules. The Game of Life was invented by Cambridge mathematician John Conway in 1970.

miscellany

2010-01-28T19:48:00.004+05:30

Today my dear wife watched our kids while I went down the hill to Dehra Dun to try to track down George Everest's theodolite. It supposedly resides at the Survey of India museum, but it turns out you have to write a letter to the Survey General at least a few days ahead of time to humbly ask permission to visit the museum, and individuals are not given entry, only groups. I thought it made a lot of sense. I mean, without this reasonable measure of having to seek the permission of a government official, we would just have kids strolling into that museum like nobody's business, learning about cartography and history and who knows what else.

I was able to buy a few maps, which have generally been difficult to find here. A letter was not necessary to buy maps, thankfully. I got a fairly nice 1:10,000 map of Mussoorie and it has my house, Redwood Cottage, marked on it and listed on the margin. Our town is pretty big but has only a few roads, so it is not very descriptive to say what road you live on. Instead the houses have names.

As I do not have any theodolite pictures to post here, I thought I would take this opportunity to show a few pictures that I have encountered which I should have posted (or maybe I did and I'm too lazy to browse the back-issues) earlier. The blog is pretty demanding in that I sometimes have a picture I would like to share, but which is a bit on the mundane side and I can't think of a caption at the time and you know how I hate to bore my readers. To begin, here is a picture of me sitting next to the campfire with my kids. We enjoy our campfire and I will miss it when we leave.
Below is a green beetle.

Here is me in a skirt with Kyle at the staff banquet last June. A lungi, actually. They are very masculine in some cultures, you'll just have to take my word for it.

Here is a pattern in the domed gateway looking into the Taj Mahal enclosure. I wondered how they decided that they would need to go from six-point stars to five-point stars.

Here is my first view of the Taj Mahal, as seen from our hotel's rooftop restaurant.

Here is my darling Annie enjoying the matching dresses that Joie made for her and "baby Dechen".

And here are my Will and Annie in their fancy hats from Auntie Dot.

jaipur - amber fort

2010-01-14T10:34:00.011+05:30

Here Will is looking through one of the many little windows in the fort. It was a maze of a place with little rooms everywhere around a big courtyard. It made us all think of hide and seek.

Here I can see Will, Joie and Eleanor across the courtyard. I think Will is showing his claws here. Lately he has been showing his tiger claws and declaring in a fierce growling voice "I am Shere Khan! Give me my man-cub!"

taj ii

2010-01-04T01:05:00.000+05:30

The taj is somewhat photogenic, to be sure.

pine marten

2010-01-04T00:45:00.003+05:30

A week or so ago I finally got a nice shot at a pine marten. These animals live somewhere on the mountainside near our house, and we seem to spot them more often than most people. They are pretty shy though, so most of my pine marten pictures are a bit like those out-of-focus shots the loch ness monster. This one was more cooperative than usual.

the taj mahal

2010-01-02T20:24:00.002+05:30

We made it! The Burchells have finally seen the Taj Mahal. I have dozens of other pictures that look like the rest of the pictures that anyone reading this has probably seen. It is certainly a grand spectacle.

activity week - kuari pass - 3

2009-11-15T21:15:00.004+05:30

This is a peak whose name I do not know. We could see it from the pass, although Nanda Devi was not visible until later. Below is Nanda Devi, a mountain held by Hindus to be very sacred. The Garhwal Himalaya is steeped in Hindu mythology. Darab says that some of the old men still refuse to believe that Nanda Devi has been climbed (it was in 1936 by Tilman and Odell) because mortals could surely not enter the dwelling place of the gods and live.

Indeed such mountains are immensely stirring and one feels an awakening need to worship. It is humbling to wander in the looming presence of such wild danger and heights. It was not a simple walk to get as high as we did, but we topped out a bit higher than the pass, at about 3800 meters or so. Nanda Devi's peak is at 7816 meters, about four kilometers above our highest point.

Below is one of the bridges we crossed. An earthquake made short work of this bridge, which did not seem very old. It is unfortunately common for such structures to be made poorly. A government contractor cuts a few corners to pocket a few rupees. This particular location is very remote and rugged, not a friendly location for a bridge. The region is earthquake prone, and we had to scramble around several broken bridges, one of which had been crushed beneath cottage-sized boulders. When I think of that power, it almost seems worth it to live there until it happens again, just to watch. The river-bed was dry, but in monsoon it evidently warrants a bridge.

And of course we had sunsets all week.

activity week - kuari pass - 2

2009-11-14T20:44:00.006+05:30

I am trying to post more pictures from my hike. The fact is, I took some 800 pictures and I am rather fond of fifty or more. The setting was very generously picturesque, which helped. The above picture was taken at our first view of Kuari Pass, which is the low spot in the top right corner of the picture, on the second ridge, directly above the tree in the foreground. This was a couple of days in to the hike.

Below is a picture of the trail right at the beginning of the hike. Our jeeps dropped us off the first night just across the river from the small village that is visible in this picture. The trail we walked is part of an ancient trade route linking Tibet and India. Much of the path (Darab figured it to be 58 kilometers that we walked) was constructed as shown in the picture, with shale or schist slabs stacked on edge to give a rough but very durable trail. It was not always easy to walk on, but I suppose that in the monsoon rains it would nicely help to meet one's need of not falling off the mountain.

We saw a great deal of this rock, schist, a metamorphic of shimmering appearance. In some places, the path was covered with a fine sparkling dust looked wet in the sunlight. Often there were heaps of it shining beautifully and crushing together like some forgotten treasure.

This is a picture of tufts of grass where people cannot be bothered to gather them. In India, it is no easy task to venture beyond the territory of the ubiquitous fodder cutters. Just when you feel like a rugged adventurer, you see some ancient lady carrying a ten-foot tall pile of grass on her back.

This is the pass from right below. We still had a few zigs and zags, but we could see victory in the blue of the sky. Of course, we still had a great deal of climbing to do after passing the pass, which inspired one student to note that reaching the pass was somewhat anticlimactic. I told him that was the whole point of a pass.

activity week - kuari pass

2009-11-08T21:22:00.005+05:30

I'm back from my activity week Himalayan trek to Kuari Pass in eastern Garhwal, where I had the privilege of beholding some fantastic views of Nanda Devi, a mountain named after a Hindu goddess. The trek was six days of invigoratingly fresh air and breathtaking altitudes. All went well and we didn't freeze or starve or fall into a gorge or get eaten by leopards or mauled by bears.

code snippets

2009-08-31T14:16:00.002+05:30

I think I have found a way to post java code snippets into blogger. Netbeans will export selected code into an html file and the code is formatted nicely. I had only to put a few lines of style (if I can't have class, at least I have style, right?) into my blogger layout. The result is a bit tidier than the screenshots and it can be copied as text, for those of you who like to do that sort of thing.

On the down side, I think that blogger decided to lose my fern picture in the title box during all of that. Conservation of formatting, no doubt.

isPrime algorithm

2009-08-29T22:08:00.010+05:30

This is something of a continuation of my previous post. I stalled out trying to post the code snippets, posted them as cropped screenshots, and finally discovered a mildly convenient way to export from netbeans to html and paste into blogger with the help of a few lines of css in my blog layout.

I find the Project Euler problems helpful because they challenge you to tidy up the algorithms. A question may be of a very approachable nature for small numbers, but the solution for large numbers relies on a solid understanding of what is necessary and sufficient to find the solution.

For example, if you want to create an algorithm that returns "true" if a number is prime and "false" if it is composite, then you might start with this:


public static boolean isPrime(int args){
    int n = args;
    for(int i=2; i<n; i++){
        if(n%i==0){
            return false;
        }
    }
    return true;
}

...which works by checking every integer in the closed interval [2, n-1] as potential factors of n. The condition n%i==0 uses the modulus "%" operator and it is true when the remainder when n is divided by i is equal to zero. If there are any factors of n on that interval, then the code will return a value of "true".

With small numbers and simple tasks, the computer calculates quickly enough to make this seem like a good algorithm. For example, I can use this algorithm to learn in the blink of an eye that 370009919 (for example) is not a prime. However, it wastes a lot of energy and in some circumstances, a lot of time. It is wasteful because it checks 370009917 potential factors. If 370009919 is a composite number, then once we have learned that 2 is not a factor, we do not need to look at other even numbers. We could use:


public static boolean isPrime(int args){
    int n = args;
    if(n==2){
        return true;
    }
    else if(n%2==0){
        return false;
    }
    for(int i=3; i<n; i=i+2){
        if(n%i==0){
            return false;
        }
    }
    return true;
}

...which still covers all of the possibilities, but after determining that 2 is not a factor, it will not look at any more even numbers because i goes up by 2's from 3. It is still quite wasteful, however.

If 370009919 is a composite number, then it must have a factor smaller than or equal to its square root, which is about 19236. If we pass the square root without finding a factor, we know the number is prime and we can stop. This effectively allows us to stop after 1/19236of the work that the previous code ordered. We can take a square root of an int using Java, but the value will be a double. We can still compare our int i to the double sqrtN, but this time we want less-than-or-equal-to "<=" instead of less-than, since the maximum value of the minimum factor may be equal to the suare root, as would be the case for the square of any prime number.


public static boolean isPrime(int args){
    int n = args;
    double sqrtN = Math.sqrt(n);
    if(n==2){
        return true;
    }
    else if(n%2==0){
        return false;
    }
    for(int i=3; i<=sqrtN; i=i+2){
        if(n%i==0){
           return false;
        }
    }
    return true;
}

The code above is correct as long as Math.sqrt(n) is never less than the actual square root of n. If rounding errors and the hidden Java algorithm that computes "sqrt" might permit such a thing, then the code would possibly be incorrect. I could (A) look at the documentation to reassure myself, (B) conservatively make the second condition of the for-loop read: i<=(sqrtN+1), or (C), test the code for all squares of known primes in the range of int values to see if it correctly detects the factor every time. Also, this code will only return a true or a false. If I want to know the factor that was found, I can add a line of code to print out the factor as soon as it is found.


public static boolean isPrime(int args){
    int n = args;
    double sqrtN = Math.sqrt(n);
    if(n==2){
        return true;
    }
    else if(n%2==0){
        return false;
    }
    for(int i = 3; i<=(sqrtN+1); i=i+2){
        if(n%i==0){
            System.out.println("found a factor: "+i);
            return false;
        }
    }
    return true;
}

The smallest factor is 1699.

I have not looked around to see how this algorithm could be further improved. It is a good idea to skip the evens after we determine that two is not a factor, but skipping the multiples of three (or five, and so on) would be, I suspect, too costly to provide an advantage. The BigInteger class has a method that determines if a number is prime, but I do not know if we are guaranteed more efficiency there, and I do not understand the cost should I need to convert to a BigInteger just to find out. I have seen references to the use of stored lists of primes, which would cut down execution time, but it does not address the programming question of primality. Also, if Java's square root algorithm is considered cheap, we could just replace "i<=sqrtN" with "i*i<=n". This would cost an extra multiplication for every value of i, but we could avoid finding the square root.

Many of the more sophisticated primality tests are designed to be quick and probabilistic, meaning there is a small chance that they will give an incorrect result. However, according to Wolfram Mathworld, the fastest general deterministic test, giving absolute certainty, is "Elliptic Curve Primality Proving", which recently took only six years of CPU time to verify the primality of a 20562-digit prime number.

to start programming

2009-08-28T21:12:00.003+05:30

Luke commented on my last post:
I took Java as an undergrad but remember none of it. IF I want to start working on these problems as well as working on programing. What would you suggest... What Language? Helpful Sites? What programs I will need? Books to buy?

Disclaimer: what follows is all very basic but (I hope) at least moderately correct. I am really not a computer programmer and I only presume as far as to say that I am learning and I do enjoy it.

I recommend using Netbeans, which is an IDE (Integrated Development Environment) that can be downloaded from here. Netbeans is the first program of this type that I have used, so I know there are people out there that would argue for other IDE's. Netbeans and the Java Development Kit (JDK) are both free, sponsored by Sun Microsystems. Netbeans is very helpful because it will alert you (without being very distracting) when you have made a syntax error, or when you have created a variable that is not used for anything. Netbeans suggests methods while you are typing, but not in a way that I feel interferes in case you do know what you want to type. It also dims out the code that is 'commented out'.

I work on a need-to-know basis. I try to challenge myself just enough to make use of what I know while learning a little bit at a time. It is a fairly experimental process, so as you try new things you need to be writing programs in which you understand all of the code except maybe one or two lines. As a task becomes more interesting and more necessary, you will find the curiosity to look up how it might be done.

For example, I am getting pretty comfortable with int's, long's, double's and BigInteger's, but there are a few problems that will require me to learn some things about dealing with text, and I have simply not gotten around to it yet. As I try to gain some new skill, I move in little steps. A programmer is supposed to begin with a "Hello, World!" program and go from there. I recommend constructing a few simple programs using the tutorials. After you have something that at least returns a result without errors, you can begin to define variables and construct algorithms.

Sun has some Java tutorials (here is an example) as well as full documentation for the Java language (here is an example). The tutorials include code and some explanations, while the documentation is very technical but it explains the properties of each instance of a given class.

When using Java, you will need to learn the differences of data types. For example, a variable which has been defined as an int cannot exceed a certain value, 2³¹-1 = 2147483647. Its minimum value is -2³¹ = -2147483648. This is because the variable is given 32 bits of disk space, one of which is a sign bit to signify (+) or (-). If your application will encounter bigger numbers than what is expressible as an int, you will need to prepare for that by initiating the variable as a long (64 bits) or a BigInteger, which has arbitrary precision. Sometimes you will need to convert between data types, and sometimes that will involve the risk of a loss of information.

Netbeans supports a trial and error learning technique. You can make small changes in your code and execute the program with a right-click of the mouse. You can track the value of a variable (to see if it is undergoing the changes you are trying to inspire) by putting this line in your code:

System.out.println("a label in quotation marks: "+variableName);

You can use this line to keep track of the progress while the program is running, to see if an anticipated condition is ever met, or to observe where the program stopped doing what you wanted it to do.

There are various tutorials and message boards around the internet, and even reading information about a different language may be helpful. Netbeans has been invaluable as I have been learning the Java language and syntax.

the prairie

2009-08-26T15:03:00.001+05:30

A few days ago I finished reading The Prairie by James Fenimore Cooper. Grandpa gave me that book a few years ago after he and Grandma took a meandering trip westward along Lewis and Clark's route through the American West. He picked it up along the way. I didn't read it until now.

The main character is in this book quite an old man, but Cooper alludes to different parts of his life as told in the previous books, of which I have read only The Last of the Mohicans. The character is one Natty Bumppo, a man of the frontier, a warrior, woodsman, and wanderer. In this book he is referred to only as "the trapper", with but a single mention of the initials of his proper name. He is above all honorable and unassuming. The bit about his identity though... In each of the books, he is something else to the people around him and he is called by their perceptions. Near the end of his life, this man of few possessions, having lived such an unfettered existence as God gives the grass of the field, asks for a stone on his grave, bearing his name. His name, used so seldom in his life--it does not appear in this book--is treasured and protected as the very center of his identity.

new hobby

2009-08-18T21:11:00.002+05:30

I have stumbled across this website off and on through my education and the forces which inspire participation have finally converged in time and space to affect my joining up.

Project Euler is a website which simply poses questions of mathematical (primarily number theory) interest which generally require a clever algorithm and expression in some computer language. The questions are well stated and challenging.

You can register and let the site keep track of which problems you have solved. When you solve a problem, you can look at a message board with comments and code snippets from people who have also already arrived at the solution. This has been a good challenge for me as I try to learn and relearn how to program.

The last problem I solved (don't worry, I won't give away the answer) asked for the sum of the prime factors of _20,000,000C_15,000,000, which is a rather large number that can be found about a quarter of the way through the twenty-million and first row of Pascal's triangle, but I rarely write that many rows.

Leonhard Euler was a Swiss mathematician who would have enjoyed these problems. Blind and hundreds of years before Java, he still would have knocked the socks off the likes of me.

between the clouds

2009-08-16T23:07:00.002+05:30

Rows and floes of angel hair

And ice cream castles in the air

And feather canyons evrywhere

I've looked at clouds that way

-Joni Mitchell

I took this picture from our neighbors' front terrace. We sipped tea with Dot and watched the drama of the firmament against the rare clarity of this particular monsoon afternoon. It was very clear when we arrived, and I took this picture just after the last fir-covered hilltop was smothered by the clouds. I have always been perplexed to find myself between two ranges of clouds which each seem to know their place. The tops of the bottom clouds and the bottoms of the top clouds seem so content with their boundaries that I stare puzzled, stuck in the middle of it all. Another fifteen minutes and our own hill was in a dense fog.

to burn or not to burn

2009-08-12T15:33:00.002+05:30

This one is for my sister Sarah. I guess I have been mulling over her blog post from May about "hyperbaking" and a more recent post of hers about finding the proportion of the benefits gleaned from biking and running. At the very heart of it she is talking about fastidiously saving calories while cooking and ravenously consuming them while exercising. Something about the juxtaposition of these ideas struck the ironical chords that reside deep within me. Cooking like Bob Cratchit thaws his fingers by a candle and exercising like Nebuchadnezzar's slave stoking the furnace.

Essentially, hyperbaking or hypercooking is a neat way of not wasting as much heat. You cook noodles, for example, by bringing water to a boil and turning off the flame, letting the heat of the water do the cooking. You throw out less-than-boiling water instead of boiling water and you have used that heat in a more productive way than usual. It is a good idea, but of course it would be a bit of work to get the timing correct for a given project. I suppose there are recipes around to help with that. On the downside, it seems counter-productive if you are just making some sort of yummy dessert to power your not-so-pragmatic running habits.

Sarah's recent post about running a mile being equivalent to biking four miles made me think of this post by Steven Levitt on the Freakonomics blog. He suggested that recreational exercise should be taxed for its consumption of fuel and ultimate acceleration of global warming. In my own twisted fashion, I could not help but wonder if running a mile is somehow equivalent to burning a styrofoam cup. Or three.

Ah, global warming... Unless of course it turns out to be global cooling, in which case we will need to be less efficient in all that we do, right? We would probably have a "cash for hybrids" program and redistribute all of those globe-warming machines that are so uselessly sitting around now. We could then feel free to start a nuclear war, and environmentally-conscious people would probably popularize hypobaking, in which one does not shut the oven door. And recreational exercise would naturally be encouraged.

the korean dmz

2009-07-17T16:14:00.014+05:30

This is Joie (and a ROK soldier) standing next to the table where the armistice was signed that ended the fighting of the Korean War.

As I mentioned in an earlier post, Joie and I had the privilege of visiting the Korean De-Militarized Zone (DMZ) that forms the border of North Korea and South Korea. The tour was organized by the USO in Seoul, you cannot just drive up to this place. The DMZ is a strip of land on either side of a very specific line of demarcation. In the picture above, the line is marked by the concrete slab between the South's gravel and the North's dirt. At the top of the stairs there is a North Korean soldier staring at the tour group with a large pair of binoculars. To his right is an open window with a very big camera lens visible. They probably have a file about me now.

The blue buildings belong to the UN and the grey buildings are for the North Koreans. They are not generally in such close proximity as these two buildings, but in the JSA (Joint Security Area) the soldiers spend time every day staring at the enemy. The JSA at Panmunjom is the only point of remotely diplomatic contact between the two nations. And it is remotely diplomatic. The rest of the border is not to be crossed under any circumstances.

The bridge in this picture is called "The Bridge of No Return" for its role in two major operations in which soldiers were allowed to return North or return South some time after the war.

The above picture shows the North Korean 'propaganda village' built near the demarcation to demonstrate (to envious South Korean eyes) the grandeur and lavish living enjoyed by the communists. In reality, much of the world fears that North Koreans starve to death in droves, suffer in poverty and probably receive little or nothing of the aid that is sent there from the rest of the world. Our guide informed us that the village is generally deserted and some of the buildings are only facades. The south has a similar village that exaggerates the green grass of democracy. A feisty little show of escalating nationalism left each village with an enormous flagpole.

For this picture, I was standing in the conference building looking over the line to South Korea, where a vigilant Republic of Korea (ROK) soldier stands in a half-exposed location. The soldiers in the DMZ (being demilitarized) are only allowed, under the provisions of the armistice, to carry single-shot weapons.

Joie and I appreciated the chance to see this situation, this relic of a recent war. It boggles the mind to witness the tension there and the profound effort that is needed to keep a peace. The line is so arbitrary and artificial, a division that does not want to be there, and so many forces press to renew and resolve, one way or the other, a suspended conflict. As Robert Frost says, "Good fences make good neighbors."

more korea pictures

2009-07-14T14:08:00.003+05:30

While in Seoul to visit Susan, a friend of ours from university, we took a brief look around the outside of the Korean War Memorial. They have a collection of American and Korean aircraft on display.

We saw several school groups there, dozens of little kids in matching t-shirts...
...and matching backpacks.

my favorite will

2009-07-14T13:42:00.005+05:30

This is my little boy watching life go on in Sarita Vihar in Delhi. He was waiting for his mother to choose some cloth from a shop there. Will goes for days on end without seeing so much as an automobile, so the chaos of the big city is captivating, to say the least. Below is Will with Dr. Varma, the neurologist in Delhi who has been such an immense help to us. Will speaks very highly of Dr. Varma and was excited to see him again.

back from korea

2009-07-12T17:10:00.002+05:30

We have returned safely from our trip to Korea. This is a picture of a South Korean soldier guarding the door to North Korea in Panmunjom. Needless to say, we were not allowed through that door. Joie and I went on a tour the the De-Militarized Zone (DMZ) and the Joint Security Area for a closeup of a very tense armistice.

calculus rhapsody video

2009-06-04T10:53:00.004+05:30

I'm not sure this will work. I feel really low-tech right now. My internet connection is really slow so YouTube is not a regular part of my life or anything, but I thought that this was well done. I showed it in class and only 4 of 22 students have ever heard the real song, Bohemian Rhapsody by Queen. Occasionally I feel old.

It reminds me of the time my high school math teacher read The Beloit List in our class. This is a list of various cultural phenomena which the current high school graduate would take for granted. Reading it to the current high school graduates is not without its irony, as the students respond (as predicted) to every statement with "yeah, so, hasn't it always been like that?" and the teacher alone has the opportunity to be amused. I am used to being the only one amused in my class.

It also made me realise that it is an ironical yet not uncommon occurrence to be exposed to a parody before you are aware of the original.

milk and honey

2009-05-14T10:40:00.005+05:30

Two nights ago, for a Will and Annie bedtime story I read about the twelve spies who went into Canaan. ...ten were bad and two were good... Caleb and Joshua were the fearless minority who considered it worth the risk. They were excited to see what God had for his people.

I can remember reading this before and thinking of Caleb and Joshua as faithful, adventurous, courageous, obedient and righteous in their attitude toward the task. The others always seemed like wet blankets. Cowards.

This time when I read it, I seemed to hear ten reasonable voices concerned about the feasibility of a plan, the numbers and dimensions of the enemies, and the strength of the strongholds. Joshua and Caleb seemed like goofy little kids saying, "Yeah, but look at the size of these grapes! Moses can we go, can we can we can we, PLEEEEEEEEEEEASE!?" And it sounded reasonable to think about not going. And it sounded ridiculous to risk your neck for a taste of milk and honey. And I realized that I have become a coward.

Numbers 13:27 They gave Moses this account: "We went into the land to which you sent us, and it does flow with milk and honey! Here is its fruit. ²⁸ But the people who live there are powerful, and the cities are fortified and very large. We even saw descendants of Anak there.
...
³⁰ Then Caleb silenced the people before Moses and said, "We should go up and take possession of the land, for we can certainly do it."³¹But the men who had gone up with him said, "We can't attack those people; they are stronger than we are."³²And they spread among the Israelites a bad report about the land they had explored. They said, "The land we explored devours those living in it. All the people we saw there are of great size."

They saw the same things, and there is nothing to indicate that they differed on their assessment of the problem, only on their assessment of the risk. Caleb and Joshua had seen the soldiers and the terrain and the fortifications. But they also saw God's promise as a tangible security, and they breathed richly of the blessings of his plans. I suspect that for Joshua and Caleb, the time in Canaan was an amazing experience, a vibrant and overwhelming exposure to God's provisions for his people, an invigorating and reassuring security. Their comrades on the same trip slept in the same hills, walked on the same paths, crossed the same rivers, ate the same fruit and paced the same valleys to survive a terrifying foray into enemy land. No, they didn't see different things. Joshua and Caleb simply knew the value of milk and honey.

God, please grant me courage to pursue your blessings.

solution: streets and avenues

2009-05-05T16:42:00.002+05:30

There are 729 ways to travel from the corner of 3rd Ave & 12th St to the corner of 8th Ave and 5th St. In the problem statement, I did not specify that north or west made the numbers go up or down. Correct diagrams could therefore differ, but I did not want to distract from the main riddle, that you must travel five blocks in one direction and seven blocks in the perpendicular direction.

Any valid shortest distance by road will need to be 12 blocks total, five of those being one direction and seven being in the other direction. In my diagram, I need to go east and north. As long as I go five blocks east, seven blocks north, zero blocks west and zero blocks south, I will arrive at the specified location. I can mix up the order...

In other words, I need to travel 12 blocks and 5 of them need to be east (the rest north). Which ones should be east? The answer is a combination, ₁₂C₅ = 12!/(5!*7!) = 729 .

solution: burning tent

2009-05-05T16:22:00.003+05:30

In the burning tent problem, we have a starting point (the camper at the moment of realization) and a finishing point (the burning tent). We want to minimize the distance from the camper to the tent via the river. A first step may be to realize that this task is not the same as minimizing the first leg or the second leg of the trip. That is, going straight to the river does not give us the shortest route, and neither does going to the point of the river closest to the tent. The answer will involve a happy medium. As such, calculus is an option.

The clever way to solve this problem is to imagine that the point along the river bank has been chosen and we are watching the reflection of the camper across the line that is the river bank. The camper and his reflection will always be the same distance from the river. In other words, if we choose a target point on the river that gives the camper the shortest distance, it also gives his reflection the shortest route. Therefore, choose the point that is on the line between the camper's reflection and the burning tent. The shortest distance is given in the comments of the question.

deadline online

2009-04-30T15:58:00.004+05:30

I used to check up on the hometown news pretty regularly, but lately the online edition of the Brodhead Independent Register is more independent and less of a register. The front page says nothing of substance except (this isn't of substance either):

"Looking for more great local news?"

No, I'm looking for SOME great local news. I clicked on the "News" link and it says:

"E-mail your news
Email your local news items to the Independent-Register at ... Deadline is noon Friday. "

So I guess it's become kind of a self-service newspaper. And I cannot help but appreciate the audacity of having a deadline for those benevolent (if unproductive) readers. Does the print edition expect a reader to draw their own pictures and call around town to figure out who is having a rummage sale?

This strikes me as a very sustainable format for a newspaper.

On that note, my blog has not had anything thought provoking, clever, witty, or otherwise worthwhile in quite some time, so get on that, people. Deadline is noon Friday.

solution: buried treasure

2009-04-21T15:34:00.005+05:30

Solution: 5*4 - buried treasure

The answer to this question is that yes, the treasure can be found without knowing the original starting point.

Solution: Let us discuss the problem using x,y coordinates, where x represents the east-west direction and y is north-south. Start at (a,b) and let the landmarks be (c,d) and (e,f). To reach the location for the first stake, you will go from (a,b) to (c,d), which takes you c-a in the x-direction and d-b in the y-direction. You then turn and go the same distance in a perpendicular direction, which means that c-a is a vertical distance and d-b is a horizontal distance.

The coordinates of the two stakes can be expressed as (c-(d-b),d-(a-c)) and (e+(f-b),f-(e-a)). Their midpoint, the location of the treasure, is ((c-d+e+f)/2,(c+d-e+f)/2), which is strikingly independent of a and b, the coordinates of the starting point. This means that if we changed the starting point but followed the same directions, we would arrive at the same location and find the treasure.

This does of course assume that the field is a plane.

I first came across this problem at http://www.techinterview.org/. This is a very neat website.

solution: chords

2009-04-21T15:33:00.000+05:30

Solution: 5*4 - chords

The shortest chord has length 16 and the longest chord is a diameter and has length 20. The in-between chords (lengths 17, 18, and 19) exist in pairs, giving us a total of eight chords of integer length passing through the given point. This is really an existence question. Constructing the actual chords is a tedious project whose aim concerns particular chords which are numerically but not geometrically of any moment.

5-star challenge #6

2009-04-17T08:46:00.004+05:30

Grades 11-12 5-Star #6
A group of 475 very democratic pirates are deciding how to distribute a massive pile of plunder. To distribute the booty, the pirates will each cast a vote for one of two options:

(A) divide the treasure evenly amongst everyone in the group, or

(B) make the youngest pirate walk the plank and have the remaining pirates vote again.

If half or more of the pirates vote to divide the plunder, the treasure will be distributed at that time. If not, the youngest pirate will walk the plank and the (slightly smaller) group will vote again. The process will be repeated as long as necessary until the treasure has been divided. How many pirates will share the treasure?

For clarification: No two pirates have exactly the same age. Although (by the pigeonhole principle) some of them must share birthdays, no two of them were born at exactly the same time. That would be inconceivable. The precise ages of all pirates are on record and each pirate has access to this information; no pirate's age is a secret. Each pirate's business savvy is exceeded only by his or her greed. Each pirate will vote only according to their selfish ambitions for wealth and not walking planks. Each pirate will make the best possible decision for himself or herself.

Grades 09-10 5-Star #6
A pirate has nine coins, identical to one another in appearance, but one of which is counterfeit. She wants to use an equal arm balance to identify the counterfeit coin, which is heavier than the others. Using an equal arm balance, what is the smallest number of weighings necessary and sufficient to identify the counterfeit coin?

Note: I have decided to extend the deadline for my students, so I will turn off the comments until the contest is properly over for them.

5-star challenge #5

2009-04-17T08:43:00.002+05:30

Grades 11 & 12 5-Star Challenge #5
A particular city's roads form a grid. The Avenues run North-South and the Streets run East-West. The Avenues and Streets are named numerically and they are in numerical order. How many distinct shortest routes exist between the corner of 3rd Avenue and 12th Street and the corner of 8th Avenue and 5th Street?

Grades 9 & 10 5-Star Challenge #5
A Woodstock student on a camping trip notices that his tent is on fire. At the moment he notices the fire, he is holding an empty bucket and standing only 10 meters away from a river (whose banks are parallel lines). The tent is 30 meters from the river and the student is 60 meters from the tent. The student and the tent are on the same side of the river. The student needs to fill the bucket with water from the river and go to the tent to fight the fire. What is the length of the shortest path to the tent via the river?

the disillusionment

2009-04-08T11:40:00.002+05:30

I just became aware that my loyal readers have been dying to see the explanation of my geeky trick. I know that there is a certain disapproval of a conjurer explaining his magic, but this cat has been out of the bag for quite some time. So breathe your sigh of relief and rest assured that your original assumptions of reality and the mechanics of our natural world are just as valid as they were a week ago, which may incidentally be less than you (by nature) think.

my impossible project

2009-03-28T14:22:00.003+05:30

Here is my project from this morning:
Yeah, it was tricky. I thought I had the calculations right, but there must have been a sign error somewhere.

solution: 7-11

2009-03-27T16:23:00.000+05:30

The question was posed in terms of tuck-shop coupons, but this problem is classically stated in terms of postage stamps. Essentially,

What is the largest number which cannot be expressed as a sum of 7's and 11's?

Consider a different (but related) question: If we have an amount that is possible, how can we increase it by one? Since there is no one-rupee stamp, we would have to trade some appropriate combination of stamps to accomplish an increase. Add a few whose sum is N, take away a few whose sum is N-1. It is also worth mentioning that the trade will only be 7's for 11's or 11's for 7's. We will not have to worry about some complicated mixture. (if we gave a 7 and received a 7 we could ignore the 7 in both cases and accomplish the same change).

The problem becomes: can we find the smallest multiple of 7 which is one less than a multiple of 11? Or, can we find the smallest multiple of 11 which is one less than a multiple of 7? Take 3*7=21, which is one less than 2*11=22. Also, 5*11=55, which is one less than 7*8=56.
What this means is that to take an amount which works and increase it by one, we can either trade three 7's or five 11's. If either of the two options is available to us, then the next amount can be expressed in terms of 7's and 11's.

The amounts which do work but cannot be increased are those numbers which are the sum of fewer than three 7's and fewer than five 11's. The largest of these numbers will have two 7's and four 11's; 2*7+4*11=58 is therefore the largest number which cannot be increased, which means that 59 cannot be constructed as a sum of 7's and 11's. The next number, 60, can be made (with one 11 and seven 7's), so every number greater than or equal to 60 can be made with 7's and 11's.

Therefore 59 is the largest number which cannot be expressed as a sum of 7's and 11's.

prisoner's dilemma

2009-03-26T12:17:00.003+05:30

This article was mentioned on the Freakonomics blog. DNA evidence collected from the scene of a jewelry robbery in Germany was matched to two people, identical twins with indistinguishable DNA. Because the evidence failed to incriminate one particular person, both brothers were released. Of course, between the two of them, they probably know which one did it (if either of them did).

The Freakonomics post about this situation alludes to the prisoner's dilemma, which I actually have been pondering lately. Not because I'm a prisoner, but because I read another chapter in my intro to game theory book while grading tests a few nights ago.

The prisoner's dilemma goes like this.

Two prisoners are held in connection with a crime. They are not able to communicate to one another. Each is given the opportunity to confess, with the understanding that the sentencing will occur as follows:

If one talks, he goes free and the other gets a maximum sentence (because there is evidence against him).
If both talk, they each get a usual sentence. (evidence against them, but less for cooperating).
If neither talks, they each get a minimum sentence. (not enough evidence for a conviction).

The outcomes can be arranged in a matrix as follows, showing how many years each prisoner gets as a result of each of the four outcomes:

	B talks	B doesn't talk
A talks	A:10, B:10	A:0, B:20
A doesn't talk	A:20, B:0	A:5, B:5

Here is the tricky part, as seen by Prisoner A (B's considerations are the same). Prisoner A is trying to make a rational decision with the sole intention of serving the least amount of jail time possible. Prisoner A knows that if B talks, A should talk (to get 10 years instead of 20). If B doesn't talk, A should talk (to get 0 years instead of 5). So while the outcome is uncertain, the strategy is clear. Prisoner A should talk, no matter what B decides.

Likewise, Prisoner B would be better off talking, no matter what A decides. Therefore, they will both be rational and talk. They will each serve 10 years.

The intriguing facet of this situation is that there exists an outcome which is better for both prisoners (they both stay quiet and each serves 5 years), but which would involve each of them acting irrationally. Of course, if their decision were made to avoid jail time, it could hardly be called irrational. That's the catch-22.

At a glance, it seems like a clever illusion and it is tempting to think that after considering the outcomes, each prisoner would be silent and thereby obtain the most favourable outcome available: 5 years and 5 years. But remember that the only reason a prisoner would be quiet is if he were confident that the other prisoner would also be quiet. And if he were confident of that, would he really pass up the chance of immediate freedom?

This leads to a mathematical discussion of why we do not see more cooperation in the world. The best outcome for everyone is usually necessarily artificially imposed or it will not happen.

My simplified rendition assumes that both prisoners are equally guilty and know that they would be convicted by the other's testimony. It also assumes that they do not like being in jail. Honor among thieves could certainly keep them quiet if the jail time were a matter of days, but as the sentence gets more substantial, the reward for informing on the other prisoner grows considerably, in addition to being the only rational choice. This means that the relative magnitudes of punishments and temptations come into play. Here is a more lengthy discussion of the dilemma that explains some of the variables.

solution: base b & b+3

2009-03-26T09:22:00.001+05:30

The number n expressed in base b has the form 364. In base (b+3), n is written as 202. Find the decimal values of n and b, which are both positive integers.

Solution:
Without knowing the base, we cannot know the number which is represented by the symbols 364. The place-value system allows us to perceive it as a number having a one's place, a b's place, and a b²'s place, just as a three digit decimal number would have a ones' place (10⁰), a tens' place (10¹), and a hundreds' place (10²).

Therefore n=3*b²+6*b+4
and n also equals 2*(b+3)²+0*(b+3)+2

so we have

3*b²+6*b+4=2*(b+3)²+0*(b+3)+2
3*b²+6*b+4=2*b²+12*b+18+2
b²-6*b-16=0
(b-8)*(b+2)=0

so b=8 or b=-2. The problem says that the base is positive, so we take b=8. Once we have found b, we can put the value into either equation for n. We get b=8, n=244.

In this problem, I offered the students an easy base that they could find by trial and error. Several students noted that since 364 contains a "6", the base had to be at least 7, a very astute observation. Also, when you solve the quadratic, you get only on positive option.

If the quadratic equation had two positive roots, we could find a number n which had the given representations (for bases b and b+3) for two different values of b. If the base b representation was four digits, there could be as many as three distinct values of b that could solve such a problem. If the base b representation was six digits, we would get a fifth degree equation, which Abel says cannot be solved for the five different b's that could exist.

another rock in the wall

2009-03-23T16:36:00.003+05:30

This is a wall we saw in the granite ruins of Hampi in South India. The wall surrounds part of the royal complex. It is hundreds of years old and probably thirty feet tall. You can tell that they got tired of hoisting the big ones as the wall grew taller.

5-star challenge #4

2009-03-23T16:18:00.005+05:30

Grades 11 & 12 5-Star Challenge #4
Captain Cooke the buccaneer buried his treasure at a small Himalayan boarding school. His legendary love of mathematical riddles inspired him to leave this clue, enigmatically encoded in the arrangement of 216 enormous stainless steel cuneiform characters in the school’s Peace Garden:

"The treasure is below Handsome Field. Start at the pile of broken tube-lights. Walk straight to the rock that looks like a flower, counting your steps. Turn 90 degrees to the right and walk the same number of steps in that direction. Put a twig in the ground where you stop. Start again at the pile of broken tube-lights and walk directly to the flower that looks like a rock, counting your steps. Turn 90 degrees to the left and walk that number of steps in that direction. Stick a twig in the ground at that point. Tie twine tautly 'tween the twain twigs and find its halfway point. The treasure is buried beneath the midpoint of the twine."

Centuries later, the clue is discovered by a visiting alumna as she ponders the 5-Star Challenge #971. After decoding the directions, the alumna rushes down to the Handsome Field Coliseum (via the Class of 2009 Memorial Escalator) to discover that the flower and the rock are still inexplicably in their places, but the pile of broken tube-lights has been removed without a trace by CLEAN! Without knowing where the broken tube-lights should be, and without digging at random all over Handsome Field, is it still possible for the alumna to find Captain Cooke's treasure? Explain.

Grades 9 & 10 5-Star Challenge #4
Point A is 6 units from the center of a circle of radius 10. How many chords of the circle have integral length and pass through point A?

5-star challenge #3

2009-03-22T20:45:00.002+05:30

Grades 11 & 12 5-Star Challenge #3
The number n expressed in base b has the form 364. In base (b+3), n is written as 202. Find the decimal values of n and b, which are both positive integers.

Grades 9 & 10 5-Star Challenge #3
The new tuck-shop coupons are printed only for 7 rupee and 11 rupee denominations. What is the largest integer number of rupees which cannot be paid using a combination of 7- and 11- rupee coupons?

solution: pair-sums

2009-03-17T15:32:00.002+05:30

Grades 9 & 10 5-Star Challenge #2
Every possible pair within a set of five numbers is summed.
The sums are 0, 3, 11, 12, 20, 21, 23, 29, 32, 41. Find the five numbers.

In this problem, the answer may be obtained by trial and error, but there are some neat aspects to the problem if one works to solve it. Let us assume that the five numbers are distinct (as they would need to be if their ten pair-sums were distinct). Let us consider the sums as F < G < H < J < K < L < M < N < P < Q. Let the five numbers be a < b < c < d < e. The order of a, b, c, d, and e offers only an incomplete understanding of the order of the sums. We have

(a+b) < (a+c) < [(b+c),(a+d)] < [(a+e),(b+d)] < [(c+d),(b+e)] < (c+e) < (d+e).

We know that F=a+b and G=a+c, and that Q=e+d and P=c+e. We do not know if the third and fourth smallest sums represent a+d and b+c or b+c and a+d, respectively.

This problem can be approached by solving all of the little equations to establish which is greater of (b+c) and (a+d). Once b+c is known, the system of three equations a+b=F, a+c=G, b+c=(H or J) can be solved for the values of a, b, and c. The values of d and e can be found shortly thereafter by substitution.

There is a less tedious way to solve the problem. By identifying that each number is summed with each possible partner, we know that each number a < b < c < d < e appears four times in the sum F+G+H+J+K+L+M+N+P+Q, so the sum of the ten pair-sums must be equal to 4(a+b+c+d+e). By adding the first and last sum, we know that F+Q=a+b+d+e, and we can therefore find the value of c.

In our problem, the pair-sums are 0, 3, 11, 12, 20, 21, 23, 29, 32, 41, whose sum is 192, which is equal to 4(a+b+c+d+e). This means that 192/4=48=a+b+c+d+e.

We know that F+Q=0+41=a+b+d+e, so c=7.

If c=7, we know that G=3=(a+c)=(a+7), so a=-4. Then F=0=(a+b)=(-4+b) tells us that b=4. P=32=(c+e)=(7+e), so e=25. Since 48=a+b+c+d+e, d must be 16.

solution: 81^2009

2009-03-17T10:37:00.008+05:30

Grades 11 & 12 5-Star Challenge #2
Find (a) the number of digits and (b) the last three (least significant) digits in the decimal expression of 81²⁰⁰⁹.

My Solution...
This problem asks for two elementary observations. If the number were written in its entirety, it would be a simple matter to obtain the number of digits and the values of the last three digits. The entire number can be generated with a couple of lines of code using Java's BigInteger class.

BigInteger myBigNumber = new BigInteger("81").pow(2009);
System.out.println("myBigNumber: "+myBigNumber);

This returns (in a fraction of a second):

myBigNumber:

1400886426099001433589715356524325994077360058833291454060331448901202858812711608791505300540205160834871243114967077782896741018493232826119962265403933221408273710682285389474611581003628765532988693839153728368550209031963049953365587333165550684382285106676049916357515617508318430039472665058505400648134895508543610798780057273663322426956561416111481073295564688756106556978220972883768498990439376379038088913512578774433394954966862795937723871349150451305665824022155706009283754336379201724142185351446653196901336911382788170275693693431585571516214076703025655694724313935072160077237502277620633564638041394172823865006761057098692874958064119778303667684220473562875130389095576612598169616570264206698883864712941599975306460033566903701203176613915280808200553190149220172757742382552424264898048660485008953744755857351182328200797217390733117879482736723725387755343016641042324605268387235071317590044699322617486518833978498750446720283318797784379982099655399049514088697411493542735966238355231518335754560005454557858737827355594203256731113150254221184261378594118900660565386875397480998177808153453972240763137726201802935458465644187508973316810617828944777550600037116120613295523595674229092486855978909224433568001602101966137609851452418351933395026445713151813690775772659529245247483081061394734160452917273402957713927998398637008649959045364542625757251174939330779763645565078645414995355444331511328545544587436720247358813300136273063929536760693736894399092656898581391777601176686638814415800605457552137755811561722992947830941361575088520921316840670291945199828480758662688749076445628397158944252925926207253727397615642302175056456168537654926096910594424358310081741719763482063248026418549153981883499716627174273667624458356479660732203553476892308402043990264919201939021708092767869280613832259386192559262656045967546526551060223597953426164586528815240198370930615443469369831397836790511704027247717287613128262970900587798085687044826070427586179676392059739626595037167328563772506422713580328669959080751912111426766214386993833793411310032618393156947481034604475623258966648528615807295339485106694016473355821675133837209206523909378952991095736019452549069793468232090700893085918364572584074859059077156795322914743377215102376435931960712370356874666922951857034687294703109308613860715556328236763180084679159342631746367233847114542831918603472299798321527157441161819532742005968777296078987636685014605803610522533976135083018079462867231622504340685958958775803013385058237912087337518155851768194473603085161979303255925564896683534521901491172404855277443706334679904563837822732299389135616888785840527710491760378625577274127165160723696855383968993354486834189433444754952751349119445485023181677760092071751879339708391793177141272161381338406604416686619826676210135423387944224584209170249358763128906780988581499912792718505330714538446491560748113617647828696553112964608607567406647137734043006583685118169281006120091135549995534312849407690524527234325455560291603364366173667248135691235606237319206864662357808314487913529150433701250314479681846670702891523598161060942612992920037173624206384607196996451365525196548440267693551682140725599199807911935490733624764556262308015951473144301853801136043280112830549607313441641080796896809627284910178979983295107307002907330430332271237131267427131349236806228355581537700780297380111225674100174944973118017202128990296866935839828010719361300152001055622795047713257519014364284326307078184203112517839087782402492861653020274288404432905972207465665985433591495473890988412099059297644788913887941133672356210069355051778825969788920325814207846046437928268923946341521305003727054700978419670287048809561516312934357723512398670440769027697003378905339200718030813912207894907559735934856589959121

Of course you could count the digits by hand or write a couple more lines of java to find the number of digits, or use the character count in a Word document. Each of these processes is admirably resourceful, but disregards a lurking elegance. I would not object to such a direct approach to the problem, but there is certainly room for some more finesse. The sheer size of the number renders useless more accessible tools like calculators and spreadsheets. Microsoft Excel has pitiful accuracy, and the students quickly find out that they cannot even use their TI calculators to find the number of digits (directly).

To find the number of digits, we need to remember that we mean 'decimal' digits. The relationship between exponents and the number of digits is clear if the bases agree. Our number, 81²⁰⁰⁹ can be written as 10^2009*log81=10^3834.146..., which must be a 3835 digit (base 10) number, as it is situated between 10³⁸³⁴ (which is the smallest 3835-digit number) and 10³⁸³⁵ (which is the smallest 3836-digit number).

The last three digits present more of a challenge (barring the availability of the number in decimal form). The last digit, one may easily observe, is 1.

As in the first part of the question, there is a simple shortcut that most would fail to find, but a few would anticipate or discover. The last three digits of the powers of 81 progress through a repeating pattern. This is because 81²⁵=515377520732011331036461129765621272702107522001. The important part is that the last three digits of 81²⁵ are 001, which means that when we multiply by 81 to get 81²⁶, we will be back at 081 for our last three digits, and we will go through the progression again for every 25 factors of 81. By this, we can conclude that 81²⁰⁰⁰ will have 001 as the last three digits, so 81²⁰⁰⁹ will have the same three-digit ending as 81⁹, which is 121. This idea can be used within Excel, but even 81⁹ is beyond the precision, so it is necessary to discard the more significant digits, through a formula like:

A2=MOD(A1*81,1000)

which gives the last three digits of the next power of 81. The calculation is always manageable for Excel, and the values never get larger than 3 digits. I do not expect my students to know about modular arithmetic, but the problem certainly supports that type of finding.

A basic high-school-math-with-scientific-calculator method for solving this problem involves the binomial theorem, and I think is very clever.

If we write 81²⁰⁰⁹ as (80+1)²⁰⁰⁹, we can expand it using the binomial theorem as:

(₂₀₀₉C₀)80²⁰⁰⁹*1⁰
+(₂₀₀₉C₁)80²⁰⁰⁸*1¹
+(₂₀₀₉C₂)80²⁰⁰⁷*1²
+...
+(₂₀₀₉C₂₀₀₆)80³*1²⁰⁰⁶
+(₂₀₀₉C₂₀₀₇)80²*1²⁰⁰⁷
+(₂₀₀₉C₂₀₀₈)80¹*1²⁰⁰⁸
+(₂₀₀₉C₂₀₀₉)80⁰*1²⁰⁰⁹

Notice that most of the 2010 numbers in this sum end in 000. Any time the exponent on 80 is at least 3, the number will have 1000 as a factor. This means that the last three digits are entirely determined by the terms (₂₀₀₉C₂₀₀₇)80²*1²⁰⁰⁷, (₂₀₀₉C₂₀₀₈)80¹*1²⁰⁰⁸, and (₂₀₀₉C₂₀₀₉)80⁰*1²⁰⁰⁹, which are:

12909030400
160720
1

...and whose sum ends in the digits 121.

This method is more general in that it does not rely on the discovery of a cycle (which may be very long for some bases) or the access to a computer brute-force approach.

pi day!

2009-03-14T10:03:00.004+05:30

Happy π Day!

5-star challenge #2

2009-03-14T09:57:00.002+05:30

Grades 11 & 12 5-Star Challenge #2
Find (a) the number of digits and (b) the last three (least significant) digits in the decimal expression of 81²⁰⁰⁹.

Grades 9 & 10 5-Star Challenge #2
Every possible pair within a set of five numbers is summed. The sums are 0, 3, 11, 12, 20, 21, 23, 29, 32, 41. Find the five numbers.

solution: circles

2009-03-11T09:10:00.004+05:30

This problem is fairly straightforward with a correct diagram. A hastily sketched diagram may not accentuate the relevant details. Segment ED is 25 units in length. The length of segment BC is 24 units. (how do we know?) To find the length of the altitude to A of triangle ABC, go 9/25 of the way from 9 to 16. Then you have the base and height of a triangle. The area comes to 138.24 square units.

The problem became more interesting when I tried to construct a diagram. A drawing will suffice if you know, for example, that the radii should be perpendicular to the tangents. The construction of a tangent to two circles (but not at their shared point) took me a few minutes to contrive. Also, triangle ABC is a right triangle for any two circles (if we do not keep the radii to 9 and 16) which are tangent in this way. (why?)

My students are always disgusted to learn that I do this stuff for fun.

This problem came from the Georgia Tech Math Olympiad Sample Test.

solution: vases

2009-03-11T08:45:00.003+05:30

Solution: The greatest number of heights that could be tested with certainty is 91.

If only one vase was available, we would have to begin at 1 foot and proceed upwards by 1 foot increments. If we skipped a height and the vase broke, we could not be certain that the maximum survival height was the last drop survived or a height which was skipped. If and when the first vase breaks, the problem will become a one-vase-scenario.

If two vases are available, we do not at first need to drop from consecutive heights, but we must choose a strategy which allows us to test the intervening heights between the first failure and the previous success.

With two vases, therefore, we can spend our first drop at 13 feet, knowing that we have 12 drops left with which to determine (in case of a failure), which of the heights 1-12 was the maximum. If the vase sustains a drop of 13 feet, we have 12 drops left and we should make our next drop from 25 feet, knowing that we will have 11 drops (in case of a failure) with which to determine the maximum that may be anywhere from 14 to 24.

Therefore, we can skip the first twelve heights, testing at 13. Then we skip the next eleven, testing at 25. By continuing this method, we can determine with certainty the safety of 91 different heights.

We plan to test at:
13 ft.
25 ft. (13+12)
36 ft. (25+11)
46 ft. (36+10)
55 ft. (46+9)
63 ft. (55+8)
70 ft. (63+7)
76 ft. (70+6)
81 ft. (76+5)
85 ft. (81+4)
88 ft. (85+3)
90 ft. (88+2)
91 ft. (90+1)

The winning student discussed 3, 4, and 5 vases. There would be a maximum, of course, if you had 14 vases and 13 drops, you would not benefit from the fourteenth vase.

Robert suspected a twist (in the comments for the question post) such as counting the first part of the fall. For example, if you drop a vase from 98 feet, it may smash at the end but it will safely fall 97 feet. However, a drop from 97 feet may also be fatal (in which case you have spent both of your vases and very little has been learned).

5-star challenge #1

2009-03-06T13:30:00.005+05:30

I am trying to start a weekly math contest at my school. I have enjoyed the task of finding problems to use. This contest finishes today for my students, so I will pass on the questions for your own enjoyment. There is no prize but the dignity and warmth that comes of mathematical triumph. I will post answers in the comments when I have the time and inclination.

Grades 11 & 12 5-Star Challenge #1

Mr. Burchell has two identical vases and he wants to know exactly how far such a vase could fall without breaking. The vases may be fairly robust. With no more than 13 drops from integer heights (in feet), what is the greatest height (in feet) which could be determined with certainty as the longest drop which such a vase could survive (without breaking)? That is, beginning at 1 foot, what is the greatest number of heights which could be proven safe for the vase?

Grades 9 & 10 5-Star Challenge #1

Circle P with radius 9 and circle Q with radius 16 intersect only once, at point A. A certain line is tangent to both circles, touching circle P at point B and circle Q at point C. Find the area of triangle ABC.

these days

2009-02-26T16:34:00.003+05:30

About a week ago I sat down with the intention of posting a note here to announce our anticipation of another child. I didn't figure out how to put it, so I abandoned the effort after a sentence or two. I thought I would come back to it. Joie had a miscarriage early Monday morning. We lost our baby that we were just beginning to love. So we are sad.

Joie has been feeling a lot better now, and we know that God is good.

I think we have a need to attribute. Part of me wants to know what caused our little bitty baby to not make it. I feel like it must be a result of some failure of mine to love it or be a better pappa for it. Anyone could argue against that, I am not saying it is rational. It is just a typical reaction of mine. Advancing beyond that a bit, I might conclude that it is not my fault, it could happen to anyone. ...statistically speaking... But I think that as a follower of Christ, there is a richer perspective.

(from John 9) As he went along, he saw a man blind from birth. His disciples asked him, "Rabbi, who sinned, this man or his parents, that he was born blind?"

Christ's disciples showed that painfully familiar reaction, to complete the picture backwards as though the blind beggar's plight should be explained in terms of the events preceding it. "Neither this man nor his parents sinned," said Jesus, "but this happened so that the work of God might be displayed in his life..." Jesus spoke in terms of what might be.

In a fallen world reeking with the perversions of God's designs, it is impossible to trace a single sorrow to its cause. Jesus did not come to damn the world for its failures, or to untangle the manifold chaos that he found. He came to save [John 3:17], to offer hope and a future [Jeremiah 29:11], and to give himself as a ransom [Matthew 20:28]. From this moment forward.

I think it means that our moment of sorrow can be a source, and not a conclusion only.

In these past few days, I have already become aware of feelings and thoughts that I would not have known otherwise. It is data, observations and findings about life, love, and the character of God. I cannot say I really understand this all of the time. But God gives good gifts to his children.

how i choose movies

2009-02-21T00:03:00.003+05:30

Well, I have survived the first two weeks of the semester. The more attentive of my students are learning how to prove trigonometric identities and navigate slope fields.

A recent Pre-calculus class took a turn for the historic and I found myself mentioning Fourier. Jean Baptiste Joseph Fourier. (He was French). That evening, I cracked open my rather intimidating (almost as thick as it is wide) Stephen Hawking book "God Created the Integers" to read the bit about Fourier. He was a revolutionary in France who narrowly escaped the guillotine. He was a scientific advisor for Napoleon's expedition to Egypt, the discoverers of the Rosetta stone. Most impressively, he wrote a paper on the behaviour of heat and made the bold assertion that every function looks like a mess of trigonometry if you can scrunch your eyes just perfectly. In my class, I was shocked to hear myself relate Fourier, and it carried with it the faintest glimmer of familiarity as I stared into the murky depths of my university memories. But mostly it felt like accidentally scratching off a scab. My Fourier is a bit rusty.

I also recently read "Abel's Proof" by Peter Pesic which is (perhaps appropriately) mostly about Niels Henrik Abel, but which mentions Evariste Galois. Galois was also a Frenchman, presumably a Gaul, by the sound of his name. Also like Fourier, I mean, not also like Abel, who was Norwegian. And also a Frenchman like Fourier, I don't really know if Fourier was a Gaul. But Asterix was. Anyway, Galois was a revolutionary in France, but then it seems like they all were. He once uttered a threat to the king in the presence of Alexandre Dumas, who evidently frequented the same seedy revolutionary locales as Galois. Galois was a world class mathematician by the time of his death at 21, and he died in a duel after a tormented night of scratching down the beautiful (and ponderously novel) math in his head that he had no time to painstakingly articulate in the volumes that others would have to write about it instead.

Anyway, these recent tangents served as sines that I must have subconsciously weighed in my decision to watch "The Count of Monte Cristo" tonight. Dumas, duels, Napoleon... you can secant you?

hindustani chai

2009-02-09T13:19:00.003+05:30

This post is for our relatives who received tools and ingredients (but not directions) for making Hindustani Chai. This recipe is, like most of my attempts to articulate what I have learned in India, confoundingly complex, badly conceived and pitifully articulated, besides being imprecise. So if you enjoy that type of recipe, read on...

Ingredients: Water, Sugar, Tea Leaves, Milk, Cardamom, Cloves, Ginger, Cinnamon.

First, the term Hindustani Chai is used to refer to a variety of concoctions that are very flexible with the spices. I have been served chai with garlic, and a different time with anise. Tradition classifies spices according to those which give warmth to the body and those which cool. The spices in chai are therefore subject to change throughout the year, as you would not consume a cooling spice in winter or a warming spice in summer. That said, we Burchells enjoy the following recipe all year round.

Cardamom is available here in little green pods which should be crushed. We use five pods for a pot of chai for our family (about 1 liter), but more if we make a large pot. If the cardamom pods are not crushed well, they will close up again when they are boiled and less flavor will be derived. Heh... derived.

Cloves should also be crushed. Cloves can have a tingly numbing effect on the mouth which I find enjoyable. Sometimes I suck on a clove while I make chai. Maybe I should not be admitting to this. Joie does not like it as much, so we generally use only five or six cloves.

Ginger should be available everywhere. We use a piece about the size of an adult big toe. Prior to writing these directions, I did not think of it in those terms. It does seem a bit morbid. We scrape the skin off the ginger before crushing it.

Cinnamon is cheaply available here, so we use plenty of it, probably about the equivalent to two of the scrolly looking sticks that sell in other hemispheres. Ours is more recognizably tree bark. If it is crushed well or boiled longer, less cinnamon could do nicely.

The spices should be crushed and put into about a liter of boiling water. This should simmer on the stove for five or ten minutes. It should smell amazing at this point.

Scraping the ginger:

Crushing the ginger:

Even Will can do it...

The boiling water and spices should look about like this:
After the spices have simmered, put about 1/3 cup of sugar into the pot and stir it so it dissolves. It will dissolve quickly, probably before you finish wondering how much of that sugar you are about to consume.

We put about four teaspoons of strong black tea leaves into the pot after the spices have simmered and the sugar has been added. The water might foam up if the flame is too high, so you should stir the tea in, making sure it does not stick to the sides uselessly.
After the tea, we add the milk. I like to give the tea a couple of minutes to brew because I imagine the milk slows that down. The milk we use is a bit more than two-percent. It is also buffalo milk. Add milk until it is creamy. The chai is now complete. I leave it on the stove for another minute or so to heat it up after adding the milk, but take care not to let it boil again after the milk has been added. Boiling the milk will make the chai stick tenaciously to the pan and the cups, and it will cause your chai to grow one of those nasty milk skins on the surface that sticks to your upper lip when you try to take your first sip. Don't boil it. The local chai is usually boiled, and I suppose that one could come to appreciate the skin.

This recipe is easily varied to taste.

When the chai is done, pour it through a sieve into cups that need not look like rhinos. The chai will become bitter if the tea leaves are left in it for too long. If you want to save chai, strain it.

the coconut man

2009-02-02T14:23:00.008+05:30

One quiet afternoon in Goa, the coconut man visited our guest house. People growing up as I did in a land devoid of palm trees frequently wonder (or at least I did): How often do coconuts really fall and hit people on the head? Cartoons lead us to believe that this may be an issue. As might be expected, the denizens of coconutty realms have contrived a protection against this danger: The Coconut Man. (People who have lived in coconutty realms will not learn anything from this and they will be amused at my ignorance.)

Each coconut palm tree has a mess of coconuts in all stages of maturity, so the coconut man squints up at the tree and decides which branches, leaves, and coconuts to knock down. He then climbs it very quickly and knocks down the dried stuff that might otherwise fall on a person or a car.

Sometimes they climbed the trees with their feet through a loop of rope, and other times they would wait until the top to use the rope. This is all very clever. Actually, I don't have any statistics about coconut men and their lifespans, so I do not really know how clever it is. Maybe it isn't clever at all. These guys looked pretty experienced though.

In the picture above, you can see the coconut man slipping his feet into the rope loop. Also, they would occasionally (and with admirable nonchalance) use their cleaver/machete to hack a little wedge out of the tree to use as a step. This heightened my respect for the coconut men.
The best part about their job was that they made the mess and somebody else scurried around cleaning it up after they had moved on to a different tree. The family that lived at the guest house collected all of the branches and coconuts, quickly sorted them into piles around the yard. Each residence had sorted piles of coconut trash. The leaves can be used to make brooms, the branches are used for fuel, and the coconuts are eaten and their fibrous shells are used to make rope. Also, the green smooth shell is the outside and if you chisel it off you can get to the hairy brown shell.

It looks like an afternoon's project, but the whole thing was probably done and cleaned up in less than an hour.

back home

2009-01-30T20:32:00.002+05:30

Well, the Burchells have survived another month-long adventure in South India. We arrived home this afternoon to find our house just where we left it. We spent a great deal of time soaking up the sun on the beach. We had some steak, some fish, and some deep-fried shrimp (but here they are 'prawns'). We saw the ruins of Hampi. At the moment it seems like we spent most of the time on trains, but that can't be right. Anyway, I have pretty much forgotten how to compose a sentence, much less type it out, so I'll quit for tonight. Here are a few pictures of spices, an ancient Hindu temple, and my beloved family. I will not insult your intelligence by telling you which is which.

more mountains

2008-12-06T22:37:00.005+05:30

These pictures were taken on the way to church last Sunday. Can you find the Yeti?

on to exams...

2008-12-04T15:49:00.002+05:30

Mr Burchell Mr Burchell, what can I do to bring up my grade?

Last day of classes this semester.

My students are in desperation mode as the exams draw near and they start to care about their grades. I have enjoyed playing a different role in exams these past few years. As a teacher, I am on the side of accuracy and validity, and that role comes with its own responsibility and stress. The student objective in the situation is to somehow or other obtain a grade which favorably misrepresents their skill level. This desire for an A transcript with C work leads to many of the crises that dominate my interactions with students lately. They are not genuinely worried about seeing an unfair exam, but the idea of a fair exam has them all a little jumpy.

In a few of my classes today, the topic of negative marking was brought up by students. Part of their exam will be multiple choice, so they are concerned about negative marking. Anyway, negative marking was a hot topic because I quickly told them that it is a great idea, which made about three more of them look up with terrified eyes at the teacher who might take away their not-so-secret weapon on multiple choice tests: blind luck. I find that topics involving grades are fertile ground for effective discussions about math. Students care about their grades more than they care about most of the other topics we discuss, which is ironic, because if they paid attention to the other topics... but that's only my humble teacher opinion.

Negative marking represents another one of those badly misunderstood tools of education that bring out the irrationality of students.

I once wrote a monograph on the subject... a mere trifle, really. The subject of negative marking, that is, not the irrationality of students. And it's not really a monograph, but a handout. I am still researching the irrationality of students.

recent reading

2008-12-03T11:47:00.004+05:30

My sister just wrote about her recent reading, so I thought that I would. I recently finished another trip through 'The Complete Sherlock Holmes' by Sir Arthur Conan Doyle. I have enjoyed it more since living in England and wandering through London a bit. I also had an analysis professor at Lancaster who will always play the role of Sherlock Holmes in my mind as I read those stories.

Sherlock Holmes is built around a supremely powerful principle of writing that is rarely used to its full advantage. Sir Arthur Conan Doyle's Holmes is bigger than the stories which define him. There are constant references to other cases, some of them which can only be hinted at by Watson, as their exposure would even now topple kingdoms. Stories frequently include some casual introduction to an unknown chapter of the detective's experience, an offhand comment about the most obscure of specialities, which implicitly releases our imaginations to wonder what the fellow does not know. I once wrote a monograph on the classification of the thirty-seven most common types of pencil lead... a mere trifle, really. We get the impression that we are seeing the slightest sample of the character. We are afforded a portrait that is less than comprehension.

A couple of years ago I inherited a fat book of Hercule Poirot short stories from some friends as they were moving. The book contained some fifty stories, and although I read them all, I was somewhat underwhelmed. Agatha Christie was supposed to be a great mystery writer and Poirot made for a poor introduction. I think that she went to such pains to make him the antithesis of Sherlock Holmes that he escaped without any personality or finesse. With Hercule Poirot, I was pretty sure that I saw the whole character and it was not very interesting.

Anyway, I just read Agatha Christie's story called 'The Witness for the Prosecution", and while it did not contain much in the way of impressive detective work, it did reveal an originality and imagination that was so wanting in poor Hercule's cases.

roof of the world

2008-12-02T15:14:00.002+05:30

We are always so refreshed when the days cool and the dust settles enough to see these mountains.
Usually we can see the nearby green mountains, but all summer long you would never guess there was a looming range of snow capped giants a little further out. We always feel like they 'come up' during the cool months, and it seems to challenge our sense of object permanence to admit their presence when we can detect no trace of them. We stand there and fill our lungs with the freshest air that this here subcontinent can offer, and it feels like it is rolling off the cold rocks and snow drifts.

These days are beautiful here in the mountains, and the evenings are sweet times of tea and woodfires with my darling, and sometimes a book. Last night it was full of grading calculus tests, and that was not as sweet.

the suicide bomber

2008-12-01T11:34:00.003+05:30

It has occurred to me that most of the people in my own country have a notion of terrorist bombings in an action movie sort of way, but probably not in a daily newspaper sort of way. They are more common than I remember thinking before living here. The first time a young person hears of crowds of people being blown to shreds in a temple or butchered by car-bombs in a marketplace, it is a vivid exposure to a very dark side of humanity. After we grow up, we realize that such things are common in some places and we begin to regard those lands as savage or violent or otherwise unenlightened, and certainly altogether foreign.

The suicide bomber has on occasion shaken the Western spirit of security by its very foundations, reminding us to have a fear that we have long ignored. Our typical suicide bombers in the West have lost all reason, we say, we hope. Things have ended badly and as their tormented spirits cry for revenge, they kill their classmates or coworkers or families. In chilling contrast, the religious suicide bomber experiences no such breakdown of reason. They are meticulous and resolute, planning for months and proudly serving a purpose. Their actions are not viewed as a desperate departure from their upbringing, but rather as a focused culmination.

Their victims are helpless, fearful and passively accepting of the fate that may be hurtling toward them in the next rickshaw.

A killer gazes lucidly,
Plunging toward eternity,
Pleased with opportunity
And brazen with impunity,
Through trudging masses tired by fear,
Unaware he's finally near.

Rickshaw goaded through the crowds
To die in stillness with two shrouds,
Of smoking canvas painted gold,
Of burning hatred ages old.

Twisted bicycle, bloodied silk,
Crumpled cans and muddied milk,
Petrol in the drainage ditch,
Swirled with blood of poor and rich
Ignorant in blasting death,
Of piety that stole their breath.

Seeing nothing, feeling less,
Wander through the filthy mess.
Wonder if the brilliant flash
Of flesh and flame and pain will pass
From memories of sons and wives,
From streets and markets and the lives
Of people in the ancient land
Who always knew the heavy hand
Of seething war and slashing hate,
And all surpassing fear of fate.

student work

2008-11-28T13:37:00.003+05:30

I do not precisely remember which topic would have led to a discussion that might have inspired this defacement of a desk in my classroom...

pirate's life

2008-11-19T14:57:00.005+05:30

I find the recent news about Somali pirates to be... well, exciting. I think that rampant economic success is more positive than all the talk about bailing people out of their failed investments. There has always been a romanticism attached to pirates, and the news that pirates have captured the largest vessel ever seized... right here in my own time... I look forward with a mathematical interest as the scenario is played out in the headlines. The economics and game theory involved makes it an intriguing situation, and certain elements are, to say the least, more interesting than the typical news about corrupt politicians and suicide bombers.

I realize that piracy is a big inconvenience for some people, and I am not saying that I would really be a pirate. I think that making a living by striking fear and terror into the hearts of people who wish I had never come into their lives is (while oddly similar to being a math teacher) probably not what Jesus would want me to do. Yet from an academic standpoint, I find it exceedingly interesting.

Evidently the perfect combination of factors has come around again to give the world a high seas piracy problem that is not easily solved. The US Navy was established to fight pirates off the African coasts, was it not? And the legitimate world approached the problem by employing unsavory characters to fight the even less savory characters, and there were official boats and captains in the Queen's employ, and some of them were practically pirates themselves and it was such a rich subject of history because: the best laid plans of the world's strongest nations were at the mercy of a few renegades.

And perhaps it does go further than the academic perspective.

The pirates make us dream because they are rich as kings and free as birds, and we in proper society only have experience with poor birds and burdened kings.

And for that reason, I think that there is within the human soul something that loves a pirate. They stir our hearts and remind us of the things that we were created to desire, a freedom and a treasure that surpasses the 'cold comfort' that is offered to those who fall in line with this world's priorities.

on strike for... something

2008-11-12T22:49:00.002+05:30

Well, I was apparently on strike today. Just as my calculus class was preparing to take another shot at LRAM, the office assistant opened the door and told everyone to gather in the auditorium, where we learned that the school was going to close its doors because of the Bandh (strike) called by the BJP. It should be noted that our doors are usually closed during classes this time of year, but this is mostly because of the chilly Himalayan air. Today their closure was symbolic, an act of independence, defiance, solidarity and resolution against the man. Or something.

The BJP is a political party, and a pretty strong one (at least in this area of the country) from what I can tell. So apparently the governing forces themselves call for institutions everywhere, school buildings full of teachers teaching and students studying, to cease business and take a day to appreciate the inanity that is Indian government. Of course, most of the government school teachers probably weren't going to come today anyway...

Our school actually had no reason to strike, and from what I can tell, the strike itself had nothing to do with education. Our school closed as a safety precaution because a strike is invariably accompanied by roving hordes of discontents, and it is not impossible that trouble could occur and our students could be at risk. I wonder if closing the school to protect the students was, in effect, working in the best interest of our students on this the day of the strike, and therefore perhaps we actually were working and struck the strike. Irregardless, nobody connected with the BJP was likely to notice our loss of productivity, so I'm not even sure it was productive. Or unproductive, or whatever it was supposed to be. (Being blindly submissive and following unknown procedures for a strike that one does not comprehend but has an obligation to join is a task not without its perplexities, and it does not seem to testify of that raging spirit of revolution that one would hope to see within the strikers.) I suppose that our school was closed down for the unlikely event that a government inspector should come around to check that we were upholding our part of keeping the children from learning. Because boy would it be terrible for the elections if the literacy rate slipped above sixty-one percent.

Now I am surely missing some of the information, but the fact remains that I and my students lost a day's progress for this unknown cause. Anyway, my confidence in the democratic process makes me certain that my sacrifice was not in vain, and I trust that the honest politicians, those bastions of integrity, were able to calmly discuss their differences (over whatever topic might be at the root of this heroic defiance) and come to a compromise. And to think that the negotiations were all made possible by shutting down schools across the region.

the problem of proof

2008-11-04T15:42:00.002+05:30

[This little rant stems from some of my contemplations of the role of proof in the processes of mathematics education. A detailed scrutiny by a guy like Bertrand Russell would make the best proof look like a pile of unjustified assumptions, which seems to result in a damning sense of futility that nullifies all scientific pretensions. Except, of course, the scientific pretension that identifies and cogently discusses the futility. That one is probably still valid and worthwhile.]

If and when the Riemann Hypothesis is finally proven, there will likely be a dozen people on the planet who understand the proof, a few more who pretend, and an excited population who believes. Is this mathematical rigor? At the very frontiers of mathematics and logic itself, we as a society, as a scientific community, are persistently and ironically burdened with the yoke of deference to an authority. On the simpler end, things are just as bad. I have a respectable degree in mathematics, and I will not pretend to follow the proof by Russell and Whitehead that 1+1=2. If I copied down their stupefying sequence of symbols from 'Principia Mathematica', I would not consider that I had proven the momentous result that 1+1=2. Nor would I consider the Riemann Hypothesis proven to me if I was expected at some point to 'take the professor's word for it'. I suspect that an argument presented as a rigorous proof is only rigorous by its delivery and is only a proof by its reception. But of course, I couldn't prove it.

elementary economics

2008-10-20T12:14:00.002+05:30

On Will’s birthday, Will and I hiked up to Sisters Bazaar and bought a few things from Prakash's, where we also bought a cold bottle of orange juice to share as we sat on the wall by the side of road. They have built short stone two-foot tall walls along the road right at the top of our trail as barriers to hold the road together and keep the cars from falling down the mountain. If you could go on the other side and look at Will, he would look just like Schroeder in one of those peculiar but nonetheless frequent Peanuts comics where a couple of the characters are just having a conversation so Schulz draws them awkwardly sitting or standing behind a stone wall and you wonder how tall the wall is and what they are sitting on. I think Charles Schulz was embarrassed about the way he drew legs. But I digress. We had a lovely time sharing our bottle of orange juice and we both enjoyed the fresh air and the view of the snowy mountains.

A few days ago Will was remembering our trip. "We went to Prakash's and he gave us orange juice." It was then that I realized that Mr Prakash was the hero of his story, not his poor father who bought the orange juice. I was just a fellow freeloader, and possibly resented for not requesting more bottles of juice, and bigger ones. Without comprehending the money thing, Will is only able to conclude that the shop keepers are very kind people who give us anything we have the presence of mind to request. Will is surely perplexed that his stupid parents do not make better and more frequent use of these institutions. Why do we leave so much good stuff in the shop? He must think we are so inept.

The day after that revelation, Will was listening to Joie order some provisions from Mr Prakash. "Can I place an order please? We would like it delivered to Redwood Cottage" Here Will was probably wishing more than anything that he knew how to use a phone. "Some boxes of milk... two boxes of cranberry juice... one yogurt... some chicken..." and Will could contain himself no longer. I could see the wheels turning in his pert little head, and he was visibly excited by the potential of such an interaction. He waited for Joie to pause between items, and in the same bold but cautious voice with which he first called me 'Nate', he chipped in "A soccer ball."

It is a delight to watch Will sort out the details of our strange life.

marvel of engineering: home security

2008-10-17T12:09:00.004+05:30

As if I needed another reason to sleep badly... We just had an alarm system installed in our home. The school decided it would help us... sleep peacefully? I always thought that alarm systems were for people in suburbs with stuff to protect or people in dodgy neighborhoods who feared for their lives. Our '95 Jetta had an alarm system that scared me once or twice. The one in our house woke us up three times during the night (here in India we say 'thrice') with all of its whooping and blaring.

The nature of such a system is full of interesting considerations. Interesting from an academic standpoint, concerning if you have to live with it. To design such a system, we have to keep altering the mechanism based on our ability to sense different things. We can ponder this and similarly inferior logic that explains the way smoke alarms beep when supper is ready, the way a forgetful person becomes locked out of their email account, the way a medical test produces false positives (sometimes much more frequently than real positives), the way an auto-flash sometimes wrecks your photograph, the way some algebra generates extraneous solutions, and the way homework grades and test grades correlate poorly. So here is the gist of the dilemma and the reason I slept badly enough last night to spend time writing something like this.

We want a home-security alarm that sounds when:

A bad person bent on evil enters my house... but this is impossible without an omniscient system that guards every conceivable point of entry, including the kitchen drain. So we settle for an alarm that sounds when:

A bad person bent on evil enters my house through the door... but this is impossible without a system that understands thoughts and intentions, and besides, the system might be confused if anyone ever wanted to hurt us and take our things with a pure motive. So we settle for an alarm that sounds when:

Any person enters my house through the door... but this is impossible for all but the most sophisticated machines. It will also require that we take the responsibility to disarm the system while innocent people are entering through the door. So we settle for an alarm that sounds when:

A door opens... this is getting more feasible, but we are suddenly unable to say with confidence that there is a person involved, or which direction that person is going through the door. Boy would it be annoying to have an alarm system that sounded when you exited your own house. The door opening seems like a simple mechanical motion, but desiring to detect it with one simple boolean variable and one simple electronic sensor will force us into a simpler condition for setting off the alarm. So we settle for an alarm that sounds when:

A small magnetic sensor on the door detects that the door has been opened... but the sensor cannot tell that the door has been opened, it can only detect the presence or absence or change of some electrical impulse. So in the end we settle for an alarm that sounds when:

A small magnetic sensor on the door detects a change in the strength of a magnetic signal. This is possible, so we install the thing and turn it on and learn that the condition that sets off our alarm is true much more often than there is a bad person with evil intentions entering my house. Ironically, we didn’t even turn the system on last night because we didn’t want to mess with it, and it still woke us up thrice.

wildlife

2008-09-29T22:17:00.004+05:30

Monsoon is finally over, which means I can walk to school without my umbrella. Gone are the torrential downpours on our tin roof, gone is the river that occasionally tumbled over the path in front of our house. The ferns are drying up to show that the oaken forest is no longer saturated with the soupy cold of rain and clouds. And we are spending more time outside, which means that we get to see fuzzy caterpillars...

...they supposedly sting, but I haven't checked yet. We see a fair number of moths, and some of them are shapes which seem unusual, but then I am not a mothist. This one seems to have antlers:
The ferns leave as quickly as they came. They are dried and brown in these pictures, but they are falling off the trees and soon all but the thicker Christmas Ferns will be gone until next monsoon.

This butterfly is kind of a dusty tan color on the outside, but the inside of the wings are an electric blue. As a photographer, I am not frantic enough to give you a blue action shot, so you will just have to take my word for it. Or visit.
Below is a small lizard of the skink type. I'm fairly sure the Latin name might be skinkus skinkus. They are soft little lizards, kind of a pewter grey with a copper shine. Their tails break off in an emergency, so sometimes we see them with crisp little cuts where their tails were. I caught one the other day and let Will try to catch it. He almost killed the poor thing, but we discussed gentleness with creatures and it was a learning experience. Skinks are pretty resilient. Today Will told me that story as I laid on the couch recovering from parent conferences. "Once a pon time... I caught a little lizard and Pappa caught a little lizard. "
And now that it is warm, the Burchell kids go for an occasional swim. They each get their own tub, and here Will is donating some of his water to Annie's tub while Annie tries to chart a course into Will's bigger tub. These two are most of the wildlife around here.

my inner economist

2008-09-29T21:00:00.002+05:30

This blog was never intended to be a serious academic source of information or news, and any observations I may make about the Wall Street drama will be pathetically under informed and laughably erroneous. So I'll give it a shot:

There are only a few people alive who know for sure why the whole thing works or if it works. If the $700 billion bailout still leaves people feeling desperate, why don't they just double the amount? It's all pretend money being pulled out of a budget that doesn't exist and creating a debt to nobody which will never be paid. There is something distinctly pretend about the world economy.

I have a sneaky suspicion that the Bernankespan guy could fix the whole thing by saying, "It's fine, guys. I fixed it." Everyone would relax and give a nervous little laugh and go about their business and it would be fine.

amusing myself

2008-09-29T20:24:00.002+05:30

Tomorrow my precalculus classes will enjoy or endure an introduction to complex numbers. This came to mind over quarter break when Joie and I spent a few minutes pulling weeds in our garden to reclaim a few inches from the encroaching jungle. Or to make our garden encroach on the jungle. Anyway, I have exploited the cuteness of my very own son to demonstrate that even the invasive weed 'kali ghas' can have both real and imaginary roots.

gym dandy

2008-09-26T17:45:00.011+05:30

They are building a gym at Woodstock School. Since the school is located in a natural forest of some sort, an ocean of paperwork held up the project for several years and the treasured bit of flat space was of very little use to the school. Construction is finally in full swing, twenty hours a day I'm told, just a very literal and very tempting stone's throw below our cozy cottage.

I thought that I would take a few pictures of the building site. Much of India is under construction at any given time, so this is not an entirely new experience for us, but people in other countries can not perhaps imagine what this implies, and we do not often take pictures of every project that we see.

If I had heard a few years ago that a gym was being constructed, I would have pictured men in hardhats and neon vests taking control of the situation and making the elements submit to the technology of mankind. They would level the site with a heavy machine and they would use a crane to lower each finely engineered steel beam into place. There are some fundamental differences when construction occurs in India.

To begin with, as seen in the picture below, the construction crew has set up shacks to live in right here on the building site. Such projects rely heavily on manpower, and India is simply teeming with that. To use a bulldozer in such an endeavor would deprive people of jobs. The bottomless supply of labor creates a very different economic situation than what we see in the Western world. In the US, salaries and availability of labor make it critical to find an efficient method--an earth moving machine is used to do a job quickly if it will save a crew of men a day of toil. Here a fundamentally different economy makes it much cheaper to be inefficient, or perhaps redefines efficiency.
The next picture shows some of the large mounds of dirt. Some is sand for concrete, carried in on the backs of mules. The nearest road access is a few hundred meters away. The materials are dumped in piles from a truck and hauled the rest of the way by mules. The brown dirt has been dug out of the holes. All of the digging is done by hand, and some of the holes are 14 feet deep, according to Nigel, the man in charge. In this picture, you can see a few of the circular pans covered in sacks that a person uses to carry gravel around. Also visible are the steel reinforcing rods that came in a few at a time on the shoulders of lines of men stepping carefully and feeling the heavy spring of all that metal.

The next picture shows a few mules making a delivery of sand and gravel for concrete. Each mule is fitted with a pair of bags that are filled to the brim with bricks or dirt or sand. The mules walk slowly and cautiously, zig-zagging up the hills to decrease the steepness (these mules know calculus). Also in this picture is a pair of men working to bend the steel bars, some of which have already been planted in forbidding little clusters in the background. This is the mixing of the concrete. The large black tank is for water, identical to the two or three that store our water here at Redwood Cottage.
This picture again shows the mixing process. Also visible is an amazing little trick that can be observed near any road project all over this fair land. The two-man rope-shovel. A rope is tied to a shovel near the blade and a second man holds the other end of the rope. The man with the wooden handle of the spade stabs it into a pile of gravel, but the lifting is done by the other man using the rope. The arrangement seems like it would prevent a few back problems, but perhaps it only addresses a shovel shortage.

The rope-shovel:

search result

2008-09-20T10:37:00.001+05:30

Someone just came to my blog after googling 'porcupine ancestry'. This very blog was evidently on the second page of results, which means that someone somewhere in this big world of ours at 5:56:49 pm on September 16 2008 had already been disappointed by the first few search results and was probably uttering words to the effect of "Lan'sakes-alive! They say this inner net is so great... I can't find ennathang about porcupine ancestry." or "Blast! Confound it all. My plan is coming to ruination in the critical moments. I shall be ruling Europe within a week if only I can establish this beast's lineage." or "Pay up, man, not even Google says they're related to sea urchins." I will be amused for the rest of the day as I speculate about the webpage that could have satisfied such a query. I'm mostly sure that my blog didn't clear anything up.

more monkeys

2008-09-01T23:47:00.004+05:30

These are the nice monkeys, the langurs. The brown monkeys that backhand their little ones are called rhesus. The langurs are lean and graceful, beautiful creatures that are an asset to the community. They enjoy eating nuts and leaves and giving humans a respectful distance. They keep their fur clean.

The rhesus are filthy stinky beasts. They are aggressive toward humans and they enjoy eating garbage and stealing from shopkeepers and pooing on the center of the trail. They fight with each other and screech and chatter from the trees. They are the ridiculous bandar-log from 'The Jungle Book'.

the monkeys on our shed

2008-08-30T20:31:00.003+05:30

This was a lucky picture taken from Will's bedroom. A rare action shot of a big monkey rejecting the playful affection of a little one. I wonder if baby monkeys have feelings. The monkeys like to sit on our shed and sun themselves and pick bugs out of one another's fur. I have to be careful lately when I take pictures of monkeys. Once I was shooting a couple right outside our door and they started grunting and chattering to the big male, who came tearing across the yard to lunge at the door and bark at me. There is a very large male monkey in the troupe recently, and rumor has it he is a renegade that was kicked out of a different troupe. He is incredibly large, probably around a hundred pounds and built like a pit-bull, so I would be scared to see the monkeys that chased him away. Yeah, we have rumors about local monkey politics.

painter guy

2008-08-24T22:19:00.003+05:30

This picture was taken in Shimla. I didn't ask if I could take his picture, but he wasn't exactly going to come down and beat me up. I do not generally approve of taking pictures of people, as I think it is mean and invasive. But I also thought it was a very vivid picture that might look neat. So here it is.

a-a-a-annie and the jets... and bigfoot.

2008-08-20T15:23:00.002+05:30

I am mortally fatigued today. Annie, whose dancing eyes of unfettered glee could warm the hardest hearts, shows a different side of herself at about three in the morning when it becomes clear that she was meticulously engineered to torment the souls of her parents. Annie does not sleep. She stands in her crib and screams, and since we do not want her to fall or wake up Will, we lose sleep over it.

Two recent news items have grabbed my attention. Brett Favre is with the Jets now, and I hope he does well. I suppose there are people in Wisconsin who do not like him any more, but I think that the Packers really made a mistake by not begging him to stay in Green Bay. He is by many measures the best quarterback who has ever played the game, and last season he was not exactly washed up. I would not mind seeing Brett Favre and Bubba Franks in the Super Bowl after the Packers don't make the playoffs. So that really annoyed me.

The second news item was the bit about the two guys in Georgia not finding a bigfoot. They sold the thing and it turned out to be fake. If you lie about something like that, aren't you supposed to produce vague and intriguing but not disprovable evidence? A blurry photo? A list of excuses that are tempting to believe? You are supposed to carry the secret to your death and become a legend or else confess (right before you die) that the picture was fake and go down as a fraud who really made a good run... You can't sell a fake bigfoot so that they figure out you were lying a day after you lied. I'm so confused. I will regard this news item as a rare and brilliant treasure for anyone who has an appreciation for the study of human nature. Under the lens of game theory, there are some intriguing decisions being made here.

A list of possibilities as far as I can see it:
1. The two men behaved irrationally. This is highly unlikely.
2. They faked their identities and they are having a good laugh over it with whatever money came in and they think that they will not be caught.
3. A more subtle mechanism of the fraud really did work, like driving people to webpages full of spyware. In this case the payoff would have to be huge and all but guaranteed.

Now if they pulled it off under fake identities, that would be pretty sweet, I guess. The only other explanation that occurs to me is that the bigfoot guys hoped to gain much more than they were certain to lose, which I imagine to be their jobs, their credibility, their self-respect, and their awesome bigfoot costume. Stupid or short-sighted is common enough, but most people avoid being completely irrational.

Sherlock Holmes would classify this under 'the grotesque'.

a few pictures

2008-08-16T13:02:00.002+05:30

Here is Mussoorie, cloaked in the monsoon fog, as seen from Sisters Bazaar.

This is a picture I took on the taxi ride back to Mussoorie after our summer traveling.

Here are some ferns growing on a tree branch near our house. When the fog sets in, we we can look down from our little yard and see nothing but murky whiteness mixed with a few of the closest trees. It is so strange to look down to see white.

I am spending a rainy Saturday trying to sort out my lesson plans for the next week or so. We are settling back into our little cottage, fighting back the mold and the scorpions for our little bit fo elbow room.

epilepsy

2008-08-03T17:00:00.001+05:30

Well, I suppose I should break the awkward silence that my blog has become. We returned from the landofthefreehomeofthebrave over two weeks ago. I was supposed to spend the time settling in and focusing on the upcoming academic year, but instead I have been thrust into a wildly unexpected adventure. Will has been diagnosed with 'Primary Generalized Epilepsy'. Needless to say, it has been a rough end to the summer.

Joie and I are slowly reacting to this new development in our life. We never imagined that we could handle such a thing, but life goes on. I don't really have more to write for now, but I thought I would throw that out there because it has certainly been consuming many aspects of my life for a while now.

my grandpa's tractor

2008-07-13T10:27:00.000+05:30

america: the forgotten blogs

2008-07-07T02:53:00.003+05:30

Before I get further behind, here are five pictures from my summer.

We got our happy little family out of Redwood Cottage and down to the Woodstock School gate by 7am to meet our taxi. Our taxi took us down the hill to Dehra Dun, where we boarded a train for Delhi, where we made our way to the Indira Gandhi International Airport, where we waited eight hours to fly to Amsterdam to Minneapolis to Omaha. I think we were traveling for about 48 hours. Here are the young'uns on the first airplane.

Some time passed and we got Will into a boat on a small lake way up in Northern Minnesota. He didn't like the idea at first, but Will is a very good sport with not nearly enough fear. We saw some big fish and 'willy pads' on that lake.

More recently, we had a fourth of July with my whole family and I got to take pictures with our new camera.

Last night I went to a Brewers game to top off my dose of America for the summer.

two years of india

2008-05-16T16:29:00.005+05:30

Well, Joie and I are finishing up our first two years of India. Today was the last regular day of classes. We have a week of exams and a few days of end of the year stuff and then we'll fly back to the US. Almost there.

Will was nine months old when we got here, and now he is a well-established toddler. He walks, talks, laughs, scowls, jokes, and is very nearly housebroken. For as long as Will can remember, power outages, monkeys, mountain climbing, and the absurdities of an Indian bazaar have all been facts of life.

Being caring parents in India means that we have to be careful about what we eat and drink. We rinse our produce in a potassium permanganate bath, and we never drink the water at restaurants. Thus, I have found myself saying to Will: "No, don't eat that, it hasn't been poisoned yet!" and, "No, you can't have any water. Drink your Fanta." Crazy, this life of ours.

In two years of living in India, I think I have learned a lot about being culturally sensitive, but there are still some things that I cannot leave unsaid:

Desserts should not taste like cumin. Not nearly as many things should taste like cumin as actually do.

I will never pay to get a full body massage from a creepy old man on a street corner.

I will never pay to get my ears swabbed by a waxy looking fellow at Connaught Place.

Un-enforced laws against plastic bags are stupid.

Nobody really believes that the traffic cops are ever going to use those massive machine guns that they tote around. We're pretty sure they are not even loaded. The guns, I mean.

If the garbage can is just going to be emptied out the door of the train, quit installing garbage cans on trains and I'll just throw my trash out the window like everyone else. Cut out the middle man.

If your country plays cricket against the same country every day for two months, it at some point stops being front-page news.

Rose is not a flavor. A flower, a color, and a scent, yes, but not a flavor.

he talks anagram indeed

2008-05-05T22:11:00.002+05:30

I duly pay my four rupees per day for the privilege of reading the Times of India. I have lately been far too busy to keep up with the crossword puzzles, but I generally try to read the comics and browse the puzzles and conundrums that spot that antepenultimate page. Today there was a Mindbender that told me to:

Rearrange these letters to give the name of a popular singer:
HE TALKS ANAGRAM

Now I enjoy anagrams as much as the next guy, (unless the next guy is blind, as Jordan would say), but this one took my feeble mind a few minutes. After failing at Mark Knopfler and Elvis Presley and Nat King Cole, my patience was running thin and I gave up in an understandably dejected manner. That's the kind of attention span I have these days. The answer is in the comments for this post.

earth day late

2008-04-23T16:16:00.002+05:30

My school assembled yesterday to watch a video about the "Agony of the Ganges". The film painted a rather grim picture of the river's future.

A main source of the Ganges, the Gangotri glacier, is melting faster than it is being replenished. Global warming is blamed for this. The glacier, they say, has been receding rapidly since about 1780, presumably the first time it was carefully surveyed. As a glacier recedes, it exposes more and more rock that heats up in the sun and melts the glacier even faster. For this reason, the recession that we witness today could be the advanced stages of a very natural melting process that has been in motion for ages. For example, 1780 falls toward the end of a period known as the Little Ice Age, which would presumably foster the growth of glaciers that were doomed to melt in the warmer weather that followed.

I am aware of a few different assessments of the glacier situation. If you read a few of the most visible sources on the web, the Gangotri glacier contributes somewhere between 25% and 75% of the water in the Ganges. The glacier is melting in an upriver direction at a rate of somewhere between 10 and 30 meters per year, and the rate is either decreasing slightly or increasing dramatically. I have no desire to join the numbers game here, since the only measurements I can access are second-hand and wildly inconsistent with one another, leaning in directions that support either of two agendas. I guess you don't get a Nobel Prize for Global Warming by being skeptical.

I am befuddled to find precise graphs of temperature dating back a thousand years. How many people have looked at this graph and never wondered how we know the average temperature of the entire earth (to the nearest hundredth of a degree or so) a full seven hundred years before the development of an accurate thermometer? Any process that leads to this graph must certainly rely on some assumptions big enough to nullify it.

To many Indians, the Ganges is not just a river, but the goddess Ganga, a giver of life and purity, possessing the ability even to wash away sins with a single drop of the holy water. Why do so many people want to worship something that washes sins away? Sadly and ironically, the water of the Ganges is hopelessly toxic and its use is linked to millions of deaths each year.

In the video, the scientists struggle to encourage a distinction between the 'symbolic purity' and the 'actual purity' of the river. Can such a distinction be? Surely a river gains human worship as a supernatural being by existing naturally as a conduit of life... might it not lose its deity by turning poisonous or by drying up? Many people who devote their lives in worship to the river are daily tainting it with their filth. The river receives the sewage of cities like Varanasi, the mortal remains of the devout, and the personal and industrial garbage of the very people who depend on it for life. They have the power to corrupt and destroy the object of their worship.

We find ourselves in a rather desperate position when once we realize that we can save or destroy our god.

Half of the wood he burns in the fire; over it he prepares his meal, he roasts his meat and eats his fill. He also warms himself and says, "Ah! I am warm; I see the fire." From the rest he makes a god, his idol; he bows down to it and worships. He prays to it and says, "Save me; you are my god." -Isaiah 44:16-17

artificial reefs

2008-04-09T16:53:00.002+05:30

Here is a link for an article on slashdot.org about sinking old New York City subway cars in the ocean to make artificial reefs for fish. Although it is presented as an environmental initiative straight out of a task force or subcommittee full of important people, I can't help but wonder if it was just a fortuitous misinterpretation...

Dirk - Well, that does it for all the boss's enemies and the barrels of green radioactive slime. So we're supposed to dump these old subway cars right about here?

Al - Yeah, that's what the boss said. I feel kinda bad about the environment though.

Skippy - U.S. COAST GUARD! PREPARE TO BE BOARDED!

Dirk and Al - [this quote censored]!!!!

Skippy - Hey guys, whatcha doin'?

Dirk - Um...

Skippy - Wow, were you fellas just getting ready to convert these old retired subway cars into artificial reefs? That's fantastic! The world would be a better place if more people like you would step up and take some responsibility. Reduce! Reuse! Recycle!

math in real life

2008-04-08T00:40:00.002+05:30

I am very tired to be writing this, so I don't know if it will work.

I recently taught a quick unit on probability for my Math 9 class. At one point I was explaining that Probability and its evil twin Statistics are very important to the industry of insurance. While I generally regard insurance with a bitter resentment worthy of tax-forms, customs officers, and the guy who broke into my truck to take the radio, my students are all ears when it comes to 'real life' topics. They were actually asking questions about insurance.

I was able to explain that insurance is placing a bet that you will die, or that your house will burn down or that your car will explode. The insurance company smiles and says no, we don't think you'll die. We bet you will live. And of course the odds and rates are all different if the probability changes, like if you are old and have seven kinds of cancer and enemies in the mafia. You answer a bunch of questions about how many cigarettes you smoke and how many ailing relatives you have and how often you go spelunking in volcanoes. I imagine that at some point in the analysis, a person at a computer looks at a number with a bunch of decimal digits that measure your life expectancy to the nearest minute. Of course they don't tell you what it is or you would breathe a raspy sigh of relief and schedule an extra trip down the volcano.

In one sense, it is utterly fascinating that insurance, a product with such an absurd premise, can take root and even stabilize an economy. The presence of insurance in a society gives almost everyone a substantial financial motive to burn down their own house, crash their car, or kill themselves. That is twisted.

porcupine a zoo

2008-04-01T18:25:00.003+05:30

This is my son's first music video, in which he uses his new blue guitar (made from the milk case) for an inspiring rendition of "Porcupine a Zoo", which is presumably inspired by a porcupine we saw scuttling along the road once on the way to Dehra Dun. We do not talk about porcupines very often, so it is a bit of a mystery to me. You all think our family is weird.

budding guitarist

2008-04-01T18:12:00.003+05:30

This is a video of Will playing his bear like a guitar. We finally broke down and made him a guitar out of a milk box (what, your milk doesn't come in boxes?) and during the process he was jamming. Once he switched teddy bears right in the middle of "Jesus Loves Me", like some sort of guitar snob who thinks that their Clapton-esque talents demand two different guitars for a song. We never suggested that he use a bear, but we are happy that he is thus far content with a guitar that makes no sound.

iWonder...

2008-03-31T15:57:00.000+05:30

A couple of evenings ago I was playing volleyball with some students down at the dorms. A handful of students were wandering around in an iTrance with wires dangling and eyes staring. A couple of them were even trying to play sports in this condition, which seemed downright dangerous. I found the whole experience to be slightly unsettling and I wanted to set them all free from their machines and tell them to climb trees and build forts and listen to the birds and smell the pine needles and engage in the world around them. Many students at my school would say they would die without their iPods, and a few of them might believe it. A few of them might be right.

I have (naturally) decided to focus on the bright side of this societal threat.

iPods, whose capitalization and pluralization is confounding me at the moment, do create stable citizens. The electronic device satisfies all of a person's emotional need for originality and romance and art and rebellion without allowing them to become original romantic artistic rebels, which are a serious threat to any society.

iPods present a thrilling business opportunity. How far are we from subliminal advertising via iPods? The people who glide through their lives of enchanted oblivion with wires hanging from their ears certainly represent a customer base (of ample proportions) waiting to be fleeced. And could we get a little video implant for a person's eye? Or both eyes? The most advanced model will plug directly into your brain and tell you what to think in pure binary.

iPods could allow politicians to amass support almost effortlessly. Is it possible that a couple of strategic podcasts from Steve Jobs could turn all of my high school students into a lethargic global army of iComrades? Are podcasts even necessary?

While a person who has read Vonnegut and seen The Matrix feel may not feel great about submitting to technology in this way, the modern high school student does.

holi shimla batman!

2008-03-26T01:09:00.006+05:30

We just returned from a trip to Shimla, a hill station located in Himachal Pradesh. The town is much like Mussoorie, but inexplicably far back into the mountains. Considering all of the empty mountains that we had to pass on our way there, one might ask why Shimla was not established closer to the plains. So I did. But I am afraid that the deed is done and no explanation is likely to satisfy me, since the truth of the matter is probably something like 'It seemed like a good idea at the time'. I remember my consternation at reaching the top of my own mountain only to see the villages peppering the mountains further into the range. Robert Frost has a theory about this here.

Shimla is trivially interesting because Rudyard Kipling lived there. In fact, his father designed a mural which was once in Christ Church. We went to Christ Church, but the mural is gone, painted over or stolen or lost or destroyed somehow. The church is a giant yellow cathedral full of brass and marble memorial plaques to British officers who died in naval battles or epidemics or wars around the world. There were a few stained glass windows left, but it seems like one breaks every couple of decades and it is replaced with a cheaper window of a lesser splendor. The organ pipes were painted matte silver like the scratch-off portions of lottery tickets. The ceiling was a fortress of hefty timbers nicely finished and well-preserved and strongly arranged--built like the inside of a boat, as my dad once pointed out to me in a church when I was little. Inside, we had to take our shoes off, like in a Sikh temple.

We were in Shimla during Holi, which is a Hindu holiday that brought back my memories of Carnivali in Italy, which in turn brought back my fear of clowns. Upon reflection, the people of this world (and often enough in the name of religion) throw themselves into the celebration of a good many holidays which exist to glorify the fleshly passions.

Anyway, Holi is celebrated by throwing colors on one another. The colors are stained water or brightly colored powders. People get drunk and stoned and do their best--and encourage their youngsters--to abandon themselves to the holiday. The bazaar was full of colorful characters, and while we declined most of the glassy-eyed invitations to have ourselves smeared with the most vivid greens, pinks, oranges, and purples you have ever seen, we did not escape unscathed, and my jeans have some faint stains.

While wandering around Shimla we found a tourist map, which we stared at for a few long minutes before discussing with one another, doubting ourselves and each other and finally, doubting the map itself. Admittedly, certain contextual cues had me doubting the map a bit anyway, but I didn't imagine it was quite as erroneous as it actually turned out to be. Maps of mountain cities are difficult, to be sure, because the terrain is rather three dimensional and does not offer any proper projection to two dimensions except through the aerial view, which distorts the distances and deprives the map of any depiction of altitude, which is a rather weighty consideration for us pedestrians. For that reason, I did not expect to make sense of the map right away. After a few minutes, however, we realized that not only was the map distorted or poorly marked, but it was entirely reflected horizontally. North was at the top and South at the bottom, but West was at the right, which only really provides a worm’s-eye view of the town, as Pete said. The transformation, though preserving betweenness, made it almost impossible to learn anything.

The map was painted onto an enormous concrete sign, and it appears to have been there since Rudyard Kipling left. How the fearless Shimlite leaders lack the pride or self-respect to fix or deface such a thing is beyond me. The map is right next to the local and state government buildings, and right across from a hut for a well-armed traffic cop who likely only pretends to possess a geographical instinct.