Thursday, April 30, 2009

deadline online

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.

Tuesday, April 21, 2009

solution: buried treasure

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

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.

Friday, April 17, 2009

5-star challenge #6

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

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?

Wednesday, April 8, 2009

the disillusionment

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.