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.