481: Fun problem of the week

100 very logical pirates need to divide 1000 gold coins. Their algorithm is as follows. The pirates are ranked from wimpiest to fiercest. First, the fiercest pirate (no. 100) proposes a distribution of the coins. Then all the pirates on board vote, and if at least half agree, then the coins are distributed and the algorithm is over. If, however, the majority votes against the proposed distribution, then the fiercest pirate is thrown overboard, and pirate no. 99 gets to propose a new distribution among the remaining 99. A vote is taken on this proposal, and if rejected then pirate 98 gets to propose a distribution among the remaining 98 pirates, etc. Question: You are pirate no. 100. What distribution should you propose?

Previous problems:

• Imagine a 6 by 6 board (like a chess board only smaller). If we have 18 dominoes, each just big enough to cover two adjacent squares of the board, it is possible to completely cover the board with the dominoes. Here is a tricky little fact: for every way of covering the entire board with the dominoes, it is possible to make a single horizontal or vertical cut through the board which will split it into two pieces, but leave all the dominoes intact. Can you prove this?
• Let's call a positive integer n "green" if it is the sum of consecutive positive integers all of which are less than itself. For example, 9=2+3+4 is a green number. Prove that a positive integer is green if and only if it is not a power of two.
• Show that any set of 100 integers contains a non-empty subset whose elements sum to a multiple of 100.
• A woman and her husband attended a party with four other couples. As is normal at parties, handshaking took place. Of course, no one shook their own hand or the hand of the person they came with. And not everyone shook everyone else's hand. But when the woman asked the other nine people present how many different people's hands they had shaken they all gave a different answer. Question: How many different people's hands did the woman's husband shake?
• Prove that from any set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint, non-empty subsets whose members have the same sum. Hint: if you prove that two different subsets must have the same sum, it is then easy to show that two disjoint subsets must have the same sum.
• Suppose a circular route is laid out and some number of gas stations, say k, are placed along the route (note necessarily equally spaced). Assume that your car has a tank big enough to hold a quantity of gas sufficient to get you around the entire route and that it consumes gas at some constant rate (miles/gallon). If you sum up all the gas available at all the gas stations, there is precisely enough to get you around the circle once. Can you demonstrate that there is some gas station where you can start so that you will be able to drive all the way around the circle without running out of gas? Assume that you start with an empty tank.