### From Fall 2010 Workshop #2

posted Feb 14, 2011, 2:10 PM by Altha Rodin   [ updated Feb 14, 2011, 2:12 PM ]
 Planet Carefully has only one landmass, a small island called Isle of Puzzles. On this island is a fearless pilot who wants to fly all the way around his watery planet. Unfortunately, planes on Planet Carefully can only carry enough fuel to get halfway around the globe. Fortunately, pilots can instantaneously transfer any amount of fuel from the tank of one plane to that of another mid-flight. What is the smallest number of planes that must participate in this adventure so that one pilot flies without stopping around the globe and all pilots make it home safely? In each round of an elimination tournament, players are randomly paired and each pair plays one game. Losers are eliminated from the tournament, but winners go on to the next round. If there are an odd number of players during a round, then one player does not play that round, but does advance to the next round. So, for example, if there are 19 players, there will be 9 games played and 10 people will go on to the next round. This continues until there is a single winner. If there are n players to begin with, how many games will be played before a winner is determined? (Note that we are counting games played as opposed to rounds played.) Two Cossack brothers sold a herd of sheep. Each sheep sold for as many rubles as the number of sheep originally in the herd. The money was then divided in the following manner: first the older brother got 10 rubles, then the younger brother got 10 rubles, and so on. At the end of the division, the younger brother, whose turn it was, saw that there were fewer than 10 rubles left, so he took what remained. To make the division fair, the older brother gave the younger one his knife which was worth an integer number of rubles. How much was the knife worth?
Ċ
Altha Rodin,
Feb 14, 2011, 2:10 PM