An Unfair Contest

Scott Moore

4/07/2002

The purpose of this paper is to provide an analysis of two games that are played as follows: A starting integer $A$ and an ending integer $B$ are chosen. Two players take turns removing one value at a time from all integers in the set [$A,B$], until there are only 2 left. In the first game, Player 1 is trying to force these two numbers to have a common factor, and in the second game, Player 2 is trying to force these numbers to have a common factor. The winner is the player who correctly matches the coprimality of the last two numbers. For example, let's suppose that Player 1 is trying to have the last 2 numbers be coprime, thus Player 2 wants them to have a common factor. They might start with the sequence 11, 12, 13, 14, 15, 16, 17, and then make the following turns:

• Player 1 removes 15
• Player 2 removes 13
• Player 1 removes 16
• Player 2 removes 17
• Player 1 removes 12

This leaves 11 and 14, which are coprime; therefore Player 1 wins. In the above case, it was pretty obvious that Player 1 had the advantage - 3 out of the 7 numbers are prime, and Player 2 only gets 2 turns! It can be shown that in any finite game without ties, one of the players can have a strategy that will allow them to win irregardless of what moves the other[s] make. In this paper, we will explore such strategies for all values of $A,B > 1$ for both games described above.