Math 160 Mathematical Problems and Games, Fall 2002
Problem Set 9: hand in one problem on Tuesday, November 19.

Instructions for this week:  As always, try all of the problems.  Then choose one to write up and hand in.

1. Ten 1’s and ten 2’s are written on a blackboard. In one turn, a player may erase any two numbers. If the two numbers were the same, they are replaced with a 2. If they were different, they are replaced with a 1. At the end of the game, one number will be left. If it is a 1, the first player wins; if it is a 2, the second player wins. Who will win, and how?

2. The numbers 25 and 36 are written on a blackboard. At each turn, a player writes on the blackboard the (positive) difference between any two numbers already on the board, as long as their difference isn’t already written there. The loser is the player who cannot write a number. Who will win, and how?  (Note:  numbers and their digits are considered separately.  For instance, even though 25 is on the board at the beginning, you could still write "2" or "5" later.  And subtracting 2 from 5 is not a valid first move.).

3. Two players start with a 9 x 10 rectangle of boxes. In each turn, a player is allowed to choose one row or column where not all of the boxes have been crossed out, and cross out that row or column. The player who cannot make a move loses. Who will win, and how?

4. There are two piles of stones. One piles has 30 stones, and the other has 20 stones. Players take turns removing as many stones as they please (at least one) from ONE pile. The player who takes the last stone wins. How should you play if you want to win? Should you go first or second?

If any of this is confusing or you want a hint, email me at ksir@math.sunysb.edu .