Generalize the Take-Away Game: (a) Suppose in a game with a

pile containing a large number of chips, you can remove any number from 1 to

6 chips at each turn. What is the winning strategy? What are the P-positions?

(b) If there are initially 31 chips in the pile, what is your winning move, if any?

The Thirty-one Game. (Georey Mott-Smith (1954)) From a deck

of cards, take the Ace, 2, 3, 4, 5, and 6 of each suit. These 24 cards are laid

out face up on a table. The players alternate turning over cards and the sum of

the turned over cards is computed as play progresses. Each Ace counts as one.

The player who rst makes the sum go above 31 loses. It would seem that this is

equivalent to the game of the previous exercise played on a pile of 31 chips. But

there is a catch. No integer may be chosen more than four times. (a) If you are

the rst to move, and if you use the strategy found in the previous exercise, what

happens if the opponent keeps choosing 4? (b) Nevertheless, the rst player can

win with optimal play. How?

Find the set of P-positions for the subtraction games with

subtraction sets (a) S = {1, 3, 5, 7}. (b) S = {1, 3, 6}. (c) S = {1, 2, 4, 8,

16, . . .} = all powers of 2. (d) Who wins each of these games if play starts at

100 chips, the rst player or the second?