Game theory by Ferguson T.S.

By Ferguson T.S.

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Consider the sum of three subtraction games. In the first game, m = 3 and the pile has 9 chips. In the second, m = 5 and the pile has 10 chips. And in the third, m = 7 and the pile has 14 chips. Thus, we are playing the game G(3) + G(5) + G(7) and the initial position is (9, 10, 14). The Sprague-Grundy value of this position is g(9, 10, 14) = g3 (9) ⊕ g5 (10) ⊕ g7 (14) = 1 ⊕ 4 ⊕ 6 = 3. One optimal move is to change the position in game G(7) to have Sprague-Grundy value 5. This can be done by removing one chip from the pile of 14, leaving 13.

A position in the game of Rims is a finite set of dots in the plane, possibly separated by some nonintersecting closed loops. A move consists of drawing a closed loop passing through any positive number of dots (at least one) but not touching any other loop. Players alternate moves and the last to move wins. (a) Show that this game is a disguised form of nim. 2, find a winning move, if any. 2 A Rims Position 9. Rayles. There are many geometric games like Rims treated in Winning Ways, Chapter 17. In one of them, called Rayles, the positions are those of Rims, but in Rayles, each closed loop must pass through exactly one or two points.

One optimal move is to change the position in game G(7) to have Sprague-Grundy value 5. This can be done by removing one chip from the pile of 14, leaving 13. There is another optimal move. Can you find it? I – 22 This shows the importance of knowing the Sprague-Grundy function. We present further examples of computing the Sprague-Grundy function for various one-pile games. Note that although many of these one-pile games are trivial, as is one-pile nim, the SpragueGrundy function has its main use in playing the sum of several such games.

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