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By Ferguson T.S.

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That is, at least one edge must be removed. Last player to move wins. Investigate this game. For example, (a) Find the Sprague-Grundy value of Sn , the star with n points. ) (b) Find the Sprague-Grundy value of Ln , the line of n edges and n + 1 vertices (at least for small values of n). (c) Find the Sprague-Grundy value of Cn, the circular graph of n vertices and n edges. (d) Find the Sprague-Grundy value of a double star. ) I – 27 (e) Who wins the 3 by 3 square lattice, (i)? Who wins the tic-tac-toe board, (ii)?

Suppose at each turn a player may (1) remove one or two chips, or (2) remove one chip and split the remaining chips into two piles. (a) Find the Sprague-Grundy function. (b) Suppose the starting position consists of one pile with 15 chips. Find an optimal first move. I – 26 6. Suppose that at each turn a player may (1) remove one chip if it is a whole pile, or (2) remove two or more chips and, if desired, split the remaining chips into two piles. Find the Sprague-Grundy function. 7. Suppose that at each turn a player may select one pile and remove c chips if c = 1 (mod 3) and, if desired, split the remaining chips into two piles.

10. Grundy’s Game. 4, for a pile of n chips for n = 1, 2, . . , 13. (b) In Grundy’s game with three piles of sizes 5, 8, and 13, find all winning first moves, if any. 11. A game is played on a finite (undirected) graph as follows. Players alternate moves. A move consists of removing a vertex and all edges incident to that vertex, with the exception that a vertex without any incident edges may not be removed. That is, at least one edge must be removed. Last player to move wins. Investigate this game.

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Game theory by Ferguson T.S.


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