In summary the set of impartial combinatorial games

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In summary, the set of impartial combinatorial games as defined in Section 1.2 is equivalent to the set of graph games with finitely progressive graphs. The Sprague-Grundy function on such a graph exists if transfinite values are allowed. For progressively finite graph games, the Sprague-Grundy function exists and is finite. If the graph is allowed to have cycles, new problems arise. The SG-function satisfying (1) may not exist. Even if it does, the simple inductive procedure of the previous sections may not su ce to find it. Even if the the Sprague-Grundy function exists and is known, it may not be easy to find a winning strategy. I – 17
Graphs with cycles do not satisfy the Ending Condition (6) of Section 1.2. Play may last forever. In such a case we say the game ends in a tie; neither player wins. Here is an example where there are tied positions. a b c d e Figure 3.4 A cyclic graph. The node e is terminal and so has Sprague-Grundy value 0. Since e is the only follower of d , d has Sprague-Grundy value 1. So a player at c will not move to d since such a move obviously loses. Therefore the only reasonable move is to a . After two more moves, the game moves to node c again with the opponent to move. The same analysis shows that the game will go around the cycle abcabc . . . forever. So positions a , b and c are all tied positions. In this example the Sprague-Grundy function does not exist. When the Sprague-Grundy function exists in a graph with cycles, more subtle tech- niques are often required to find it. Some examples for the reader to try his/her skill are found in Exercise 9. But there is another problem. Suppose the Sprague-Grundy function is known and the graph has cycles. If you are at a position with non-zero SG-value, you know you can win by moving to a position with SG-value 0. But which one? You may choose one that takes you on a long trip and after many moves you find yourself back where you started. An example of this is in Northcott’s Game in Exercise 5 of Chapter 2. There it is easy to see how to proceed to end the game, but in general it may be di cult to see what to do. For an e cient algorithm that computes the Sprague-Grundy function along with a counter that gives information on how to play, see Fraenkel (2002). 3.5 Exercises. 1. Fig 3.5 Find the Sprague-Grundy function. 2. Find the Sprague-Grundy function of the subtraction game with subtraction set S = { 1 , 3 , 4 } . I – 18 1 0 0 0
3. Consider the one-pile game with the rule that you may remove at most half the chips. Of course, you must remove at least one, so the terminal positions are 0 and 1. Find the Sprague-Grundy function. 4. (a) Consider the one-pile game with the rule that you may remove c chips from a pile of n chips if and only if c is a divisor of n , including 1 and n . For example, from a pile of 12 chips, you may remove 1, 2, 3, 4, 6, or 12 chips. The only terminal position is 0.

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