1 mod 3 then k 2 k 4 and k 7 have sg values of 1 0

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1(mod3).Thenk-2,k-4, andk-7 have SG values of 1, 0, and 0,respectively, by modular arithmetic; therefore,n=kwill have an SG number of 2.Suppose thatk2(mod3).Thenk-2,k-4, andk-7 have SG values of 0, 2, and 2,respectively, by modular arithmetic; therefore,n=kwill have an SG number of 1.We have shown in all cases thatg(k) accurately describes the SG numbers ofk, given that it isaccurate forn= 8,9, . . . , k-1. Therefore, by the mathematical principle of strong induction, theg(x) described above is our SG function.Since any point with an SG number of 0 is aP-position, ourP-positions for this specific subtrac-tion game are at{0,1}∪{3k|k2, kZ}, and all other points areN-positions ({2,3,4,5,7,8,10,11, . . .}).
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Stat 155, Spring 2017Homework 1Problem 2 (KP 1.1)Consider a game of Nim with four piles, of sizes 9, 10, 11, 12.(a) Is this position a win for thenextplayer or thepreviousplayer? Describe the winning firstmove.
(b) Consider the same initial position, but suppose that each player is allowed to remove at most9 chips in a single move. Is this anN- or aP-position?

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