I'm hoping someone computes the period for various values of k and finds a pattern (though as Adam points out there need not be one). Jim On Tuesday, June 27, 2017, Adam P. Goucher <apgoucher@gmx.com> wrote:
The Sprague-Grundy number S(n, i) corresponding to a particular score deficit n and top face i can be obtained as:
mex{S(n-j, j) : j neq i, 2k+1-i}
where 'mex' = 'minimum excluded ordinal'. So all S-G numbers are bounded above by 2k-2, and each one is a function of the 4k^2 previous ones (in the lexicographical order on N x [1, 2k]). Consequently, we can say that they are eventually periodic with period at most:
(2k-2)^(4k^2)
I don't think that there is any deep reason for why the period is 9, or any other number -- there must be *some* period, by the above argument, and it's quite arbitrary what it turns out to be. Dawson's Chess, for instance (begin with an empty line of squares, and take turns filling in squares not adjacent to existing filled squares), has a period of 34.
Best wishes,
Adam P. Goucher
Sent: Tuesday, June 27, 2017 at 5:49 AM From: "James Propp" <jamespropp@gmail.com <javascript:;>> To: math-fun <math-fun@mailman.xmission.com <javascript:;>> Subject: [math-fun] Philip Yasskin wants to know
Philip Yasskin of Texas A&M taught me a fun game yesterday. (He learned it from his son, who learned it at a Renaissance Faire.)
You roll a die, and the number it shows becomes the current score. The first player then rotates the die by 90 degrees, so that one of the four faces adjacent to the old top face becomes the new top face. The score increases by this amount. The second player does the same. Play alternates in this fashion until either the score becomes exactly 31 (in which case the player who just played wins) or the score exceeds 31 (in which case the player who just played loses).
In some ways it's more natural to look at the "score deficit" (31 minus the score) as opposed to the score, especially if one wants to consider other versions of the game in which some number other than 31 is the "target".
Philip noticed that when you look at the behavior of the game as a function of the score-deficit, you observe periodicity with period 9 (maybe he said "eventual periodicity" --- I forget), and he wondered why the 9 shows up. Here's a proposed variant that might shed light on Philip's question (or might further enlarge the darkness).
For each k, we can consider a combinatorial game whose states are specified by ordered pairs (n,i), where n is a non-negative integer and i is between 1 and 2k inclusive. (For concreteness you might want to focus on k=3, and to think of i as the number shown on the top face of a 6-sided die.)
A legal move goes from (n,i) to (n-j,j) as long as n-j is non-negative and j is neither i nor 2n+1-i. (For our concrete case k=3, j can be any number that is adjacent to i on the die.)
Does this game exhibit eventual periodicity with period 3k for all values of k? That is, can we say that (n,i) is a P-position iff (n-3k,i) is, for all sufficiently large n?
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