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A047708
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Diagonal of Sprague-Grundy function for Wyt Queens (Wythoff's game).
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3
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0, 2, 1, 6, 7, 8, 3, 5, 4, 16, 14, 15, 10, 9, 11, 20, 13, 21, 12, 25, 17, 18, 19, 30, 31, 38, 35, 36, 22, 23, 43, 45, 48, 49, 24, 26, 27, 28, 29, 33, 60, 32, 61, 57, 66, 37, 63, 34, 64, 67, 40, 39, 41, 42, 82, 44, 74, 79, 47, 46, 87, 86, 50, 95, 96, 52, 101, 51, 102, 53, 54
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OFFSET
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0,2
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COMMENTS
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Since Wythoff(m,n) <= m+n, Wythoff(n,n) <= 2n. It is not known whether there is an efficient (linear in log(m)+log(n)) strategy to compute Wythoff(m,n). Each single row is "easy" in the sense that a+n-Wythoff(a,n) is eventually periodic. - Howard A. Landman.
It is impossible for any integer to appear twice in this sequence because of the way it is constructed. Thus to prove that it is a permutation of the integers, we need only show that every value g appears at least once.
Suppose this were not true; then there must be some g such that for any value of n, G(n,n) is not = g. Since G(n,n) is defined to be the smallest number not found as a G(k,n), G(n,k), or G(k,k) for k < n, this can only happen in one of 2 ways: either there is a number g' smaller than g which is chosen (this can occur at most g times) or g already appears as both G(n,k) and G(k,n) for some k < n (because G(n,k) = G(k,n)) (this can happen at most n/2 times).
Thus we have n <= n/2 + g, or n <= 2g; if g has not appeared within the first 2g terms we have a contradiction. Therefore not only must every integer g appear in the sequence, but it must appear within the first 2g terms (and no sooner than term g/2, since G(n,n) <= 2n). Conversely, this also proves that n/2 <= A(n) = G(n,n) <= 2n. (End)
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REFERENCES
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E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.
Howard A. Landman, "A Simple FSM-Based Proof of the Additive Periodicity of the Sprague-Grundy Function of Wythoff's Game", in R. Nowakowski (ed.), More Games of No Chance.
Howard A. Landman and Tom Ferguson showed that this is a permutation of the integers at the Jul 24-28 2000 MSRI workshop on combinatorial games.
W. A. Wythoff, "A Modification of the Game of Nim". Nieuw Arch. Wiskunde 8, 199-202, 1907/1909.
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LINKS
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MATHEMATICA
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mex[list_] := mex[list] = Min[Complement[Range[0, Length[list]], list]];
move[Wnim, {a_, b_}] := move[Wnim, {a, b}] =
Union[Table[{i, b}, {i, 0, a - 1}], Table[{a, i}, {i, 0, b - 1}],
Table[{a - i, b - i}, {i, 1, Min[a, b]}]];
SpragueGrundy[game_, list_] := SpragueGrundy[game, list] =
mex[SpragueGrundy[game, #] & /@ move[game, list]];
Table[SpragueGrundy[Wnim, {i, i}], {i, 0, 64}] (* Birkas Gyorgy, Apr 19 2011 *)
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PROG
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(PARI) See Links section.
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CROSSREFS
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Main diagonal of square array in A004481. Sequences A000201 and A001950 give the m and n coordinates of the zeros of Wythoff (i.e., the P-positions of the game, where the previous player has won).
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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