A002188 Sprague-Grundy values for Grundy's game. (Formerly M0044 N0014)

0, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 2, 1, 3, 2, 1, 3, 2, 4, 3, 0, 4, 3, 0, 4, 3, 0, 4, 1, 2, 3, 1, 2, 4, 1, 2, 4, 1, 2, 4, 1, 5, 4, 1, 5, 4, 1, 5, 4, 1, 0, 2, 1, 0, 2, 1, 5, 2, 1, 3, 2, 1, 3, 2, 4, 3, 2, 4, 3, 2, 4, 3, 2, 4, 3, 2, 4, 3, 2, 4, 5, 2, 4, 5, 2, 4, 3, 7, 4, 3, 7, 4, 3, 7, 4, 3, 5, 2, 3, 5, 2, 3, 5, 2, 3

Offset

0,6

References

C. Berge, Graphs and Hypergraphs, North-Holland, 1973; p. 324.

R. K. Guy, Fair Game: How to play impartial combinatorial games, COMAP's Mathematical Exploration Series, 1989; see p. 96.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Eric M. Schmidt, Table of n, a(n) for n = 0..10000

Achim Flammenkamp, Sprague-Grundy Values of Grundy's Game

P. M. Grundy, Mathematics and games, Eureka (The Archimedeans' Journal), No. 2, 1939, pp. 6-8. [Annotated scanned copy. My former colleague and coauthor Florence Jessie MacWilliams (nee Collinson), who was a student at Cambridge University in 1939, gave me this journal. - N. J. A. Sloane, Nov 17 2018]

Richard K. Guy and Cedric A. B. Smith, The G-values of various games, Proc. Cambridge Philos. Soc. 52 (1956), 514-526. See Table 4.

Gabriel Nivasch, The Sprague-Grundy theory of impartial games

Gabriel Nivasch, The Sprague-Grundy theory of impartial games [archived version]

Eric Weisstein's World of Mathematics, Grundy's Game

Formula

"Mike Guy has computed ten million values, but a discernible pattern remains elusive" [Guy, 1989]. - N. J. A. Sloane, Jan 03 2016

Mathematica

mex[list_] :=  mex[list] = Min[Complement[Range[0, Length[list]], list]];
move[grundygame, list_] := move[grundygame, list] = Union@Flatten[Union[Table[ Sort@Join[Drop[list, {i}], {list[[i]] - j, j}], {i, Length[list]}, {j, Floor[(list[[i]] - 1)/2]}], Table[Sort@Join[Drop[list, {i}], {list[[i]] - j, j}], {i, Length[list]}, {j, Ceiling[(list[[i]] + 1)/2], list[[i]] - 1}]], 1];
SpragueGrundy[game_, list_] :=  SpragueGrundy[game, list] =
   mex[SpragueGrundy[game, #] & /@ move[game, list]];
Table[SpragueGrundy[grundygame, {i}], {i, 0, 42}] (* Birkas Gyorgy, Apr 19 2011 *)

Prog

(C++)
#include <algorithm>
#include <array>
#include <iostream>
int main() {
    constexpr int bound = 10000;
    std::array<int, bound+1> gnumbers;
    std::array<bool, bound/2+1> excluded;
    for (int i = 0; i <= bound; ++i) {
        auto e_begin = excluded.begin();
        auto e_end = e_begin + i/2;
        std::fill(e_begin, e_end, false);
        for (int j = 1; j < (i+1)/2; ++j) {
            int const k = i - j;
            excluded[gnumbers[j] ^ gnumbers[k]] = true;
        }
        gnumbers[i] = std::find(e_begin, e_end, false) - e_begin;
    }
    for (int i = 0; i <= bound; ++i)
        std::cout << i << ' ' << gnumbers[i] << '\n';
} // Eric M. Schmidt, Jan 04 2017

Crossrefs

See A036685 for indices of zero terms.

Sequence in context: A025656 A194517 A110658 * A128313 A283486 A330759

Adjacent sequences: A002185 A002186 A002187 * A002189 A002190 A002191

Keyword

nonn,easy,look,nice

Author

N. J. A. Sloane

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 11 2004

Status

approved