A131469 Grundy numbers of one pile short global nim.

0, 1, 1, 2, 3, 3, 2, 4, 5, 5, 6, 7, 7, 6, 4, 8, 9, 9, 8, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 14, 18, 19, 16, 17, 18, 20, 10, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 24, 26, 27, 27, 28

Offset: 0

Author

Mark Schlatter (mschlat(AT)centenary.edu) and Jeffery James (jjames(AT)centenary.edu), Jul 26 2007

Keywords

• nonn

Comments

The game short global nim is identical to nim except that the last move cannot be repeated. (For example, if 2 stones were taken last turn, any number except 2 may be taken this turn.) The i-th term is the Grundy number of a pile of size i.

References

• R. K. Guy and R. J. Nowakowski, Unsolved Problems in Combinatorial Games, More Games of No Chance, MSRI Publications, Volume 42, 2002, pp. 457-473, problem 22.

Links

• Urban Larsson, Simon Rubinstein-Salzedo, Aaron N. Siegel, Memgames, arXiv:1912.10517 [math.CO], 2019.

Programs

Maple

mex := proc (list) local testn; testn := 0; while evalb(`in`(testn, list)) do testn := testn+1 end do; testn end proc
nextmoves := proc (move) local i, j, k, l, list1, list2, list3, list4, list5, list6; i := move[1]; j := move[2]; k := move[3]; list1 := `minus`({seq([n, j, i-n], n = 0 .. i-1)}, {[i-k, j, k]}); list2 := `minus`({seq([i, n, j-n], n = 0 .. j-1)}, {[i, j-k, k]}); convert(`union`(list1, list2), list) end proc
sgnimgrundy := proc (move) local nmoves, i, j, k; option remember; nmoves := nextmoves(move); i := move[1]; j := move[2]; k := move[3]; if i = 0 and j = 0 then 0 elif i = 0 and j = 1 and k = 1 then 0 elif i = 1 and j = 0 and k = 1 then 0 elif i = 1 and j = 1 and k = 1 then 0 else mex({seq(apply(sgnimgrundy, nmoves[i]), i = 1 .. nops(nmoves))}) end if end proc