A028933 Table of winning positions in Tchoukaillon (or Mancala) solitaire.
0, 1, 0, 2, 1, 2, 0, 1, 3, 1, 1, 3, 0, 0, 2, 4, 1, 0, 2, 4, 0, 2, 2, 4, 1, 2, 2, 4, 0, 1, 1, 3, 5, 1, 1, 1, 3, 5, 0, 0, 0, 2, 4, 6, 1, 0, 0, 2, 4, 6, 0, 2, 0, 2, 4, 6, 1, 2, 0, 2, 4, 6, 0, 1, 3, 2, 4, 6, 1, 1, 3, 2, 4, 6, 0, 0, 2, 1, 3, 5, 7, 1, 0, 2, 1, 3, 5, 7
Offset: 0
Examples
The rows of b(n,i) begin n\i 1 2 3 4 5 6 1 1 2 0 2 3 1 2 4 0 1 3 5 1 1 3 6 0 0 2 4 7 1 0 2 4 8 0 2 2 4 9 1 2 2 4 10 0 1 1 3 5 11 1 1 1 3 5 12 0 0 0 2 4 6 13 1 0 0 2 4 6 14 0 2 0 2 4 6 15 1 2 0 2 4 6 16 0 1 3 2 4 6 17 1 1 3 2 4 6
Links
- D. Betten, Kalahari and the Sequence "Sloane No. 377", Annals Discrete Math., 37, 51-58, 1988.
- D. M. Broline and _Daniel E. Loeb_, The combinatorics of Mancala-Type games: Ayo, Tchoukaillon and 1/Pi, J. Undergrad. Math. Applic., vol. 16 (1995), pp. 21-36.
- Brant Jones, Laura Taalman and Anthony Tongen, Solitaire Mancala Games and the Chinese Remainder Theorem, Amer. Math. Monthly, 120 (2013), 706-724.
Programs
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Mathematica
s[list_] := Module[{x = Append[list, 0], i = 1}, While[x[[i]] =!= 0, x[[i]] = x[[i]] - 1; i = i + 1]; x[[i]] = i; If[Last@x == 0, Most[x], x]]; Prepend[Flatten@NestList[s, {}, 20],0] (* Birkas Gyorgy, Feb 26 2011 *)
Formula
Let p(n) be the minimum j such that b(n,j) = 0. (This is A028920.)
Directly from the rules of Tchoukaillon, we find b(n+1,i) = (b(n,i) - 1 for 1 <= i < p(n), i for i = p(n), and b(n,i) for i > p(n)).
Also, b(n,i) = (n - Sum_{j=1..(i-1)} b(n,j)) mod (i+1).
Extensions
Formulas added by Brant Jones, Oct 14 2013
Comments