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

0,4

Comments

Table read by rows where b(n,i) = the number of counters in the i-th position from the store of the unique winning Tchoukaillon board having n total counters.

Links

Table of n, a(n) for n=0..87.

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.

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).

Example

   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

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 *)

Crossrefs

Cf. A002491, A028920, A007952, A028931, A028932, A028933.

Sequence in context: A063574 A144515 A178650 * A190491 A364835 A143352

Adjacent sequences: A028930 A028931 A028932 * A028934 A028935 A028936

Keyword

nonn,tabl

Author

N. J. A. Sloane.

Extensions

Formulas added by Brant Jones, Oct 14 2013

Status

approved