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Stones in Cups

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Stones in Cups
Other Names: Cipra's Problem
1388, CSCP, Cups and Stones
Counting Problem
Inventor: Barry Cipra, 1992
Ranks: One
Sowing: Reverse
Region: USA

Stones in Cups, also called Cipra's Problem 1388, is a solitaire mancala game. Closely related games are Circular Composition and Montreal Solitaire. The game was invented in 1992 by Barry Cipra, a resident of Northfield, Minnesota (USA), who proposed it as a mathematical problem in Mathematics Magazine. The game was independently solved by Kay P. Litchfield (Farmington, Utah, USA) and David Callan (University of Wisconsin, Madison, USA) in 1993. The solutions use inverse moves, that is reverse sowing. The game was also briefly described by Paul J. Campbell and Darrah P. Chavey in 1995.


The game is played with n cups that are arranged in a circle. At the beginning there are k stones placed in each cup.

Stones in Cups ini

Possible Set-up

The first move may start from any cup. Later, a move begins at the cup, which was filled last.

Each move the contents of a cup are distributed clockwise, one by one, into the succeeding cups.

The game ends when all the stones wind up in the original cup (ie. the first hole). The next move would restore the original position. The number of steps to reach this result is called a(kn). The task is to predict the number of steps.


Let's try it with two holes and one seed per hole (k=1, n=2).

Stones in Cups exam1

Now, all stones are in the original cup.

Stones in Cups exam2

The initial position, after all stones were in the original cup, is reached again after 4 steps.

You can try it for other values of k and n.


The first values are given in the following table:

k\n 1 2345
1 1 4 15 12 75
2 1 6 21 164 115
3 112 45 164 260
4 1 8132 1243825
5 1 6 4815801966

External Links


Callan, D., Cipra, B. & Litchfield, K. P. et al. 
Stones in Cups (Solutions). In: Mathematics Magazine 1993; 66 (1): 58-59.
Campbell, P. J. 
Tchuka Ruma Solitaire. In: The UMAP Journal 1995; 16 (4): 343-365.
Cipra, B. 
1388. In: Mathematics Magazine 1992; 65: 56.
Ettestad, D. & Carbonara, J.
Fractal Properties of the Matrix for the Cups and Stones Counting Problem. In: International Journal of Pure and Applied Mathematics 2006; 29 (1): 81-106.
Servedio, R. & Yeh, Y.-N. 
A Bijective Proof on Circular Compositions. In: Bulletin of the Institute of Mathematics Academia Sinica 1995; 23: 283-293.


© Wikimanqala.
By: Ralf Gering & Víktor Bautista i Roca
Under the CC by-sa 2.5 license.

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