Stones in cups
From Wikimanqala
Stones in cups
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| Cipra's problem 1388 |
| © 1992, Barry Cipra |
| USA |
| Published rules |
| This game is a solitaire |
| One cycle |
| Multiple lap |
| n holes per row |
| One row |
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.
Contents |
Rules
The game is played with n cups that are arranged in a circle. At the beginning there are k stones placed in each cup.
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| Possible set-up |
The first move may start from any cup. Later, a move begins at the cup, which was filled last. It is, as some sort of multiple lap, but also continuing the sowing with a singleton.
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. The next move would restore the original position. The number of steps to reach this result is called a(kn). Your task is to predict the number of steps.
Example
Let's try it with two holes and one seed per hole (k=1, n=2)
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| Now, all stones are in the original cup |
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| And the initial position, after all stones were in the original cup, is reached after 4 steps. |
You can try it for other values of k and n.
Results
The first values are given in the following table:
| k\n | 1 | 2 | 3 | 4 | 5 |
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| 1 | 1 | 4 | 15 | 12 | 75 |
| 2 | 1 | 6 | 21 | 164 | 115 |
| 3 | 1 | 12 | 45 | 164 | 260 |
| 4 | 1 | 8 | 132 | 124 | 3825 |
| 5 | 1 | 6 | 48 | 1580 | 1966 |
References
- Callan, D. & Litchfield, K. P.
- (1993) 'Stones in Cups (Solutions)', in Mathematics Magazine; 66. Pages 58-59.
- Campbell, P. J.
- (1995) 'Tchuka Ruma Solitaire', in The UMAP Journal; 16 (4). Pages 343-365.
- Cipra, B.
- (1992) '1388', in Mathematics Magazine; 65. Page 56.
- Servedio, R. & Yeh, Y.-N.
- (1995) 'A Bijective Proof on Circular Compositions', in Bulletin of the Institute of Mathematics Academia Sinica; 23. Page 283-293.
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