| Bulgarian Solitaire |
| Other Names: Deterministic Bulgarian Solitaire |
| Inventor: Martin Gardner, 1983 |
| Ranks: None |
| Sowing: Reverse |
| Region: USA |
Bulgarian Solitaire, also known as Deterministic Bulgarian Solitaire, was invented by the famous American recreational mathematician Martin Gardner in 1983.
Many variants were created, such as Austrian Solitaire (Ethan Akin & Morton David Davis (USA), 1985), Montreal Solitaire (Chris Cannings & John Haigh (England), 1992), Random Bulgarian Solitaire (Serguei Popov (Brazil), 2003) and Two-handed Bulgarian Solitaire (Tim Bancroft (USA), 2004). Gardner's game is based on a traditional Bulgarian game that is known as Carolina Solitaire. His game and its variants are extensively researched in Combinatorial Game Theory (CGT). It employs reverse sowing.
Prof. Su Dorée of Augsburg College, Minneapolis, USA, called Bulgarian solitaire "a somewhat distant relative of the two-player African pebble games Mancala".
Rules
The game is played by just one person.
In the game, a group of N cards is divided into several decks (or "piles").
Then one card is removed from each deck.
The removed cards are collected together to form a new deck (piles of zero size are ignored).
The decks are not ordered, so it doesn't matter in which order the cards are being removed or where the new pile is placed.
The game ends when the same position occurs again.
Mathematics
If N is a triangular number (that is, N = 1 + 2 + ... +k for some k), then it is known that Deterministic Bulgarian Solitaire will reach a stable configuration in which the size of the piles is 1, 2, ... k. This state is reached k² − k moves or fewer. If N is not triangular, no stable configuration exists and a limit cycle is reached.
References
- Akin, E. & Davis, M.
- Bulgarian Solitaire. In: American Mathematical Monthly 1984 (2); 92: 237-250.
- Bending, T.
- Bulgarian Solitaire. In: Eureka 1990; 50 (April): 12-19.
- Bentz, H.-J.
- Proof of the Bulgarian Solitaire Conjectures. In: Ars Combinatoria 1987; 23: 151-170.
- Etienne, G.
- Tableaux de Young et Solitaire Bulgare. In: Journal of Combinatorial Theory Series A 1991; 58: 181-197.
- Gardner, M.
- Mathematical Games. (a.k.a Bulgarian Solitaire and Other Seemingly Endless Tasks). In: Scientific American 1983; 249: 8-13 or 12-21.
- Griggs, J. R. & Ho, C.-C.
- The Cycling of Partitions and Compositions under Repeated Shifts. In: Advances in Applied Mathematics 1998; 21: 205-227.
- Gwihen, E.
- Tableaux de Young et Solitaire Bulgare. In: Journal of Combinatorial Theory 1991 (2); 58: 181-197.
- Hobby, J. D. & Knuth, D.
- Problem 1: Bulgarian Solitaire. In: A Programming and Problem-Solving Seminar. Department of Computer Science, Stanford University, Stanford (USA) 1983 (December): 6-13.
- Hopkins, B. & Jones, M. A.
- Shift-Induced Dynamical Systems on Partitions and Compositions. In: Electronic Journal of Combinatorics 2006; 13 #R80.
- Hopkins, B. & Sellers, J. A.
- Exact Numeration of Garden of Eden Partitions. In: Integers: Electronic Journal of Combinatorial Number Theory 2007; 7 (2), #A19.
- Igusa, K.
- Proof of the Bulgarian Solitaire Conjecture. In: Mathematical Magazine 1985 (5); 58: 259-271.
- Nicholson, A.
- Bulgarian Solitaire. In: Mathematics Teacher 1993; 86: 84-86.
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By: Ralf Gering
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