Mancala World

Mancala games which are using the move mechanism of reverse sowing (or: reaping, inverse sowing) instead of regular sowing are called "reverse mancala". If applied to existing games, reverse sowing is a meta-rule, which defines according to Jeff Erickson a peculiar sub-group of "sowing games" (in French mancala games are also called jeux de semailles, literally "sowing games"). This term was proposed by John Conway and Richard Guy as an alternative for "mancala" (the generic term for our games in social studies) in Combinatorial Game Theory (CGT).

Sowing can be defined as distributing the contents of a pit into the following pits, usually one by one. In reverse sowing the seeds are picked up, usually one from each pit, and then put into another pit (usually an empty one). If the direction of the move is changed as well, it's like travelling into the past with a time machine.


Single Lap Sowing





Direction of move from left to right.





Direction of move from right to left.

Reverse Sowing





Direction of move from right to left.





Direction of move from left to right.

Mathematically these two move mechanisms are related like positive and negative numbers, physically like matter and antimatter.

Short History of Reverse Sowing

The oldest game based on reverse sowing was introduced in 1980 by an anonymous train passenger to the mathematical community. It is called Bulgarian Solitaire. Also known as Deterministic Bulgarian Solitaire, it has become a favorite pastime of mathematicians and you'll find hundreds of pages about this game on the web.

Other solitaire variants include Austrian Solitaire (Ethan Akin & Morton David Davis (USA), 1985), Montreal Solitaire (Chris Cannings & John Haigh (England), 1992), Carolina Solitaire (Andrej Andreev (Bulgaria), 1997) and Random Bulgarian Solitaire (Serguei Popov (Brazil), 2003). Two-person games have also been suggested such as Reaping (Jeff Erickson (USA), 1996), an unnamed game by David Eppstein (David Eppstein (USA), 1999), and Two-handed Bulgarian Solitaire (Tim Bancroft (USA), 2004).

The mathematical solution for Stones in Cups involves reverse sowing.

Reversing Moves in "Normal" Mancala Games

Reverse sowing is used by experienced mancala players in Oware and Toguz Kumalak to find a move that would effect a capture. In your mind, you first travel into the future by looking at the pit, where you want to end the move, not yet knowing if such a move really exists. Then you use the "time machine" again, this time for slowly travelling back into the present by counting back (= undoing the imaginative move) until you find a hole on your side of the board that exactly has the number of seeds needed to turn this science fiction into reality.

Alexander Johan de Voogt conducted experiments in Rejesha Bao ("to return bao") to obtain insight into the influence of direction changes upon the calculating abilities of Bao masters. All masters perfomed well on this task, which can prove very complicated, and one master, Abdulrahim Muhiddin Foum, could even undo moves blind and simultaneously.

See also



Akin, A. & Davis, M. D. 
Bulgarian Solitaire. In: American Mathematical Monthly 1985; 92 (4): 237-250.
Cannings, C. & Haigh, J. 
Montreal Solitaire. In: Journal of Combinatorial Theory Series A 1992; 60 (1): 50-66.
Dorée, S. 
Bulgarian Solitaire Bibliography. Minneapolis (USA) 2005.
Erickson, J. 
Sowing Games. In: Nowakowski, R. J. (Ed.). Games of No Chance. Mathematical Sciences Research Institute Publications 29. Cambridge University Press, Cambridge (England) 1996, 287-297.
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.
Popov, S. 
Random Bulgarian Solitaire. Sao Paulo (Brazil) 2003.
Voogt, A. J. de
The Blind Bao Experiment. In: De Voogt, A.J. Limits of the Mind: Towards a Characterization of Bao Mastership. CNWS Publications, Thesis Rijksuniversiteit Leiden (Netherlands) 1995, 96-100.


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By: Ralf Gering.
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