Reverse sowing
From Wikimanqala
Some mancala-like games are using reverse sowing (or: reaping, inverse sowing) instead of regular sowing. If applied to existing games, reverse sowing is a metarule, 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.
Contents |
Examples
Single lap sowing
|
| before |
|
| after |
Direction of move from left to right.
|
| before |
|
| after |
Direction of move from right to left.
Reverse sowing
|
| before |
|
| after |
Direction of move from right to left.
|
| before |
|
| after |
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
In America the oldest game based on reverse sowing was introduced by Martin Gardner in 1983. He called his game 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. The Bulgarian Andrey Andreev reported in 1998 that Gardner's game is a generalized version of a traditional game played in Bulgaria. Gardner's game has an unordered set of pits, while the game in Bulgaria uses an ordered topology like mancala games.
The original game from Bulgaria has been described as Carolina solitaire in the USA. Other solitaire variants include Austrian solitaire (Ethan Akin & Morton David Davis (USA), 1985), Montreal solitaire (Chris Cannings & John Haigh (England), 1992) 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 D. 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 general mancala games
Reverse sowing is used by experienced mancala games players, e.g. in oware or toğız qumalaq, 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 that you are searching, 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 in your area that exactly has the number of seeds needed to turn this science fiction into reality.
Alexander J. 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.
References
- Akin, A. & Davis, M. D.
- (1985) 'Bulgarian Solitaire', in American Mathematical Monthly; 92 (4). Pages 237-250.
- Cannings, C. & Haigh, J.
- (1992) 'Montreal Solitaire', in Journal of Combinatorial Theory Series A; 60 (1). Pages 50-66.
- De Voogt, A. J.
- (1995) 'The Blind Bao Experiment', in De Voogt, A.J. Limits of the Mind: Towards a Characterization of Bao Mastership, Leiden: CNWS Publications. Pages 96-100.
- Dorée, S.
- (2005) Bulgarian Solitaire Bibliography. [Pdf document]
- Erickson, J.
- (1996) 'Sowing Games', in Nowakowski, R. J. (Hg.). Games of No Chance. Mathematical Sciences Research Institute Publications 29. Cambridge: Cambridge University Press. Pages 287-297.
- Gardner, M.
- (1983) 'Mathematical Games. (a.k.a Bulgarian Solitaire and Other Seemingly Endless Tasks)', in Scientific American; 249. Pages 8-13 or 12-21.
- Griggs, J. R. & Ho, C.-C.
- (1998) 'The Cycling of Partitions and Compositions under Repeated Shifts', in Advances in Applied Mathematics; 21. Pages 205-227.
- Popov, S.
- (2003) Random Bulgarian Solitaire. [Pdf document]
