Mancala Games

1. Mancala Games Jeroen Donkers 18 october 20000

2. Mancala Games • Played on a "board" with some rows of pits • By (mostly) two players • With a collection of equal counters • Players own rows, not counters • Sometimes additional pits (stores) are used

3. Mancala Games • Moves are made by "sowing" (demo) which is a type of counting • After sowing, a "capture" might take place (demo) • hence: count - and - capture games • The game is over if one player has captured the most counters

4. Literature • Murray, H.J.R., 1952. A history of board games other than chess. London: Oxford at the Clarendon Press. • Russ, L., 2000. The complete Mancala games book. New York: Marlowe & company.

5. Mancala Games: Their Age • Mancala games are very old. • Mostly, boards are made of wood or just in the soil, which do not last very long • Rows of pits are found cut out of rocks near the pyramid of Gizeh • "Neolithic" boards are found in Africa

6. Mancala Games: Their Origin • Possibly, mancala games date back to the early civilisations in the middle east • Counting became important in early agricultural societies • Counting larger amounts has some type of unpredictibality or "magic" (iene-miene-mutte) • "Magic counting" can be used in clairvoyance • This could lead to the game of Mancala

7. Mancala Games: Spreading • The game is traditionally known all over Africa and most of Asia • It has been brought by slaves to the Americas • There is a medieval european version (Trysse) • In modern times, the game became popular in the United States and Europe (Kalah, Awari) • Computerized versions are widely available

8. Variation • There exist hunderds of variants of Mancala with differences in: • board size and shape (rows, pits, stores) • number of players (1 - 4) • initial position (number of counters) • special counters • sowing procedure and direction • capture rules • end-of-game rules, multiple rounds, special rules

9. Mathematics of Mancala • A mancala game seems very simple: • it is deterministic, has perfect information, there are not many choices per move • But it is difficult to predict if a position leads to a win or a loss • Only some special situations can be solved mathematically (by reasoning)

10. Tchoukatlion Positions • Tchoukatlion is an artificial solitair variant, related to Tchuka Ruma (Broline & Loeb) • collect all your counters in the store, but only moves that end in the store are allowed. • The positions that can be solved are unique and can be constructed using a simple algorithm

11. Tchoukatlion Positions 1: 0 0 0 0 0 1 2: 0 0 0 0 2 0 3: 0 0 0 0 2 1 4: 0 0 0 3 1 0 5: 0 0 0 3 1 1 6: 0 0 4 2 0 0 7: 0 0 4 2 0 1 8: 0 0 4 2 2 0 9: 0 0 4 2 2 1 10: 0 5 3 1 1 0 11: 0 5 3 1 1 1 12: 6 4 2 0 0 0 • These positions play a role in many mancala games (Tchuka Ruma, Awari, Bao, Dakon) • Recognizing them can be important

12. Tchuka Ruma • Solitair mancala game from South-east asia. • Goal: collect all counters in the store. • Sowings appear in laps. If you end in an empty pit, you loose. • Losses can sometimes be proven, wins never. • If there are n pits, then you certainly loose when you start with kn counters per pit. • Try Tchuka Ruma on the computer!

13. Bao • Two-player game from Africa, two rows of pits, no stores, variable number of pits and stones per pit. Sowings in laps. African style capture rule. • In Bao infinite sowings can happen that have quite large periods. • This is probably also the case in more mancala games without stores. • Uiterwijk

14. Dakon • Two-player game from SE Asia, two rows of pits, two stores, n pits per row and n stones per pit. Sowings in laps. Multiple moves per turn. Asian style capture rule. • There are winning openings in Dakon. • The beginning player can capture so many stones in the first turn that the opponent cannot play anymore. • Cure: both players start simultaneously

15. Dakon • These winning openings are very large, (93 moves for Dakon-8) but people discovered them by hand. • For a computer this is an easy task. • We tried to emulate man-found solutions: the players tend to prefer moves that have limited effect (small laps). • Donkers, de Voogt, Uiterwijk Board Game Studies

16. Mancala in AI • Two mancala games have a long history in AI: Kalah and Awari. • Kalah has recently been solved, Awari is expected to be solved in the next two years. • Standard game search techniques can be used, but end-game databases can be created more efficiently than in many other games. (van der Meulen, Lithidion)

17. Awari • African 2-player game, 2 rows, 6 pits, 4 counters per pit, no stores. African style capture (2-3 counter rule), single-lap sowings. • Played at the Computer Olympiad. • Very large end-game database are being comstructred (35 stones) (Lincke, ICGA)

18. Kalah • Modern 2-player game. 2 rows of pits, 6 pits per row, 3-6 stones per pit, 2 stores. Asian style capture, single-lap sowings. stones (Irving, Donkers, Uiterwijk, ICGA) pits

19. Future Research • General principles of mancala games • effects of rules on complexity • heuristic evaluation • find more mathematical facts • model / investigate human play • handle special rules and uncertainty • robot play