Mancala Games

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Mancala Games

• Jeroen Donkers
• 18 october 20000

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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

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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

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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.

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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

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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

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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

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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

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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)

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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

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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

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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!

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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

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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

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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

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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)

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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)

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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
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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

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