Mancala World
Mancala Solitaire
Other Names: Mancala-game
Inventor: Roland Schröder,
Variant of Stones in Cups
Ranks: One
Sowing: Single laps
Region: Germany

Mancala Solitaire, also known as the Mancala-game in the On-Line Encyclopedia of Integer Sequences™ (OEIS™), was invented in 2005 by OStR a.D. Roland Schröder, a retired math teacher in Celle (Germany). He was inspired by Klaus Hasemann's game Türme Bauen (i.e. "Building Towers"), a combinatorial math problem for 1st and 2nd graders. Research on the game's mathematical properties demonstrated that the lower Wythoff sequence can be constructed by playing it. Results were published in the OEIS™. Knott and Schröder also formulated in 2010 two assumptions for a game played on an infinite board.

Knott's assumption:

"After move n – 1, the hole #n contains rounded down n ∙Phi.jpg stones with Golden ratio-1.jpg, the only positive solution of the golden ratio in the set of algebraic irrational numbers."

Schröder's assumption:

"If only the first n holes are filled and the following ones are empty, the order of contents will be exactly mirrored (and, of course, moved to the right) after rounded down n - Phi.jpg moves."

The game inspired Rüdeger Baumann to create a close variant called Montenegrinisches Mancala in 2010.


Basic Rules

The game is played by one person on a board of n holes arranged in one row.

The first hole (starting from the leftmost one) contains one stone, the second hole two stones, the third hole three stones and so on.

A move consists of distributing the counters of the leftmost occupied hole, one by one, to the right into the ensuing holes.

The object of the game is to discover mathematical laws.

Special Rules

The game can be played starting from different set-ups.

Roland Schröder gave the following positions:

  • A finite board with all holes initially occupied. The last legal sowing finishes in the rightmost hole.
  • A finite board with only the first holes occupied, the following ones being empty. Again the game stops, when the last stone of a sowing falls into the rightmost hole.
  • An infinite board with all holes being occupied.


Infinite Set-up

External Links


Hasemann, K., Leonhardt, U. & Szambien, H.
Denkaufgaben für die 1. und 2. Klasse. Cornelsen Verlag, Berlin (Germany) 2006.


"Actually it isn't a game, but the result of a playful activity. Of course, you can change the problem as you want and each time analyze its mathematical laws. Then it might be possible to call it a game. More precisely a playful matrix for mathematical discoveries."

Roland Schröder (2011)


© Ralf Gering
Under the CC by-sa 2.5 license.