Solitaire Mancala Games and the Chinese Remainder Theorem

TL;DR

This paper analyzes the solitaire mancala game Tchoukaillon, establishing a connection with the Chinese Remainder Theorem, and introduces algorithms for reconstructing winning boards and a graph-theoretic generalization.

Contribution

It provides a new mathematical framework linking Tchoukaillon to the Chinese Remainder Theorem and offers algorithms for board reconstruction and a novel graph-theoretic extension.

Findings

Characterization of all winning Tchoukaillon boards of a given length

An analogue of the Chinese Remainder Theorem for Tchoukaillon boards

An algorithm to reconstruct boards from partial information

Abstract

Mancala is a generic name for a family of sowing games that are popular all over the world. There are many two-player mancala games in which a player may move again if their move ends in their own store. In this work, we study a simple solitaire mancala game called Tchoukaillon that facilitates the analysis of "sweep" moves, in which all of the stones on a portion of the board can be collected into the store. We include a self-contained account of prior research on Tchoukaillon, as well as a new description of all winning Tchoukaillon boards with a given length. We also prove an analogue of the Chinese Remainder Theorem for Tchoukaillon boards, and give an algorithm to reconstruct a complete winning Tchoukaillon board from partial information. Finally, we propose a graph-theoretic generalization of Tchoukaillon for further study.