New Kalah study shows how hard the game can get on larger boards

14th November 2026, 16:55 CET

A new mathematical result about Kalah asks what happens when the board can have more than the familiar six small pits per player. In a preprint posted on 12 November, Ethan Dickey and Ethan Mader prove that deciding whether a player can force a win from a supplied position becomes a demanding computational problem when the width of the board can vary.

The researchers examine positions with stones distributed unevenly across the pits. The question is whether one player has a plan that ends with more stones in their store, whatever the opponent does. A draw does not count as a win. Read the preprint.

At a glance

• The result concerns boards of varying width and positions supplied to the solver.
• The proof covers two common capture rules, each with and without extra turns.
• Earlier computer analyses solved particular openings on standard-width boards. The new paper asks how the problem scales.

What changes when the board grows?

On a regular Kalah board, each player has six small pits and a larger store. A turn starts by picking up all the stones from one of your pits and dropping them one at a time around the board. Where the final stone lands can earn another turn or let you capture stones. Playiro’s Mancala rules walk through the standard four-stone setup and show how captures work.

The study keeps the two rows and stores but lets the number of pits vary. The proof works even when both stores start empty. One ingredient is familiar from the tabletop game. Adding a stone to a pile shifts the place where that pile’s next sowing move will finish, which can change what may be captured later.

What the result says about solving Kalah

The researchers classify the decision problem as PSPACE-complete when stone counts are written with one mark per stone. That puts it among the hardest problems that can be solved with a quantity of working memory bounded by a polynomial in the size of the input. The classification describes how difficult the hardest positions can be as the input grows. Individual positions can still have an obvious answer.

Earlier programs analysed particular standard-width openings, including the familiar six-pit board with four stones per pit. Those analyses start from an exact layout. Dickey and Mader’s proof applies to supplied layouts, and the paper leaves open how the difficulty scales if every pit must start with the same number of stones. MathWorld’s overview of Kalah