Is Mancala Solved?

Yes. The standard mancala game (Kalah: six pits a side, four stones each) was solved in 2000 by Geoffrey Irving, Jeroen Donkers, and Jos Uiterwijk, who proved the first player wins with perfect play. The margin depends on one capture rule: 29-19 under the rule they used, 28-20 under the stricter standard rule this site plays.

So if both players make the best possible move every single turn, whoever moves first wins. Nobody plays like that, which is why the proof matters less at your kitchen table than the headline suggests. It also comes with a rule detail that most summaries leave out.

What “solved” actually means

Game researchers use three levels. A game is ultra-weakly solved when someone proves who wins from the starting position, without necessarily showing how. It’s weakly solved when there’s a known strategy that gets that result from the start against any defense. It’s strongly solved when the best move is known from every position that can occur, including positions after someone has already blundered.

Kalah sits at different levels depending on the setup. The 2000 paper strongly solved small boards by building complete databases of every reachable position, and weakly solved the full-size game with an optimized search that proved the result from the opening. For your purposes the distinction is small: the standard game’s outcome under perfect play is known, and so is a perfect line.

The 2000 Kalah proof

The paper is “Solving Kalah,” published in the ICGA Journal (volume 23, issue 3, September 2000). Irving was a student at Caltech; Donkers and Uiterwijk were at Universiteit Maastricht in the Netherlands. They solved Kalah for boards up to six pits per side; on the full six-pit board, up to five stones per pit.

The headline results for the six-pit board:

The computing numbers are almost funny now. On an 800 MHz AMD Athlon with 256 MB of RAM, the four-stone game took 35 seconds to prove and the five-stone game took 4.7 hours. The six-stone game beat them: after 4.2 days, their program still hadn’t finished its first full-depth pass.

That gap closed in 2011, when Anders Carstensen proved in a master’s thesis at the University of Southern Denmark (supervised by Kim Skak Larsen) that the first player also wins six-stone Kalah.

The capture rule that changes the margin

The 2000 paper used a capture rule where your last stone landing in an empty pit on your side always captures, even when the pit opposite is empty (you bank your own lone stone). Kalah analysts call that the “empty capture” variant.

The version usually called standard Kalah, and the one the game on this site uses, is stricter: no capture unless the opposite pit holds stones. (The capture rules page covers that house-rule split.) Mark Rawlings of Gaithersburg, Maryland, wrote an analysis program that handles both versions, and in 2015 he published the value of every opening move for the four- and five-stone games under each rule:

Same verdict, different margin. If someone tells you perfect mancala ends 29-19, they’re right about the rules that paper used and two stones off for the version you probably play.

Which opening does perfect play choose?

The third pit’s four stones land exactly in your store, so your first turn can go ten different ways: five single moves, plus the third pit followed by one of its five follow-ups. Rawlings’ tables give the perfect-play value of all ten under the standard rule:

• Third pit, then sixth pit: win by 8. This is the best first turn.
• Third pit, then fifth pit: win by 4.
• Sixth pit alone: win by 4.
• Third pit, then fourth pit: win by 2.
• Third pit, then second pit: tie.
• Any other first turn: a loss. Opening with the first pit loses by 14.

Only four of the ten first turns keep the win, so the famous third-pit opener isn’t enough by itself: follow it with your first pit and a perfect opponent beats you, or with your second pit and the best you get is a tie. And one choice on move one is worth a 22-stone swing, from winning by 8 to losing by 14. The first move guide covers the opener in more depth. The perfect game printed in the 2000 paper (under its own capture rule) opens with the same two moves: third pit, then sixth.

Which other mancala games are solved?

Oware (awari): solved, and it’s a draw. John Romein and Henri Bal of the Vrije Universiteit Amsterdam strongly solved awari in 2002, computing the best move and outcome for all 889,063,398,406 positions that can occur. With perfect play, either side can force a draw. They ran it on a cluster of 72 dual-processor machines for 51 hours and reported it in the ICGA Journal as “Awari is Solved.” Oware’s rules vary by region, so the result applies to the rule set they analyzed. More on the game itself at oware.

Dakon and ohvalhu: first-player wins. The 2000 Kalah paper notes that Dakon has winning openings that end the game on the very first turn, found by Donkers, Alex de Voogt, and Uiterwijk. Mancala World’s list of solved games records the Maldivian game ohvalhu as a first-player win.

Bao, omweso, sungka, congkak: not solved. None appear on Mancala World’s list of solved mancala games. The four-row games like bao have much bigger boards and turns that relay through occupied pits, and games with simultaneous openings, like sungka and congkak, don’t fit the turn-by-turn search that cracked Kalah; the 2000 paper flags simultaneous moves as one of the rules that puts a mancala game outside that kind of analysis.

Is Avalanche mode solved?

We couldn’t find any published solution for Avalanche, the chain-sowing mode from GamePigeon. The practical answer is blunter than for Kalah, though.

In 2017, Anoush Najarian showed on a MathWorks blog that under relay-sowing rules close to Avalanche’s, the first player can take all 48 stones on the opening turn. We checked the question against the exact rules this site plays by running every possible first turn through our game engine. There are 26,286 distinct ways the first turn can play out (each store landing gives you another move, so turns branch). In 20,595 of them, the first player banks 25 or more stones, which is already a guaranteed win because the opponent can’t get past 23. The best first turn banks 43.

That’s our own computation, not a peer-reviewed paper. It does mean Avalanche is far more lopsided than Kalah: a player who has memorized a winning chain can decide the game before you touch a stone. The capture vs Avalanche comparison covers what that means for choosing a mode.

Why being solved doesn’t ruin the game

The proof tells you the result when nobody makes a mistake. Humans make mistakes constantly. The 2000 paper estimated standard Kalah’s game tree at around 6 x 10^18 lines of play, in the same class as Connect Four. You can’t memorize that; you can only count well.

The proof also shows how little room for error there is. That 22-stone swing on move one shows how sharply a single choice changes a perfect game, and the same is true all game long: one miscounted pit hands a won position to your opponent. In practice, the solved result gives you two useful rules and not much else. Going first is a real edge, so alternate who goes first in a series. And the opening has a known best line, which you now know.

Everything after that is the actual game. The strategy guide covers extra-turn chains, hoarding, and endgame counting. To check a real position, the mancala calculator ranks every move available to you in Capture or Avalanche mode.