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Conclusion

In conclusion, then, the experiment was a success.  We were able to develop a computer program to solve the game tree through the backward-induction path, and we identified most of the opening moves as being backward-induction winners or losers.

We did not find a backward-induction sure-win strategy that resulted in such a quick, clear and decisive win that memorizing a series of plays several levels deep into the game would yield a practical way of consistently winning without having to apply soft strategy.  Thus, the relevance to a real player of our proof of mancala’s triviality is small.  In this regard, the triviality of mancala is more akin to that of chess than to that of tic-tac-toe.

Somewhat surprisingly, we discovered a high correlation between look-ahead strategies on the one hand, and both soft strategies and backward induction strategies on the other.  The higher the advantage shown in a look-ahead strategy, the more highly it seems to correlate with backward induction wins.  The closer the look-ahead advantage is to zero, the more difficult that move was to solve with our backward-induction solver.  The greater the depth of the look-ahead, the higher the correlation it seemed to have with backward induction results.  Also, look-ahead advantage appears, at least from one data point, to be not only persistent but also cumulative.

Finally, we were pleased to discover that soft strategies which seem to make sense according to defined principles and observed patterns really do prove, in many cases, to be good plays by backward induction.  Without some such correlation, mancala would be a game that did not reward skill, insight, depth of thinking, or strategy.  If this were the case, one would not expect the game to have survived the ages.  Likewise, if some opening strategy, such as our CF-?-[A/E] play, proved to be too easy a sure win, one would also not expect the game to have survived the test of time.  Consistent with our findings, mancala seems to be “just hard enough” to keep it interesting and practically non-trivial, but “just easy enough” that strategy, insight, and concentration are not easily overridden by randomness and luck.  Our verdict is that, despite our findings, mancala will continue to survive.

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