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    "For example: 4 can be divided into [3], [1, 2] and [1, 1] games."
    What about [2] ? IIII -> IIXX

    Can you please explain this idea of xor'ing results of sub-games? I'm not sure about it.

    Btw, how do you deal with duplicate sub-games? Fro example, [3,3] can became both [2,3] and [3,2]. and if they are win-states, their xor will be 0 as if both of them be fail-states. What's the poin of xoring here?