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class Solution:
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    def validTicTacToe(self, board: List[str]) -> bool:
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        # The two criteria for a valid board are:
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        #   1) num of Xs - num of Os is 0 or 1
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        #   2) X is not a winner if the # of moves is even, and
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        #      O is not a winner if the # of moves is odd.
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        d = {"X": 1, "O": -1, " ": 0}  # transform the 1x3 str array to a 1x9 int array
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        s = [
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            d[ch] for ch in "".join(board)
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        ]  # Ex: ["XOX"," X ","   "] --> [1,-1,1,0,1,0,0,0,0]
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        sm = sum(s)
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        if sm >> 1:
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            return False  # <-- criterion 1
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        n = -3 if sm == 1 else 3  # <-- criterion 2.
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        if n in {
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            s[0] + s[1] + s[2],
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            s[3] + s[4] + s[5],
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            s[6] + s[7] + s[8],
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            s[0] + s[3] + s[6],
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            s[1] + s[4] + s[7],
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            s[2] + s[5] + s[8],  # the elements of the set are
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            s[0] + s[4] + s[8],
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            s[2] + s[4] + s[6],
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        }:
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            return False  # the rows, cols, and diags
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        return True  # <-- both criteria are true