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| | Plato mathematical game |
 | | Let me define a 'game sequence' i_1, i_2,..., i_n as a game the lady plays in which she fist makes a move on pot i_1, then on i_2, and so on until after she makes a move on pot i_n, at which point she finishes with at most one bean in each pot. |
 | | This modified second game is still a valid game, since the only thing that's not the same as before was that we performed j_k' earlier, and we know that that pot had at least two beans in it, so we're alright. |
 | | Inductively apply this procedure now, and before long we will have a valid game which is a permutation of B which has the first n moves in common with game A. But game A ended after those very n moves, so there are no more legal moves. |
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