The game of 31 was a favourite of con artists who rode the railroads in days of yore. The game is played with a deck of 24 cards: four labelled each of 1, 2, 3, 4, 5, 6. (That is, there are four cards labelled '1', four cards labelled '2', and so on.) Initially all of the cards are spread, face up, on a table and the "discard pile" is empty. The players then take turns. During each turn, a player picks up one unused card from the table and lays it on the discard pile. The object of the game is to be the last player to lay a card such that the sum of the cards in the pile does not exceed 31. Your task is to determine the eventual winner of a partially played game, assuming each player plays the remainder of the game using a perfect strategy.
For example, in the following game player B wins:
Player A plays 3 Player B plays 5 Player A plays 6 Player B plays 6 Player A plays 5 Player B plays 6
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B B A A A