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problem:simple_shift_repetition_game
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problem:simple_shift_repetition_game [2013/11/21 11:18] tarafife created |
problem:simple_shift_repetition_game [2013/11/21 11:18] (current) tarafife |
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| ==== Problem ==== | ==== Problem ==== | ||
| Consider the set of simple shift permutations $H=\{\phi_n\mid n\in \mathbb{Z}\}$ on a 26 letter alphabet. We've shown that there are 26 different functions in this set. Consider the following game. | Consider the set of simple shift permutations $H=\{\phi_n\mid n\in \mathbb{Z}\}$ on a 26 letter alphabet. We've shown that there are 26 different functions in this set. Consider the following game. | ||
| - | * The first player picks an element $\sigma_1\in H$. They then remove from $H$ the span of $\sigma_1$ (so everything generated by $\{\sigma_1\}$). | + | |
| - | * The second player now chooses an element $\sigma_2\in H$ that hasn't yet been removed. They then remove from $H$ any element in the span of $\{\sigma_1, | + | * The second player now chooses an element $\sigma_2\in H$ that hasn't yet been removed. They then remove from $H$ any element in the span of $\{\sigma_1, |
| - | * Players alternate taking turns, choosing an element $\sigma_k\in H$ that hasn't been removed in a previous stage, and then removing from $H$ any element in the span of $\{\sigma_1, | + | * Players alternate taking turns, choosing an element $\sigma_k\in H$ that hasn't been removed in a previous stage, and then removing from $H$ any element in the span of $\{\sigma_1, |
| - | * Whoever take the last element of $H$ wins. The game can also be played as a misere game. | + | * Whoever take the last element of $H$ wins. The game can also be played as a misere game. |
| Answer the following questions. | Answer the following questions. | ||
| - Player 1 takes $\phi_6$. | - Player 1 takes $\phi_6$. | ||
problem/simple_shift_repetition_game.1385050689.txt.gz · Last modified: 2013/11/21 11:18 by tarafife
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