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Problem 23 (Permutation Scoring On S 3)
Consider the game of permutation scoring on the set $S_3$, where $S_3$ represents the set of all permutations of $X=\{1,2,3\}$. We already know that there are 6 elements in $S_3$.
- Play the game of permutation scoring on $S_3$. Play it a few times.
- Can player 1 win by choosing a single permutation in $S_3$?
- Does player $1$ have a winning strategy? What is it?
- Pick an element in $S_3$ other than the identity, and call it $\sigma$. Let $S=\text{span}(\sigma)$. What is the span of $S$?
- Repeat part 4 with a different element of $S_3$. You can use Sage to make this part fast (first compute the span of $\sigma$ with Sage. The list below the graph gives you the elements of the span. Then create a graph using these elements.
- Make a conjecture.
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