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Problem 14(The Span Of A Simple Shift)
Suppose that our alphabet $S$ consists of only 12 letters $S=\{a,b,c,d,e,f,g,h,i,j,k,l\}$. Let $H_{12}=\{\phi_n\mid n\in \mathbb{Z}\}$ be the set of simple shift permutations on this 12 letter alphabet (we wrap around from $l$ to $a$).
- For each $n\in \{0,1,2,\ldots,11\}$, list the elements in $\text{span}(\{\phi_n\} ) $.
- For which $n$ does $\text{span}(\{\phi_n\})=H_{12}$.
- Make a conjecture about any patterns you see above and their relation to the size (12) of the alphabet.
- Whenever you make a conjecture, you should always test your conjecture on an example you have not yet considered. Pick another integer $k\neq 12,26$, and look at the set $H_k$ of simple shift permutations of an alphabet consisting of $k$ letters. Then check if your conjecture holds.
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- Problem.TheSpanOfASimpleShift
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- Schedule.AllProblems
- Solution.TheSpanOfASimpleShiftJason