Table of Contents

The Span Of A Simple Shift

Problem

Suppose that our alphabet consists of only 12 letters . Let be the set of simple shift permutations on this 12 letter alphabet (we wrap around from to ).

  1. For each , list the elements in .
  2. For which does .
  3. Make a conjecture about any patterns you see above and their relation to the size (12) of the alphabet. When you report your preparation, please tell me your conjecture.
  4. Whenever you make a conjecture, you should always test your conjecture on an example you have not yet considered. Pick another integer , and look at the set of simple shift permutations of an alphabet consisting of letters. Then check if your conjecture holds.

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Remarks

$\LaTeX$ version

problem.the_span_of_a_simple_shift.tex
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\begin{problem}
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 $a$). 
\begin{enumerate}
\item For each $n\in \{0,1,2,\ldots,11\}$, list the elements in $\text{span}(\{\phi_n\} ) $.
\item For which $n$ does $\text{span}(\{\phi_n\})=H_{12}$. 
\item Make a conjecture about any patterns you see above and their relation to the size (12) of the alphabet. When you report your preparation, please tell me your conjecture.
\item 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. 
\end{enumerate}
\end{problem}