## The Rotationally Distinct Ways to Label a Die

Posted in Math by pgadey on 2013/07/24

I’m giving a talk at the Canadian Math Camp this year. I’ll be showing the kids of how to count the number of ways to label a six sided die up to the rotational symmetries of the cube. Here is the handout for the talk with questions about dice labellings, the 15-puzzle, and permutation groups.

For the curious the labellings are below the cut. Please note that there are typos in the table below. Alex Fink kindly pointed them out and they will be fixed eventually. For now they are an exercise in keen observation.

$\displaystyle \begin{array}{c|c|c} \cline{2-2} & 3 & \\ \hline \multicolumn{1}{|c|}{5} & 6 & \multicolumn{1}{c|}{2} \\ \hline & 1 & \\ \cline{2-2} & 4 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 4 & \\ \hline \multicolumn{1}{|c|}{5} & 6 & \multicolumn{1}{c|}{2} \\ \hline & 1 & \\ \cline{2-2} & 3 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 2 & \\ \hline \multicolumn{1}{|c|}{5} & 6 & \multicolumn{1}{c|}{3} \\ \hline & 1 & \\ \cline{2-2} & 4 & \\ \cline{2-2} \end{array}$

$\displaystyle \begin{array}{c|c|c} \cline{2-2} & 4 & \\ \hline \multicolumn{1}{|c|}{5} & 6 & \multicolumn{1}{c|}{3} \\ \hline & 1 & \\ \cline{2-2} & 2 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 2 & \\ \hline \multicolumn{1}{|c|}{5} & 6 & \multicolumn{1}{c|}{4} \\ \hline & 1 & \\ \cline{2-2} & 3 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 3 & \\ \hline \multicolumn{1}{|c|}{5} & 6 & \multicolumn{1}{c|}{4} \\ \hline & 1 & \\ \cline{2-2} & 2 & \\ \cline{2-2} \end{array}$

$\displaystyle \begin{array}{c|c|c} \cline{2-2} & 3 & \\ \hline \multicolumn{1}{|c|}{1} & 6 & \multicolumn{1}{c|}{2} \\ \hline & 5 & \\ \cline{2-2} & 4 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 4 & \\ \hline \multicolumn{1}{|c|}{1} & 6 & \multicolumn{1}{c|}{2} \\ \hline & 5 & \\ \cline{2-2} & 3 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 2 & \\ \hline \multicolumn{1}{|c|}{1} & 6 & \multicolumn{1}{c|}{3} \\ \hline & 5 & \\ \cline{2-2} & 4 & \\ \cline{2-2} \end{array}$

$\displaystyle \begin{array}{c|c|c} \cline{2-2} & 4 & \\ \hline \multicolumn{1}{|c|}{1} & 6 & \multicolumn{1}{c|}{3} \\ \hline & 5 & \\ \cline{2-2} & 2 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 2 & \\ \hline \multicolumn{1}{|c|}{1} & 6 & \multicolumn{1}{c|}{4} \\ \hline & 5 & \\ \cline{2-2} & 3 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 3 & \\ \hline \multicolumn{1}{|c|}{1} & 6 & \multicolumn{1}{c|}{4} \\ \hline & 5 & \\ \cline{2-2} & 2 & \\ \cline{2-2} \end{array}$

$\displaystyle \begin{array}{c|c|c} \cline{2-2} & 3 & \\ \hline \multicolumn{1}{|c|}{2} & 6 & \multicolumn{1}{c|}{1} \\ \hline & 5 & \\ \cline{2-2} & 4 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 4 & \\ \hline \multicolumn{1}{|c|}{2} & 6 & \multicolumn{1}{c|}{1} \\ \hline & 5 & \\ \cline{2-2} & 3 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 1 & \\ \hline \multicolumn{1}{|c|}{2} & 6 & \multicolumn{1}{c|}{3} \\ \hline & 5 & \\ \cline{2-2} & 4 & \\ \cline{2-2} \end{array}$

$\displaystyle \begin{array}{c|c|c} \cline{2-2} & 4 & \\ \hline \multicolumn{1}{|c|}{2} & 6 & \multicolumn{1}{c|}{3} \\ \hline & 5 & \\ \cline{2-2} & 1 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 1 & \\ \hline \multicolumn{1}{|c|}{2} & 6 & \multicolumn{1}{c|}{4} \\ \hline & 5 & \\ \cline{2-2} & 3 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 3 & \\ \hline \multicolumn{1}{|c|}{2} & 6 & \multicolumn{1}{c|}{4} \\ \hline & 5 & \\ \cline{2-2} & 1 & \\ \cline{2-2} \end{array}$

$\displaystyle \begin{array}{c|c|c} \cline{2-2} & 2 & \\ \hline \multicolumn{1}{|c|}{3} & 6 & \multicolumn{1}{c|}{1} \\ \hline & 5 & \\ \cline{2-2} & 4 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 4 & \\ \hline \multicolumn{1}{|c|}{3} & 6 & \multicolumn{1}{c|}{1} \\ \hline & 5 & \\ \cline{2-2} & 2 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 1 & \\ \hline \multicolumn{1}{|c|}{3} & 6 & \multicolumn{1}{c|}{2} \\ \hline & 5 & \\ \cline{2-2} & 4 & \\ \cline{2-2} \end{array}$

$\displaystyle \begin{array}{c|c|c} \cline{2-2} & 4 & \\ \hline \multicolumn{1}{|c|}{3} & 6 & \multicolumn{1}{c|}{2} \\ \hline & 5 & \\ \cline{2-2} & 1 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 1 & \\ \hline \multicolumn{1}{|c|}{3} & 6 & \multicolumn{1}{c|}{4} \\ \hline & 5 & \\ \cline{2-2} & 2 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 2 & \\ \hline \multicolumn{1}{|c|}{3} & 6 & \multicolumn{1}{c|}{4} \\ \hline & 5 & \\ \cline{2-2} & 1 & \\ \cline{2-2} \end{array}$

$\displaystyle \begin{array}{c|c|c} \cline{2-2} & 2 & \\ \hline \multicolumn{1}{|c|}{4} & 6 & \multicolumn{1}{c|}{1} \\ \hline & 5 & \\ \cline{2-2} & 3 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 3 & \\ \hline \multicolumn{1}{|c|}{4} & 6 & \multicolumn{1}{c|}{1} \\ \hline & 5 & \\ \cline{2-2} & 2 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 1 & \\ \hline \multicolumn{1}{|c|}{4} & 6 & \multicolumn{1}{c|}{2} \\ \hline & 5 & \\ \cline{2-2} & 3 & \\ \cline{2-2} \end{array}$

$\displaystyle \begin{array}{c|c|c} \cline{2-2} & 3 & \\ \hline \multicolumn{1}{|c|}{4} & 6 & \multicolumn{1}{c|}{2} \\ \hline & 5 & \\ \cline{2-2} & 1 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 1 & \\ \hline \multicolumn{1}{|c|}{4} & 6 & \multicolumn{1}{c|}{3} \\ \hline & 5 & \\ \cline{2-2} & 2 & \\ \cline{2-2} \end{array} \quad \quad \quad \begin{array}{c|c|c} \cline{2-2} & 2 & \\ \hline \multicolumn{1}{|c|}{4} & 6 & \multicolumn{1}{c|}{3} \\ \hline & 5 & \\ \cline{2-2} & 1 & \\ \cline{2-2} \end{array}$

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