Parker Glynn-Adey

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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One Response

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  1. Alex Fink said, on 2013/08/02 at 16:09

    You’ve missed the ways to have 5 opposite 6, and duplicated the ones with a counterclockwise (6,5,1) corner instead!


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