05A15-CrazyDice.tex

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\pmtitle{crazy dice}
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\pmtype{Definition}
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\begin{document}
It is a standard exercise in elementary combinatorics to \PMlinkescapetext{calculate} the number of ways of rolling any given value with 2 fair 6-sided dice (by taking the \PMlinkescapetext{sum} of the two rolls).  The below table \PMlinkescapetext{enumerates} the number of such ways of rolling a given value $n$:

\begin{center}
\begin{tabular}{|c|c|}\hline
$n$&\# of ways\\
2&1\\
3&2\\
4&3\\
5&4\\
6&5\\
7&6\\
8&5\\
9&4\\
10&3\\
11&2\\
12&1\\
\end{tabular}
\end{center}

A somewhat (un?)natural question is to ask whether or not there are any other ways of re-labeling the faces of the dice with positive integers that \PMlinkescapetext{generate} these sums with the same frequencies.  The surprising answer to this question is that there does indeed exist such a re-labeling, via the labeling
\begin{align*}
\mbox{Die }1&=\{1,2,2,3,3,4\}\\
\mbox{Die }2&=\{1,3,4,5,6,8\}\\
\end{align*}
and a pair of dice with this labeling are called a set of \emph{crazy dice}.  It is straight-forward to verify that the various possible \PMlinkescapetext{sums} occur with the same frequencies as given by the above table.
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