Henry Ernest Dudeney/Puzzles and Curious Problems/314 - Card Shuffling/Solution

by : $314$

 * Card Shuffling

Solution

 * $14$ shuffles are needed for $14$ cards.

Proof
Each shuffle can be represented as a permutation in $S_{14}$, the symmetric group on $14$ letters.

In two-row notation, this can be written as:


 * $\begin {pmatrix}

1 & 2  & 3  & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\ 14 & 12 & 10 & 8 & 6 & 4 & 2 & 1 & 3 & 5  & 7  & 9  & 11 & 13 \end {pmatrix}$

which, when expressed in cycle notation, would be:
 * $\begin {pmatrix} 1 & 14 & 13 & 11 & 7 & 2 & 12 & 9 & 3 & 10 & 5 & 6 & 4 & 8 \end {pmatrix}$

which is a single (disjoint) $14$-cycle.

Thus this permutation has order $14$.

This means that $14$ shuffles are required to get the cards back to their original arrangement.