Henry Ernest Dudeney/Puzzles and Curious Problems/287 - The Handcuffed Prisoners/Solution

by : $287$

 * The Handcuffed Prisoners
 * Nine dangerous convicts needed to be guarded.
 * Every day except Sunday they were taken out for exercise, handcuffed together in groups of three, as in the diagram:


 * Dudeney-Puzzles-and-Curious-Problems-287.png


 * On no day in any one week were the same two men to be handcuffed together.
 * If will be seen how they were sent out on Monday.
 * Can you arrange the nine men in triplets for the remaining $5$ days?


 * It will be seen that No. $1$ cannot be handcuffed to No. $2$ again, but $1$ and $3$ can subsequently be so.

Solution
The following is a solution.

Each block of three is an arrangement of the prisoners such that each prisoner will have been handcuffed to every other prisoner once and only once:


 * $\begin{array} \\ 1-2-3 \\ 4-5-6 \\ 7-8-9 \end{array}

\qquad \begin{array} \\ 2-6-8 \\ 5-9-1 \\ 3-7-4 \end{array} \qquad \begin{array} \\ 6-1-7 \\ 9-4-2 \\ 8-3-5 \end{array} \qquad \begin{array} \\ 1-4-8 \\ 2-5-7 \\ 6-9-3 \end{array} \qquad \begin{array} \\ 7-2-9 \\ 3-6-4 \\ 8-1-5 \end{array} \qquad \begin{array} \\ 4-3-1 \\ 5-8-2 \\ 9-7-6 \end{array}$

Also see

 * Fifteen Schoolgirls Puzzle, of which this is a relative