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

Puzzles and Curious Problems by Henry Ernest Dudeney: $287$

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:

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:

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

Also see

• Fifteen Schoolgirls Puzzle, of which this is a relative

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $287$. -- The Handcuffed Prisoners
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $464$. The Handcuffed Prisoners