Henry Ernest Dudeney/Puzzles and Curious Problems/153 - A Military Puzzle/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $153$

A Military Puzzle

An officer wished to form his men into $12$ rows, with $11$ men in every row,

so that he could place himself at a point that would be equidistant from every row.

"But there are only one hundred and twenty of us, sir," said one of the men.

Was it possible to carry out the order?

Solution

Place the men along the sides of a regular dodecahedron with the officer in its centre.

You will find that the men at the vertices of this dodecahedron will be in two rows of $11$ men at the same time.

Hence $120$ men are needed for this arrangement.

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $153$. -- A Military Puzzle
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $266$. A Military Puzzle