Henry Ernest Dudeney/Modern Puzzles/136 - A New Match Puzzle/Solution

by : $136$

 * A New Match Puzzle

Solution
$36$ matches.

Proof
We can form:
 * the triangle and the square with $12$ and $24$ respectively


 * the triangle and the pentagon with $6$ and $30$ respectively


 * the triangle and the hexagon with $6$ and $30$ respectively


 * the square and the pentagon with $16$ and $20$ respectively


 * the square and the hexagon with $12$ and $24$ respectively


 * the pentagon and the hexagon with $30$ and $6$ respectively.

There are different arrangements for all except the $4$th and $6$th of these.

The triangle and hexagon require a number divisible by $3$.

The square and the hexagon require an even number.

Therefore the number must be divisible by $6$.

But this condition cannot be fulfilled for the pentagon and hexagon with less than $36$ matches.