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

Modern Puzzles by Henry Ernest Dudeney: $136$

I have a box of matches.

I find that I can form with them any given pair of these four regular figures, using all the matches every time.

This, if there were eleven matches, I could form with them, as shown, the triangle and pentagon

or the pentagon and hexagon, or the square and triangle (by using only three matches in the triangle);

but could not with eleven matches form the triangle and hexagon,

or the square and pentagon, or the square and hexagon.

Of course there must be the same number of matches in every side of a figure.

Now, what is the smallest number of matches I can have in the box?

$36$ matches.

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.

$\blacksquare$