Henry Ernest Dudeney/Modern Puzzles/136 - A New Match Puzzle/Solution
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Modern Puzzles by Henry Ernest Dudeney: $136$
- A New Match Puzzle
- 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
- 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?
Solution
$36$ matches.
Proof
We can form:
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.
$\blacksquare$
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $136$. -- A New Match Puzzle
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $494$. A New Match Puzzle