# Henry Ernest Dudeney/Puzzles and Curious Problems/157 - Counting the Matches/Solution

*Puzzles and Curious Problems* by Henry Ernest Dudeney: $157$

- Counting the Matches

*A friend bought a box of midget matches, each one inch in length.**He found he could arrange them all in the form of a triangle whose area was just as many square inches as there were matches.**He then used up $6$ of the matches,**and found that with the remainder he could again construct a triangle whose area was just as many square inches as there were matches.*

*And using another $6$ matches he could again do precisely the same.**How many matches were there in the box originally?**The number is less than $40$.*

## Solution

First we recall Heron's Formula for the area of a triangle with sides equal to $a$, $b$ and $c$:

- $\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$

where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.

With $36$ matches you can make a triangle with sides $17$, $10$, $9$ whose area $\AA$ is calculated by Heron's Formula to be:

\(\ds \AA\) | \(=\) | \(\ds \sqrt {18 \paren {18 - 17} \paren {18 - 10} \paren {18 - 9} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \sqrt {18 \times 1 \times 8 \times 9}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \sqrt {1296}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 36\) |

With $30$ matches you can make a triangle with sides $13$, $12$, $5$, which is the $5$-$12$-$13$ Pythagorean Triangle whose area $\AA$ is calculated by Area of Triangle in Terms of Side and Altitude to be:

\(\ds \AA\) | \(=\) | \(\ds \dfrac 1 2 \times 5 \times 12\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 30\) |

With $24$ matches you can make a triangle with sides $10$, $8$, $6$, which is the $3$-$4$-$5$ Pythagorean Triangle scaled up by a factor of $2$, whose area $\AA$ is calculated by Area of Triangle in Terms of Side and Altitude to be:

\(\ds \AA\) | \(=\) | \(\ds \dfrac 1 2 \times 6 \times 8\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 24\) |

$\blacksquare$

## Sources

- 1932: Henry Ernest Dudeney:
*Puzzles and Curious Problems*... (previous) ... (next): Solutions: $157$. -- Counting the Matches - 1968: Henry Ernest Dudeney:
*536 Puzzles & Curious Problems*... (previous) ... (next): Answers: $510$. Counting the Matches