Counting integer triangles of perimeter 15

Text

With $15$ equal toothpicks, how many different triangles can be formed using a whole number of toothpicks per side? (The sides must satisfy the triangle inequality: each side is less than the sum of the other two.)

Solution

Look for the triples $a\ge b\ge c\ge 1$ with $a+b+c=15$ and $a<b+c$, i.e. $a<7.5$, so $a\le 7$:

• $a=7$: $b+c=8$, with $b\le 7$ and $b\ge c$: $\ (7,1),(6,2),(5,3),(4,4)$ → $4$
• $a=6$: $b+c=9$, with $b\le 6$ and $b\ge c$: $\ (6,3),(5,4)$ → $2$
• $a=5$: $b+c=10$, with $b\le 5$: $\ (5,5)$ → $1$

In total $4+2+1=\boxed{7}$ triangles.