It’s a sample chapter from MATHEMATICS GALORE!, published by the MAA in 2012.
( Check out the book here!)
Enjoy the mathematics of triangles with integer side-lengths and see the proof developed by high-school students establishing that the number of integer triangles with perimeter N is N*N/48 rounded to the its nearest integer if N even, and (N+3)(N+3)/48, again rounded, if N is odd.
For example: How many different triangles can you make with 20 toothpicks?
Answer: round(400/48) = round(8.333…) = 8. (This is the same count as the number you can make with just 17 toothpicks.)