Integer triangles
A farmer has 27 meters of fence and wants a triangular pen whose three sides are each a whole number of meters.
How many differently shaped triangular pens can be built? (Turning or flipping a pen over does not make it a new one.)
Order the sides, then keep only the ones that close up
- We want triangles with whole-number sides adding to 27, counting a triangle and its mirror as one.
- Put the sides in order, a ≤ b ≤ c. Three lengths only close into a triangle when a + b > c — the two shorter sides must out-reach the longest. A lopsided split like (1, 1, 25) fails that and is not a triangle at all.
- Walk the shortest side up from 1, and for each one take the middle side as far as it can go with b ≤ c. Keeping just the splits that pass the rule: (1, 13, 13), (2, 12, 13), (3, 11, 13), (3, 12, 12), (4, 10, 13), …, (9, 9, 9).
- That is 19 different triangles.
How it works.
- 01
We want triangles with whole-number sides adding to 27, counting a triangle and its mirror as one.
- 02
Put the sides in order, a ≤ b ≤ c. Three lengths only close into a triangle when a + b > c — the two shorter sides must out-reach the longest. A lopsided split like (1, 1, 25) fails that and is not a triangle at all.
- 03
Walk the shortest side up from 1, and for each one take the middle side as far as it can go with b ≤ c. Keeping just the splits that pass the rule: (1, 13, 13), (2, 12, 13), (3, 11, 13), (3, 12, 12), (4, 10, 13), …, (9, 9, 9).
- 04
That is 19 different triangles.