Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 12G / b / i / Solution

Count red-green segments modulo two. On $DE$, part (a) says their number is odd; the other two outer sides contain no red-green segment. Every interior segment belongs to two small triangles and therefore contributes zero modulo two, while each boundary segment belongs to one. Thus the sum, over all small triangles, of their numbers of red-green edges is odd.

A triangle with all three colours has exactly one red-green edge. A triangle using only red and green has zero or two, and any other nontrichromatic triangle has zero red-green edges. Hence the parity sum counts precisely the trichromatic triangles modulo two, proving that their number is odd.