How to Break a Math Problem Down from First Principles

This Oxford University entrance exam question illustrates the systematic, first principles reasoning that all good mathematicians use, and which GPT4 still struggles with

Many of us are familiar with the rules of the sides of a right angled triangle. By the Pythagorean Theorem, if a and b are the two smaller sides, and c is the longest side, then a² + b² = c². Well known integer solutions to this are known as Pythagorean Triples. Examples include (3, 4, 5) and (5, 12, 13).

A more general construct is a Triangular Triple. Any trio of positive integers which can form the lengths of the sides of a triangle is called a triangular triple. Obviously, integer Pythagorean triples are triangular triples. But in general, an integer triple (a, b, c) is a triangular triple if, when listed in (non-strictly) increasing order, the sum of the first two integers is strictly larger than the third.

As an example, lets take the triple (4, 2, 3). These can be the lengths of the sides of a triangle because 2 + 3 > 4. Similarly (2, 2, 2) is a triangular triple. But (1, 3, 1) is not a triangular triple, as it is not possible to construct a triangle with these side lengths (1 + 1 ≯ 3).