values x = 3, y = 4 and z = 5 turn out to be a solution of the kind we are
looking for because 3^2 + 4^2 = 9 + 16 = 5^2. From the converse, a triangle with
dimensions 3, 4 and 5, must include a right angle. This is the mathematical fact
that the builder uses to build his walls at right angles.
In this case we can break up a 3 × 3 square, and wrap it around a 4 × 4
square to make a 5 × 5 square.
There are other whole number solutions x^2 + y^2 = z^2. For example x = 5, y =
12 and z = 13 is another solution because 5^2 +12^2 = 13^2 and in fact there are an
infinite number of solutions to the equation. The builder’s solution x = 3, y = 4
and z = 5 holds pride of place since it is the smallest solution, and is the only
solution composed of consecutive whole numbers. There are many solutions
where two numbers are consecutive, such as x = 20, y = 21 and z = 29 as well
as x = 9, y = 40 and z = 41, but no others with all three.