Here’s a quick one for you. Consider the sequence of triangular numbers
and the set of square numbers
It seems that there are some numbers which are both squares and triangles! For example, 1 and 36 get to be both shapes – we will call these square-triangles from now on.
Actually, you can very easily prove that there are infinitely many square-triangles via the following exercise.
Exercise. Let be a square-triangle number. Prove that and are both square numbers. Finally, prove that is a square-triangle.