Athena, the Olympiad Goddess of wisdom, had a test for her disciples - Odysseus and Perseus. She told them:
I am thinking of a triangle whose sides are integers.
Perseus, I will tell you its perimeter.
Odysseus, I will tell you its area.
You have to figure out the side lengths of the triangle.
Since they were both vying for Athena’s favor, they were hesitant to reveal too much information, and guarded what they said. They made the following statements sequentially:
Now, your task is: how many non-congruent triangles could Athena be thinking of?