Alice is playing with a new tetrahedral die. Each face has a different positive integer on it, but the numbers are peculiar, in that the numbers on the three exposed faces always sum to a perfect square.

Assuming that Alice’s tetrahedral die uses the smallest possible numbers, what are they?

Alice claps her hands in delight. 'How splendid it would be if all four sides also summed to a square!' she declares.

Is that possible?

(Adapted from Science 2.0)