The Math-e-Monday Puzzle: Squares from a Tetrahedral Die

It isn’t Monday, but I’m puzzled every day of the week.

Alice is puzzled too; she’s playing with a new tetrahedral die. Each face has a different positive integer on it, but the numbers are peculiar. Alice quickly notices that if she rolls the die and adds up the numbers on the three exposed faces, she always comes up with a perfect square. “That’s clever!” thinks Alice.

Assuming that Alice’s tetrahedral die uses the smallest possible numbers, what is the sum of the numbers on all four faces?

Alice is puzzled too; she’s playing with a new tetrahedral die. Each face has a different positive integer on it, but the numbers are peculiar. Alice quickly notices that if she rolls the die and adds up the numbers on the three exposed faces, she always comes up with a perfect square. “That’s clever!” thinks Alice.

Assuming that Alice’s tetrahedral die uses the smallest possible numbers, what is the sum of the numbers on all four faces?

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