Sunday Times Teaser 3084 – Face Value
by Susan Bricket
Published Sunday October 31 2021 (link)
Plato: I have written a different whole number (chosen from 1 to 9 inclusive) on each of the faces of one of my regular solids and have labelled each vertex with the product of the numbers on its adjacent faces. If I tell you the sum of those eight vertex labels, you can’t deduce my numbers, but if I rearrange the numbers on the faces and tell you the new sum, then you can deduce the numbers.
Eudoxus: Tell me the new sum then.
Plato: No, but I’ll tell you it’s a 10th power.
Eudoxus: Aha! I know your numbers now.
Plato: Yes, that’s right. But if I now told you the original sum, you couldn’t work out which numbers were originally opposite each other.
What was the original sum?