A Strange Die

A PuzzleCritic Original:

A cube has a different positive integer painted on every face, so that every pair of numbers on adjacent faces has a sum divisible by 7. Let T be the sum of all the numbers that appear. Find, with proof, the smallest possible value of T.

Suppose a, b and c are three numbers all surrounding the same vertex P. Since a+b and b+c are both divisible by 7, their difference a-c is also divisible by 7. But a+c is also divisible by 7, so

(a+c)+(a-c) = 2a

is divisible by 7, and thus a is also divisible by 7.

However, there is nothing special about a; it must be the case that every number is a different multiple of 7. Since any multiples of 7 will do, the smallest possible value of T is

7 + 14 + 21 + 28 + 35 + 42 = 147.

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