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.