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cannot each vertex of a cube$ 8$ distinct numbers from 1-12

Source: 2010 Romania District VIII p1

September 1, 2024
combinatoricsnumber theory

Problem Statement

a) Prove that one cannot assign to each vertex of a cube 8 8 distinct numbers from the set {0,1,2,3,...,11,12}\{0, 1, 2, 3, . . . , 11, 12\} such that, for every edge, the sum of the two numbers assigned to its vertices is even.
b) Prove that one can assign to each vertex of a cube 88 distinct numbers from the set {0,1,2,3,...,11,12}\{0, 1, 2, 3, . . . , 11, 12\} such that, for every edge, the sum of the two numbers assigned to its vertices is divisible by 33.