Question 56
a) Prove that one cannot assign to each vertex of a cube 8 distinct numbers from the set $\{0, 1, 2, 3, \ldots, 11, 12\}$ 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 8 distinct numbers from the set $\{0, 1, 2, 3, \ldots, 11, 12\}$ such that, for every edge, the sum of the two numbers assigned to its vertices is divisible by 3.
b) Prove that one can assign to each vertex of a cube 8 distinct numbers from the set $\{0, 1, 2, 3, \ldots, 11, 12\}$ such that, for every edge, the sum of the two numbers assigned to its vertices is divisible by 3.