Problem #12

It is easy to see that there is no permutation of 1,2,...,n such that the sum of two adjacent elements is always even, since at some point there must be an odd number adjacent to an even one. This month's problem is to investigate a number of related questions.

• For which n is there a permutation of 1,2,...,n such that the sum of three adjacent elements is always even? For such an n, how many permutations are there?

• For which n is there a permutation of 1,2,...,n such that the sum of four adjacent elements is always even? For such an n, how many permutations are there?

• For which n is there a permutation of 1,2,...,n such that the sum of three adjacent elements is always a multiple of three? For such an n, how many permutations are there?

The solution will be posted shortly.

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