Using your Head is Permitted

June 2010 solution

The answer is that this is possible iff n is odd.

Let Z_i be the permutation mapping each x to x+i mod n.

For an odd n, A=B=Z₁, C=Z₂ is a valid solution.

For even n, consider the sign of permutations A, B and C. All cyclic permutations can be described as a name-change over any other cyclic permutation. So, for example, for permutation A there is some one-to-one function P_A, such that A=P_A^-1Z₁P_A. Regardless of what the sign of P_A is, the sign of A must equal the sign of Z₁, which is -1. More generally, the sign of all cyclic permutations, including all of A, B and C, must be -1.

However, the sign of BοA is -1*-1=1, so it cannot be a cyclic permutation.

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