## June 2010 riddle

One of my all-time favorite "Using your Head is Permitted" riddles was the December 2007 riddle. This month's riddle is a follow-up to it.

Let n be an integer greater than 1, and let A, B and C be one to one functions, mapping the elements [0..n-1] to themselves.

All three functions satisfy the property that they are cyclic permutations, in the sense that for any x:0≤x<n the list x, A(x), A(A(x)), A(A(A(x))), etc., goes through all n possible elements before returning to x. This is true not only for A, as in the example, but also for B and C.

The question: for which n do such A, B and C exist, where for all x, B(A(x)) = C(x)?

For each n, either provide a proof of impossibility or an example of such an A-B-C triplet.

### List of solvers:

Christian Blatter (1 June 23:10)
Zilin Jiang (2 June 15:15)
Jin Ruizhang (2 June 23:49)
Oded Margalit (3 June 15:38)
Phil Muhm (4 June 00:25)
Hongcheng Zhu (4 June 17:50)
Lee Siu Hung (7 June 01:53)
Daniel Bitin (7 June 07:19)
Itsik Horovitz (7 June 08:23)
Jan Fricke (21 June 22:42)
Erick Wong (23 June 10:08)
Omer Moussaffi (24 June 06:31)