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Hi J. If the grader thinks it is the proper answer it doesn't mean it is. I tried it for a several k and there is no cicle except fo k=2. Try for k=7, and goes and goes without end, never in cicle.
Some ppl think there is a simple answer to the problem. To be honest I do not beleive it. For instance let m=1 and k=171. Why not? The original state was not defined in the problem, so it just might be that way. Will the "answer" work? I am sure it won't.
I think it should look like (k-2)^x + ... ------------- k^x where x = m+1 but i have failed for find oput more :-(