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Thanks for thinking about it D, but I don't think the teaser is looking for cases where B and C can be swapped in the expression. Surely it is the the expression itself that is ambiguous.
A^B^C can be read as (A^B)^C or A^(B^C) which give very different results in most cases.
For (A^B)^C to equal A^(B^C), as (A^B)^C is equivalent to A^(B*C), the cases we need to find are where B^C = B*C which is possible for a few positive fractions.
The most trivial case is B=4 ; C=1/2 B^C=B*C=2
But not many of these B,C pairs are both proper fractions.
Indeed, the only cases I can find are where for fraction C the difference between the denominator and the numerator is 1. These are:
B C Effective power
4 1/2 2
27/8 2/3 9/4
256/81 3/4 64/27
3125/1024 4/5 625/256
etc.
Of these the 27/8, 2/3 case with effective power 9/4 is the only one where both B and C are proper fractions and there is a possible "answer" Z value < 1,000,000
As I said, I never did find any other cases except those where C is of the form n/(n+1) and suspect there aren't any, but haven't proved that. I would be interested to hear of any.
Of course, I might be completely misinterpreting the original question.