hyperpower

Consider the equation xxx.. = 2.
How do you find x in this iterated exponentiation, a.k.a hyperpower?
The solution is quite easy: Consider the 'tower' above the bottom x, ie. xx...
But it is the same as the original 'tower', so we know it is 2.
Therefore x2 = 2 and it follows that x = sqrt(2).

But there is a problem if we now consider the equation xxx.. = 4.
Using the same argument we find x4 = 4, with the same solution x = sqrt(2).
Clearly there is a contradiction; it seems that we just proved that 2 = 4 if we equate the 'towers' of sqrt(2).

I found this math exercise and the solution to the puzzle here.

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