Congruence Classes: Solve the Hard Problem!

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Here is a fun problem, it's hard to write out clearly, but I'll try to do it w/ little confusion.
Is it, or is it not true that
(2^2^...^2)(n times)=(2^2^...^2)(n-1 times) mod n
so for example, when n=2, 2^2=2 ->
4=2 mod 2.
 
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He he sorry.
 
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Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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