Find three different 2 digit prime numbers where the average of any
two is a prime and the average of all three is a prime.

WebSearcher, does 5 count as a 2-digit prime number??????? I guess if
you write it as 05, maybe, but somehow I doubt that this would be
considered legitimate.

I don't think that answer meets all the conditions of the initial
riddle. Namely, it seems in the riddle that all three prime numbers
are to be two-digit primes; that would preclude "5" from being one of
those three numbers...

How about 11, 47, and 71?

(11 + 47) / 2 =  58/2 = 29
(11 + 71) / 2 =  82/2 = 41
(47 + 71) / 2 = 118/2 = 59

(11 + 47 + 71) / 3 = 129/3 = 43

11 47 71

To find them, note that all three primes must be the same [mod 4]
(otherwise the average of two would be even), and also the same [mod
3] (otherwise the average of all 3 would not be an integer).  This
devides the 2-digit primes into 4 groups of size 4, 5, 5, and 6.  The
only solution is in the 6-member group, which has the primes that are
3[mod 4] and 2[mod 3].

Ansel - yeah, that's it - out of curiosity, why is it that you posted
this under the "comments" section rather than the "answer" ?

aj9989,

Only an official Google answerer can post under the answer section. 
The rest of us can only post comments.