Power Tower Luck
Tommy incorrectly thinks that while Jenny correctly knows that
For how many of the 64 ordered triples of positive integers less than or equal to 4, will Tommy's incorrect method get the correct answer?
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1 solution
If a = 1 then both sides will equal 1 for b , c ∈ { 1 , 2 , 3 , 4 } , resulting in 4 2 = 1 6 triples.
If c = 1 then both sides will equal a b for a , b ∈ { 1 , 2 , 3 , 4 } , resulting in 4 2 = 1 6 triples.
However, this will result in the double-counting of the 4 triples for which a = c = 1 , b ∈ { 1 , 2 , 3 , 4 } , and so we thus far have 1 6 + 1 6 − 4 = 2 8 solution triples.
Next, we exploit the fact that b 2 = b b has solutions b = 1 and b = 2 . Having already dealt with the b = c = 1 case, we look next at b = c = 2 , in which case
a b c = a 2 2 = a 4 = a 2 × 2 = ( a 2 ) 2 = ( a b ) c .
As a result we must add the triples ( a , b , c ) = ( 2 , 2 , 2 ) , ( 3 , 2 , 2 ) , ( 4 , 2 , 2 ) to our solution list, giving us a total of 2 8 + 3 = 3 1 solution triples.