August 2011 Brain Teaser Solution

How many integers less than or equal to 1,000,000 are both perfect squares and perfect cubes?

Answer: 10
Anything that is both a perfect square and a perfect cube will be a number to the 6th power.
The lowest one is 1 = 1^2 and 1^3
2^6 = 64 = 8^2 and 4^3
3^6 = 729 = 27^2 and 9^3
4^6 = 4096 = 64^2 and 16^3
5^6 = 15625 = 125^2 and 25^3
6^6 = 46656 = 216^2 and 36^3
7^6 = 117649 = 343^2 and 49^3
8^6 = 262144 = 512^2 and 64^3
9^6 = 531441 = 729^2 and 81^3
10^6 = 1,000,000 = 1000^2 and 100^3

For an example of why this works, let's examine 2^6  which is 2 x 2 x 2 x 2 x 2 x 2 then these 2s can be split into 2 groups of 3 as (2 x 2 x 2)^2 which can be rewritten as 8^2
or 3 groups of 2 (2x2)(2x2)(2x2) = 4^3