Find a list of squares and cubes:
Cubes: 1, 8, 27
Squares: 1, 4, 9, 16, 25
Notice '25' and '27'.
If his age was '25' last year, it would be a perfect square 5 * 5. If his age next year will be '27', it will be a cube, 3 * 3 * 3.
So his age now is 26.
