N=2a+b, where a is a two-digit square number and b is a two-digit cube number. What is the smallest possible value of N?

Considering the smallest possible value of N will mean finding the smallest possible values of a and b to give the minimum N. As a must be a square number, let's consider the square numbers: 1 squared is 1; 2 squared is 4; 3 squared is 9; 4 squared is 16. Here we have reached the smallest two-digit square number, as al square numbers up to 16 (1, 4, 9) are all one-digit. So a must be 16. Considering b; b must be a cube number. So think of all the cube numbers in increasing order, as we did with the squares: 1 cubed is 1, 2 cubed is 8, 3 cubed is 27. Here we have reached a 2-digit number. So, as 27 is the smallest two-digit cube number, b must b 27. So inserting our values for a and b into the equation for N gives 2a+b=2(16)+27=59, so the answer is 59.