What is the least number of cubes that must be added to this stack to make it a solid cube?

The stack is 4 cubes high and 4 rows deep, so the smallest solid cube around it is $4\times 4\times 4=64$. The stack has $4+3+1+2+2+1+1+1+1=16$ cubes, so $64-16=48$ must be added. (32 fills a 3 by 4 by 4 box, which is not a cube; 11 uses a 3 by 3 by 3 cube, which is too small to hold the stack.)