TITLE: Is it possible to divide a cube into $5772$ cubes of varying sizes?
QUESTION [5 upvotes]: Is it possible to divide a cube into $5772$ cubes of varying sizes? I'm pretty sure this riddle has to do with algebra and group theory, but so far everything I've tried has led me nowhere. Any help would be appreciated!

REPLY [8 votes]: Yes you can !
When you have a cube, you can divide that cube into $n^3$ smaller cubes by cutting it into $n$ equal slices along each side. Hence this operation adds $n^3-1$ cubes.
$$5772=1+(17^3-1)+(9^3-1)+(5^3-1)+(2^3-1)$$
So take a cube, and in any order :

Choose one cube and cut it into $17^3$ cubes
Choose one cube and cut it into $9^3$ cubes
Choose one cube and cut it into $5^3$ cubes
Choose one cube and cut it into $2^3$ cubes