I like problems like these. I was posed one at the end of a 'Constructive Mathematics' course I took at Stanford. Eight and nine are right next to each other (+

I like problems like these. I was posed one at the end of a 'Constructive Mathematics' course I took at Stanford. Eight and nine are right next to each other (+-1). One is a cube, and the other is a square. Do a cube and a square land next to each other ever again? Note it's not asking if they land on each other, that is trivial - 1000000 is 100^3 and 1000^2. My solution was to form an equation, divide by x and solve the resulting quadratic equation. It produce new solutions: 0 and 1, 0 and -1! As well as 2 and 3. So I was encouraged to say no, those were the only solutions.