Take, say, 73 little cubes (blocks or ordinary D6 dice). Could you arrange them into three perfect cubes?
I’ll wait while you try to solve this problem…
OK, you probably started by taking 64 of them to make a 4×4×4 cube. That left 9, which it’s easy to fashion into a 2×2×2 cube and a 1×1×1 cube.
Or you might treat it as a number theory problem, somehow finding 73 = 43 + 23 + 13.
Of course there’s nothing special about 73. But several questions naturally arise:
- Is there a second solution for 73?
- If so, how many solutions are there?
- Is this true of all primes?
- Is this true of all integers?
- Are there any integers that cannot be represented as the sum of three cubes?
Before we go any further, there’s one small complexity: we have to allow “negative cubes,” more easily written as something like –43 or (–4)3 rather than portrayed as physical cubes.
It turns out that exploring these questions takes us rapidly into the realm of very large numbers. But you’ll have to watch Brady Haran’s entertaining and informative five-minute video in order to find out. Can a number literally be “astronomically large”? Watch the video and you’ll learn what galactic proportions really are!