[civilized]

I was confused about this too - why should we believe that the numbers should be equal after summing?

There is an explanation based on symmetry, which I outlined here: https://news.ycombinator.com/item?id=32531692

Interestingly, this approach does not seem to require any induction. The argument just works directly for all powers and all dimensions.

[woevdbz]

That's a great explanation, thanks!