[woevdbz]

It looks super neat! But if I'm honest with myself I don't actually "see" the numbers being equal after summing these tetrahedra (or even the triangles). I believe it, but I'd find a regular induction proof more believable. Visually, the trick in dimension 1 is a lot more convincing.

[kilovoltaire]

Yeah I agree, it's not 'visually' obvious that the combined entries are all the same. But once I started writing it I had to see where it went :p

(Footnotes 2 and 3 at the end explain why the entries all end up the same. And the same argument works for simplices in any number of dimensions—there's always two directions that change the value by +1 and -1, and the rest are all parallel / keep it the same. In higher dimensions there are more ways to be parallel!)

[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!