As Jessicata pointed out, in 3-dimensions or higher, our n-hedra don’t split up as nicely as in the 2d case.
That isn’t the only issue: many of the surfaces of connected blocks may not correspond to the type of 2d grid that we just proved this result for and it doesn’t seem trivial to figure out how to characterise what kind of grids we need to extend our result too (it will be even worse in higher dimensions)
I’ve found two results: firstly that if you remove a triangular face and replace it with three others (imagine allowing the surface to jut out of the page), then the trichromatic parity will be preserved. Secondly, that we can replace two triangle with four triangles
Given what I’ve discussed above, I’d be keen for a hint as learning enough geometry to make progress on this problem would seem to be taking me pretty far afield from maths useful for ai-risk.
Some preliminary thoughts on q9:
As Jessicata pointed out, in 3-dimensions or higher, our n-hedra don’t split up as nicely as in the 2d case.
That isn’t the only issue: many of the surfaces of connected blocks may not correspond to the type of 2d grid that we just proved this result for and it doesn’t seem trivial to figure out how to characterise what kind of grids we need to extend our result too (it will be even worse in higher dimensions)
I’ve found two results: firstly that if you remove a triangular face and replace it with three others (imagine allowing the surface to jut out of the page), then the trichromatic parity will be preserved. Secondly, that we can replace two triangle with four triangles
Given what I’ve discussed above, I’d be keen for a hint as learning enough geometry to make progress on this problem would seem to be taking me pretty far afield from maths useful for ai-risk.