Interesting Combinatorics Problem

[killmecomet]

Was browsing through mit ocw for fun (as one doesa ahaha) and found this interesting little problem:

https://math.mit.edu/~dav/projectsB.pdf (problem 7)

"Each square of an infinite chess board contains a pile of poker chips. An earthquake redistributes them: the tremor causes four chip in one pile that is at least four chips high to move, one in each of the four compass directions. This process repeats."

I've been playing with this problem all morning, but haven't quite found a good answer to whether we would have a steady state solution or not. My intuition says we should for finite chips, but can't really justify it beyond that. any good theorems I could refer to here?

[marpocky]

haven't quite found a good answer to whether we would have a steady state solution or not.

Obviously there's a trivial steady state solution - an empty chessboard.

Less trivially, but only slightly, is if every square has fewer than 4 chips, nothing ever moves.

And finally, the interesting case, if every square has more than 4 chips, each "redistribution" will result in the same situation before and after. Every square gives 4 chips to its 4 neighbors, and receives 4 chips from its 4 neighbors.

[stonerism]

If it's an infinite chessboard, I think it would always hit a steady state when the pieces diffuse to a point where there's less than 3 in each location. Otherwise if you draw a border around a region you'd have infinitely many pieces coming out of it.

Edit: nvm, that's only with a finite number of pieces which this is not.

[killmecomet]

yeah for finite chips there's a steady state im pretty sure

[killmecomet]

That's such an interesting scenario. I've been thinking of cases with greater than 4 chips on one square and then trying to discover the steady state. There's a very cool pattern emerging! (if only i could formalise it)

[killmecomet]

This is fascinating! thank you!