Interesting puzzle that I haven't been able to solve or find a solution to.
An ant rearranges a line of stacks of seeds as follows:
With each iteration, the ant goes to each stack in order and grabs a single seed from each; then, when reaching the end of the line, places the seeds it has accumulated at the end of the line as a new stack. [for example: given the following line $(3,4,2,4)$ after one iteration: $(3,4,2,4)\to (2,3,1,3,4)$]
Note that when a stack is reduced to zero, it is considered as removed: $$(4,1,3) \to (3,0,2,3) = (3,2,3)$$
Prove that, given that the sum of seeds in the line is a triangular number (i.e $T_n = 1 + 2 + 3 + \cdots + n$), the ant's infinite rearrangement will ultimately converge to a steady state of $(1,2,3,...,n)$.
Additional example:
$$(4,2) \rightarrow (3,1,2) \rightarrow (2,1,3) \rightarrow (1,2,3)$$
I have been able to accumulate some thoughts on this puzzle but nothing too concrete to hint towards a solution, I would love to discuss possible approaches for this puzzle and find out the solution.
Thanks!
Edit: My thoughts on the puzzle.
Firstly, any permutation on $[n] := \{1,2,...n\}$ will eventually converge to the steady state. This is easy enough to see. So it is sufficient to prove that any initial state of the puzzle (with the sum being a triangular number) will eventually become a permutation on $[n]$.
Second, although it is not a strong indication, it might be a nice direction to ponder in: if the length of the line is $n$ and, the elements are distinct, the line is a permutation on $[n]$. So this is some weaker version of the statement about which might be easier to approach.
Third, I have been trying to model this puzzle as an operator on vectors/matrices/series, but with no success. But thinking of the ant's iterations as an operator $F$, then $F$ is not reversible, since eventually it reaches a steady state. So applying a group-like operation or model on this puzzle might not work.
Fourth, a clear invariant in this puzzle is the sum of seeds, and seems to be almost the only one (if not the only). So thinking of invariant-preserving operators might come in handy, although I'm not familiar with enough and haven't been able to identify useful operators (and respective spaces) to help me model this puzzle efficiently.
Currently, this is about it.