Invariants
https://codeforces.com/contest/2210/problem/D
In this problem, we can show two invariants. Then, we can show that for every configuration with a given invariant, it can transformed to a “fixed point” in that space. Then, because the operation is reversible, this means all configurations that share the two invariants can transform to one another.
https://atcoder.jp/contests/arc215/tasks/arc215_e
See my ARC215 editorial for notes on this problem.
https://codeforces.com/contest/2237/problem/D