What are some puzzles that are solved by using invariants?

The entire first chapter of Engel's famed olympiad training book, Problem-Solving Strategies is on "The Invariance Principle". There are several illuminating examples followed by sixty problems and their solutions. The second chapter in the book is on "Coloring Proofs," a part of which is about parity, and that often also involves invariants (and showing that some configuration, say on a chessboard, is impossible because it violates the parity invariant).

I recommend also looking into monovariants which change in some predictable way, unlike invariants which stay constant; monovariants are sometimes mistakenly grouped under invariants.

Pranav Sriram's free online book on Olympiad Combinatorics contains material on advanced olympiad problem solving using invariants and monovariants. It takes some effort to find a PDF that contains all the chapters though; I recall that there are nine chapters but they were posted separately at different times.

Looking up "invariant" in puzzling.SE yielded results so you could try that.

I heard of this in early 2000's: given a cube with number as each of it's vertices. Permitted operation is selecting an edge and adding same number to the both vertices at the ends of the edge. Is a state, where all numbers at the vertices are equal, achivable from given state (which may be arbitrary, but known) using only the permitted operation?
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