Logical constraints and invariants

Build a model, justify why it applies and check the result.

Turn each statement into a precise condition and test it on a candidate arrangement.

Use a table of exclusions for a one-to-one matching and explain why each cell is ruled out.

Find a quantity that every allowed move preserves and use it to show that a target cannot be reached.

Prove a minimum in two parts: a construction that reaches the value and an argument that nothing smaller works.