MATH.11:1 - Problem frame

Use this pattern when a mathematical construction can take many steps and you need a relation that survives every allowed step. Such a relation can rule out a proposed result, constrain a search or derive a formula for a quantity that the steps accumulate.

Here an invariant is a function whose value is unchanged by each allowed transformation. The method constructs that function from the transformations. It starts with a small family of expressions, derives conditions on their coefficients and uses the resulting expression in a proof. Conserved weighted totals, polynomial relations and residues provide different ways to carry out that construction.

First useful move: write one allowed change, substitute it into a candidate expression and calculate what changes. Derive coefficient conditions that make the change vanish, then include the remaining allowed rules. Once preservation holds for them all, compare the invariant at the starting and proposed states.

The reader needs substitution, elementary algebra and the arithmetic used by the example. The polynomial branch uses collection of like terms and linear equations; the residue branch explains arithmetic modulo two. An available invariant can be used directly after checking that it fits the allowed transformations. When a short explicit sequence already answers the question, constructing an invariant may add no useful result.