MATH.9:1 - Problem frame

Use this pattern when a mathematical rule must choose an answer and transforming the input should transform that answer in the corresponding way. Examples include selecting an element after relabeling a configuration, returning a direction after rotating geometric data, and choosing a solution from problems related by a symmetry.

Start with one transformation that leaves the supplied input unchanged. Ask which permitted answers it leaves unchanged too. This can identify an admissible answer or expose an impossible choice before a selector is implemented.

The method then constructs a rule from one suitable answer for each represented input orbit. It also uses uniqueness, when established, to restrict a solution to the points fixed by the input’s symmetries. A useful result is a compatible choice rule, a restricted candidate to calculate, or a requirement that must change.

The reader needs sets, functions, composition and elementary algebra. MATH.8 supplies group actions, orbits and stabilizers; the definitions needed for the choice argument are restated here. The plane example uses vector length and rotation. If a supplied rule already has the required correspondence, apply it. If the task accepts any answer in the given coordinates and requires no symmetry relation, use an ordinary selection method.