MATH.8:1 - Problem frame

Use this pattern when a mathematical solution is known and transformations that preserve its defining conditions can produce further useful solutions. You may need related roots of an equation, arrangements considered equivalent under rotation, or answers to problems whose data have been permuted.

Start with the known solution and one proposed transformation. Apply it and establish which problem the result solves. This can return another answer immediately, expose changed data, or identify a condition that the transformation fails to preserve.

A symmetry is an invertible transformation preserving the structure named by the problem. Its orbit at an object is the family of objects reached by the chosen group of transformations. The construction below obtains that family, distinguishes repeated objects, and explains what is still missing from a claim to have found all solutions.

The reader needs sets, functions, elementary equations and composition. The required group-action rules are stated below. The polynomial example uses factorization; the finite-arrangement example needs only binary strings and rotation. Use a direct calculation when it already supplies the needed answer. A transformation that changes the problem can still transfer a solution to the changed problem when that is useful.