MMP.10:1 - Problem frame

Use this pattern when you know conditions that a possible object or situation must satisfy, but still need to express its possibilities mathematically. You may be looking for an arrangement, a quantity, a shape or a rule of action. Choosing variables already makes decisions: a number can express an amount, but an ordered list, a set of members and an unknown function permit different operations and different possible answers.

Start by describing one candidate object and what would make it admissible. Choose a way to represent it, then distinguish two kinds of condition: those needed for the representation to denote such an object, and those expressing the requirements on that object. This produces a mathematical problem that can be reasoned about or handed to an appropriate computational method.

The result consists of variables with their domains, joint conditions, the requested operation on their solutions and a way to interpret the answer. It can be enough to find one feasible case; another question may require all possible values, a preferred case or a count. The chosen representation must support that particular result.

You need to understand the objects and requirements in the original question, elementary logical conditions and the mathematics used to express them. A mathematical collaborator can construct the expressions while you resolve their intended meaning. When a suitable formulation already exists, use it. When the missing contribution is a physical relation, an observation law or a rule of change, obtain that relation through the relevant method before translating it. B.5.FM supplies the broader first-model construction; MMP.7 and MMP.9 develop observation and reduced-evolution constructions.