MMP.10:4.2 - Construct variables that represent the object

Choose an expression from which a candidate can be recovered. Give each variable a domain and any unit or reference point needed by its operations. A machine label ranges over names; arithmetic on the label requires a separate meaning. A count ranges over nonnegative integers; an amount may be divisible. State a finite bound when the task supplies one. Adding a bound solely to finish a search restricts the question to that bound.

For a structured object, compare representations by the operations you need. A function on a finite set can use one output variable for each input. Alternatively, a table of Boolean indicators can say which input-output pairs belong to its graph. The first expression makes function evaluation easy to state. The second makes some relations among pairs visible, but needs conditions to make the table a function. For a partial function, represent undefinedness as well as defined values.

Keep a shared quantity shared. If several equations use the same unknown offset, one offset variable must occur in all of them. Introducing a separate offset in each equation creates additional possibilities. Conversely, equating genuinely separate values can remove possibilities.

For a function or shape over an infinite domain, choosing finitely many coefficients also chooses a family. Identify that family and whether it expresses the intended possibilities or is a deliberate restriction. For example, the conditions on a continuous function may allow curved solutions even when no affine function satisfies them. A useful restricted family can be sufficient for finding a witness; failure inside it leaves the larger family unresolved.