Mirrored from Sudopedia, the Free Sudoku Reference Guide

Constraint Subsets

Constraint subsets is an abstraction of subsets and fish. By itself, it is not a solving technique that can be employed by human players, but the broad range of subsets that could be created may give birth to new and exciting solving techniques.

A constraint is a restriction in a Sudoku puzzle that has only room for a single candidate. Each cell is such a constraint. When the puzzle is solved, there is only one candidate left in the cell. Each house is a collection of 9 constraints. There can only be one of each digit in each house.

When we isolate a set of constraints from the remaining puzzle, we are creating a constraint subset. Although it is possible to combine any number of constraints into a set, most of these sets will not help us solve the puzzle. There must be something in common for the selected constraints to form an effective set. For example, we could create a set that contains a number of cells which belong to the same house. Another effective set, which forms the basis for all fish strategies, is a set of constraints for a single digit.

Having something in common is not the only restriction to form a constraint subset. The constraints must also have something not in common: candidates. A constraint set is useless if there are candidates which belong to 2 or more constraints in the set. That may sound like an impossible demand, but we only need to look at the available candidates. When all common candidates between 2 constraints have been eliminated, we can combine them in a constraint subset.

By themselves, constraint subsets are not very interesting. The fun begins when we can compare them with another constraint subset of equal size. When all candidates of one set are also members of a second set, the first set, which we call the defining set, is imposing a limit on the number of ways we can complete the secondary set.