Subsets

Subsets are a family of techniques that eliminate candidates by looking at how a group of candidates is distributed among unfilled cells.

They are based on a simple fact: in a house (row, column, or box), if n candidates are confined to exactly n cells, then regardless of how those n digits are eventually arranged within those n cells, those digits must occupy those cells as answers. This leads to two kinds of deductions:

• If those n cells contain only those n digits as candidates, with no extra candidates in the same cells, then none of those n digits can appear in any other cell in the house. No matter how the digits are arranged, each of them must already appear once in those n cells. Therefore, those digits can be eliminated from all other cells in the house. This case is called a "naked subset."

• If a certain set of n candidates is confined to n cells, but those cells still contain additional candidates, then those extra candidates can be eliminated from the n cells. Otherwise, some of the confined digits would have nowhere left to go, which would violate Sudoku rules. This case is called a "hidden subset."

This kind of candidate elimination based on finding "n digits in n cells" is called a subset. The number of digits, n, is also called the size of the subset. Since larger subsets are harder to spot, different sizes are usually treated as different techniques.

• Naked subset techniques (eliminate subset digits from other cells):

Naked Pair

Naked Triple

Naked Quadruple

• Hidden subset techniques (eliminate other candidates from subset cells):

Hidden Pair

Hidden Triple

Hidden Quadruple

In theory, the subset size n can be any integer from 1 to 9. If n=1, the subset degenerates into a single. If n=9, it is just a completely empty house. In practice, when people talk about subsets, n usually means one of {2,3,4}. For cases with n ≥ 5, there is usually a smaller complementary subset in the same house. For example, even if all nine cells in a house are empty, a size-5 subset automatically implies a size-4 subset formed by the remaining four digits in the remaining four cells.

A subset does not require every one of the n digits to appear in every one of the n cells. For example, in a triple with digits a, b, and c, the three cells do not all need to contain {a,b,c}. They could instead be {a,b}, {a,c}, and {a,b,c}. In other words, it is enough that the union of the candidates in those cells contains {a,b,c}.