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.

Subset size $n$	Naked subset (eliminate subset digits from other cells)	Hidden subset (eliminate other candidates from subset cells)
$2$	Naked Pair	Hidden Pair
$3$	Naked Triple	Hidden Triple
$4$	Naked Quadruple	Hidden Quadruple

In theory, $n$ 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 $n\ge 5$, 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 $\{a,b,c\}$.