Techniques / Subsets

Naked and hidden subsets: pairs, triples and quads

N cells that between them hold exactly N digits.

10 techniques in this family, introduced at tier 2, 10 with a worked example from a real board. How to read a sudoku grid explains the notation used below.

What these have in common

Counting. If two cells in a house can only hold 4 and 7 between them, then 4 and 7 are used up by those two cells whichever way round they go, and neither digit is available anywhere else in that house. Stretch the same count to three cells and three digits, or four and four, and nothing about the argument changes.

Why the logic holds

A house has nine cells and nine digits, and the match is exact. So any N cells that can only draw from N digits must consume all N of them, and any N digits that can only go in N cells must fill exactly those. Both halves are the pigeonhole principle, and the eliminations run in opposite directions: the first clears those digits out of the rest of the house, the second clears the other digits out of those cells.

When to reach for it

Once you have pencil marks and singles have stopped. A naked pair is often visible at a glance, and it is the technique that makes medium puzzles tractable.

How the members differ

Naked and hidden are complements, and this is worth internalising because it halves the work. In a house with five unsolved cells, a naked pair among two of them is the same fact as a hidden triple among the other three. Whichever is easier to see is the one to look for. Locked pairs and triples are subsets whose cells also share a second house, so they eliminate in the box and the line at once.

The mistake to avoid

Hunting quads. A naked quad in a house with six unsolved cells is a hidden pair among the other two, and the hidden pair is far easier to spot. If you find yourself checking four-cell combinations by hand, look at the complement instead.

Every technique in this family

• Hidden PairTwo digits in one group can only go in the same two squares.
• Hidden QuadFour digits in a group can only go in the same four squares.
• Hidden Subset (general form)Some numbers in a group have only a few homes left.
• Hidden TripleThree digits in a group can only go in the same three squares.
• Locked PairA pair that shares two groups at once, usually a row and a box.
• Locked TripleA triple that sits in two groups at once, usually a row and a box.
• Naked PairTwo squares in the same group both hold only the same two digits.
• Naked QuadFour squares in one group share the same four digits between them.
• Naked Subset (general form)Find some squares in one row, column or box.
• Naked TripleThree squares in one group share the same three digits between them.