Naked Triples: A Deep Dive

Naked triples are pairs' big cousin. Three cells in the same unit whose combined candidate set covers exactly three digits. The three digits commit to those three cells, and every other cell in the unit can have them eliminated.

Triples are more common than they look and harder to spot than pairs because the three cells do not need identical candidate sets. That last bit is what trips up most solvers. The pattern can hide in plain sight.

Why the triple is locked

The unit needs all nine digits. If three cells between them can only hold three specific digits, those three cells fill those three slots. The other six digits live elsewhere in the unit. Conclusion: no other cell in the unit can hold any of those three digits as a candidate.

That conclusion is the technique. Find the triple, then strip the three digits from every other cell in the unit. The strip almost always produces immediate placements somewhere.

The two shapes

The "easy" shape is three bivalue cells: {1,2}, {1,3}, {2,3}. Each cell has only two candidates, and the union is exactly three. These announce themselves visually if your pencil marks are clean.

The harder shape mixes bivalue and trivalue cells: {1,2,3}, {1,3}, {2,3}. The combined union is still three. The technique still applies. Most beginners miss this version because they are looking for symmetry, and the symmetry is not there.

r1c1 {4,8}       r1c2 {2,4,5,7,8}   r1c3 {2,7}*
r1c4 {2,5,7}*    r1c5 {1,6,9}       r1c6 {1,5,6,9}
r1c7 {2,5,7}*    r1c8 {1,9}          r1c9 {1,5,9}

Look at columns 3, 4 and 7. Their candidates are {2, 7}, {2, 5, 7} and {2, 5, 7}. The union is exactly {2, 5, 7}. That is a naked triple. The digits 2, 5 and 7 are now confined to those three cells, which means every other cell in the row can have them eliminated.

r1c2 loses 2, 5 and 7. r1c5 loses no candidates because it has none of them. r1c6 loses 5. r1c9 loses 5. Several cells shrank. Singles often follow.

How to scan for them

Triples hide best in units with four to six empty cells. Once the unit has only three empties, the triple is trivially the answer. Once it has seven or more, the candidate noise is too high to spot the pattern.

Work from bivalue cells outward. Find two cells with two-candidate sets that share at least one digit. If a third cell in the unit completes the union to exactly three digits, you have a triple. That third cell can be bivalue or trivalue.

The hidden-triple distinction

Naked triples look at cells. Hidden triples look at digits. Naked means three cells whose candidates cover three digits. Hidden means three digits whose candidate positions cover three cells. The two techniques find the same kind of pattern from opposite angles. Most puzzles have one or the other; some have both hiding in the same unit.

The mistake that wastes a minute

Including a fourth cell. If the union of your three candidates spills to four digits, the pattern is not a triple. Discard and scan elsewhere. The amount of energy spent trying to force a non-triple to be a triple is a significant fraction of every solver's wasted time.

Trust the union test. Three cells, exactly three digits combined. Anything else is not a triple.

For practice, try a expert puzzle and look for triples after pairs have run out. The formal page is at naked triples.