Naked triples
A naked triple is three cells in one row, column or box whose candidates, taken together, are only three different digits. Those three digits must fill those three cells, so they can be removed from every other cell in the group. The cells do not each need all three digits.
On this site naked triples are first needed in Hard puzzles.
A pair, one size up
A naked pair is two cells sharing two digits. A naked triple is three cells sharing three digits, with one important loosening: the cells do not have to carry identical lists. What matters is the union. If three cells in a group between them mention only the digits 3, 5 and 6, nothing outside that set can be placed in any of the three, and the three digits are all used up by the three cells.
The diagram above is a real position from this site's Hard pool, drawn by the technique engine. Its caption says that in row 1, the cells r1c2, r1c3 and r1c7 hold only the digits 3, 5 and 6 between them, so those digits belong to those cells and can be removed from the rest of row 1. Notation is row first, then column, counted from the top left.
The shapes a triple can take
Because only the union matters, a naked triple can look quite different from cell to cell. Using the digits 3, 5, 6 as an example, any of these combinations works:
- Three cells each holding 3/5/6.
- Cells holding 3/5, 5/6 and 3/6, a chain where each pair overlaps.
- Cells holding 3/5/6, 3/5 and 5/6.
- Cells holding 3/5/6, 3/5/6 and 3/6.
Each cell must contain at least two of the three digits, or the group would have a cell with one candidate and a simpler technique would have found it. A cell holding only a single digit from the set is a naked single, and a cell with a digit outside the set breaks the triple.
How to find them
- Choose a group with several empty cells and fill in candidate notes.
- Collect the cells that have two or three candidates. Cells with four or more are rarely part of a triple.
- Look for three of them whose combined candidates make a set of exactly three digits.
- Remove those three digits from every other cell in the group.
The difficulty is step three. A pair is spotted by matching two lists; a triple needs you to hold three lists and take their union. It is easier if you start from a cell with three candidates, then look for two other cells in the group whose candidates fit inside the same three digits.
Why it works
Three cells, three digits, and no other digit allowed in any of those cells. The three cells need three distinct digits and only three are available to them, so the cells use all three. No digit of the set is left for the rest of the group. The argument does not care how the digits are arranged among the three cells, which is why loose, overlapping lists still count.
Common mistakes
- Requiring identical lists. Insisting that each cell shows all three digits means missing most real triples.
- Counting four digits. If the union of the three cells has four digits, it is not a triple. Count carefully.
- Looking in the wrong group. The three cells must share a row, column or box; the elimination is limited to that group.
- Giving up after one failed search. A group can hold a triple that only becomes visible after another elimination. Re-check the group each time its notes change.
- Using a cell that is already a single. A one-candidate cell is a naked single and should be placed, not folded into a set.
Where the site's grading needs it
Naked triples are third-tier, along with X-Wing. In the committed frequency data over 1,000 stored classic puzzles per tier, a naked triple was used in 0 Easy and 0 Medium puzzles, 578 Hard, 76 Expert and 85 Evil puzzles. The counts fall at the higher tiers; the data show that harder puzzles are not simply more of the same technique, and this page does not claim a cause. Each puzzle is solved cheapest technique first, and a technique counts if used at least once. See how difficulty is graded and the frequency guide.
Naked triples versus hidden triples
A hidden triple is the digit-based version: three digits confined to the same three cells. As with pairs, it is hard to see and the engine here does not implement it. When a group has a hidden triple, the other cells form a naked set whose size is the group size minus three, which is often a better thing to look for. In practice, if you suspect a hidden triple, count the leftover cells and see whether they make a naked set.
In the variants
On the 16×16 board the groups are sixteen cells long, which leaves more room for several cells to overlap on the same few digits. On the hyper board the extra windows add groups to search. On the killer board cage sums often cut candidate lists to two or three digits, which makes triples easier to spot.
Check your understanding
- Three cells in a row hold 1/4, 4/7 and 1/7. What can you remove from the other cells in the row?
- Three cells hold 1/4, 4/7 and 1/8. Is this a triple?
- Why is a triple where each cell has only two candidates still valid?
The answers: remove 1, 4 and 7 from the rest of the row; no, since the union is 1, 4, 7 and 8, which is four digits; and because only the union matters, not how many candidates each cell has.
A worked example
Follow the diagram. Row 1 has several empty cells. Three of them, r1c2, r1c3 and r1c7, list candidates drawn only from 3, 5 and 6. They need not match: perhaps r1c2 holds 3/5, r1c3 holds 5/6 and r1c7 holds 3/6, or any other mix with the same union. The row must contain a 3, a 5 and a 6, and these three cells are the only ones allowed to take them, because no other digit is possible in those cells. All three digits are therefore spoken for. Every other empty cell in row 1 can lose any 3, 5 or 6 it still lists. After the cleanup, check the row for new singles: a cell that was down to 3/9 and has lost its 3 is now a naked single 9.
How the hint explains it
If a naked triple is the cheapest step on your board, the level-three hint spells out the three cells, the three digits and the group, in the same words as the diagram caption, and lists the cells that lose a candidate. The engine works out candidates from the digits you have placed, so it needs no notes from you. It tests pairs before triples, so a triple is only shown when no pair or pointing pattern is available.
Practice
Play a Hard puzzle on the main board with notes on. If you are stuck, use the level-two hint to see whether a triple is available. The daily puzzle varies in difficulty across the week.
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