Hidden triple: three digits squeezed into three cells

What is a hidden triple?

A hidden triple is three digits that, inside one row, column or 3×3 box, appear as candidates only in the same three cells. Each digit must appear in at least two of the three cells, but it does not need to appear in all three. The cells can hold other candidates too.

Once you find it, remove all candidates other than the three digits from those three cells. The rest of the unit is not affected, since the three digits already appear nowhere else in it.

Why it works

Each of the three digits must be placed somewhere in the unit, and all three are limited to the same three cells. Three digits need three different cells, and there are exactly three to go around, so every one of those cells is filled by one of the three digits. No cell in the group has room for anything else, which makes every extra candidate impossible.

How to find hidden triples

Pick a unit and, for each unplaced digit, write down the cells where it can still go. Ignore digits that have four or more possible cells; they cannot be part of a hidden triple. Keep the digits that have two or three possible cells.

Now look for three of those digits whose positions, combined, cover exactly three cells. For example, digit 2 in cells A and B, digit 3 in A, B and C, and digit 4 in A and C. Combined they use only A, B and C, which is a hidden triple.

Boxes with many candidates are the best hunting ground. If a box has two or three cells with long candidate lists while the other cells are short, check which digits are rare in the box.

A worked example

In the example board, singles, locked candidates, pairs and naked triples have run out. Look at box 7, the bottom-left box. Its empty cells are R7C1 (row 7, column 1) {5,7,9}, R7C3 {7,9}, R8C1 {2,3,4,5,9}, R8C2 {5,8,9}, R8C3 {2,3,9}, R9C1 {3,4,5,7} and R9C2 {5,8}.

Track the digits 2, 3 and 4. The 2 can only go in R8C1 or R8C3. The 3 can go in R8C1, R8C3 or R9C1. The 4 can go in R8C1 or R9C1. All three are confined to the highlighted cells R8C1, R8C3 and R9C1, so these cells form a hidden triple {2,3,4}. We can remove 5 and 9 from R8C1, 9 from R8C3, and 5 and 7 from R9C1.

The useful removal is the 7 in R9C1. In row 9, the only cells that could take a 7 were R9C1 and R9C4. With R9C1 now limited to {3,4}, R9C4 is the only place for 7 in row 9. That hidden single is shown in gold.

Common mistakes

Requiring each digit in all three cells. In the example, 2 and 4 each appear in only two of the three cells. What matters is that the combined positions of the three digits are exactly three cells.

Including a digit that has a fourth position. If the 4 could also go in R7C3, the positions would cover four cells and the triple would be gone. Check each digit against the whole unit, not just the cells you are looking at.

Removing the triple digits from the rest of the unit. That is what a naked triple does. A hidden triple only cleans up the three cells it lives in.

Hidden triples and related techniques

A hidden triple is the three-digit version of the hidden pair, and the same pattern extends to hidden quads. It is also the mirror of the naked triple: whenever a unit contains a hidden triple, its remaining empty cells form a naked subset of the remaining digits. In the example, box 7 has seven empty cells, so the other four cells hold a naked quad of {5,7,8,9}. The hidden view is far easier to see here.

On Quietdoku, hidden triples are part of the Hard toolkit, together with naked pairs, hidden pairs and naked triples. When one is the next logical step, a hint will point to it and name the technique.