Hidden Triple

Expert

Description

Choose a row, column or 3x3 block. If three missing digits have all their possible positions within just three cells, those cells form a Hidden Triple. They must contain the three digits, one in each cell, so remove every other candidate from those three cells.

The group needs all three digits, and none can go elsewhere in it. That reserves the three cells even though their order is still unknown. As with a Hidden Pair, extra notes can obscure the pattern. You are looking at where the digits can go, rather than looking for small candidate lists.

Each digit need not appear in all three cells. For example, one may occur in the first and second cells, another in the first and third, and the last in the second and third. What matters is that all possible positions for the three digits, taken together, fit into three cells in one group. Keep the existing candidates for those digits; do not add a missing candidate to make the lists match.

A Hidden Triple eliminates candidates. It may reveal a Hidden Single or a Naked Single, but further eliminations may be needed before any cell can be filled. The demo and the third practice puzzle show such continuations.

We use pencil notes throughout. Green candidates stay and red candidates are removed. Yellow highlights show every other occurrence of the selected digits, and the group being discussed is shaded.

Example 1. A Hidden Triple reveals a single

In row 2, every 4, 5 and 9 candidate lies at columns 1, 4 and 6. Keep those green notes. Remove the red 3 and 6 at column 1, and the red 6 and 7 at column 4. — Check row 2 (shaded). The only positions for 4, 5 and 9 are columns 1, 4 and 6. Not every cell allows all three digits: column 1 has 5/9, column 4 has 4/5/9, and column 6 has 4/5, among their notes. Keep the green candidates and remove the four red notes.

After the triple removes 7 from row 2, column 4, block 2 has only one 7 candidate: the green 7 at row 1, column 4. Every other 7 candidate is highlighted. — Now check block 2 (shaded). Removing 7 from row 2, column 4 leaves just one place for 7 in the block: row 1, column 4. That cell still has other notes, so this is a Hidden Single. This example gives an immediate placement.

Example 2. Two views of the same elimination

These pictures show the same position before any notes are removed. A hidden subset and a naked subset can describe the same restriction. Understanding this also helps when Hint chooses a different name for your deduction.

In column 7, digits 1, 4 and 6 can go only at rows 3, 7 and 8. Their green candidates form a Hidden Triple. Remove the red 3s and 9s at rows 3 and 7, and the red 3 and 8 at row 8. — In column 7 (shaded), 1 can go only at rows 3 and 8, 4 only at rows 7 and 8, and 6 only at rows 3 and 7. Together they occupy three cells. Reserve those cells for 1/4/6 and remove the six red notes from them.

In the same unchanged column 7, the other four empty cells at rows 2, 4, 5 and 6 allow only 3, 5, 8 and 9. This Naked Quadruple removes exactly the same six red candidates from rows 3, 7 and 8. — The other four empty cells, at rows 2, 4, 5 and 6, allow only 3/5/8/9 between them. This is a Naked Quadruple: the same idea as a Naked Triple, with four cells and four digits. It removes exactly the same red notes. These are alternative explanations, not two successive moves.

Compare the directions: the Hidden Triple removes other digits from its own cells; the naked subset removes its digits from other cells. Either view can establish the same eliminations. Apply them once, then inspect the updated notes.

Demo

Click on arrow buttons or use arrow keys on your keyboard to navigate between slides. Tap on arrow buttons to navigate between slides.

How to find Hidden Triples

Use up-to-date pencil notes. First check for singles, Pointing, Claiming and pairs.
Choose one row, column or block. Look for missing digits with only two or three possible positions there.
Try combining three such digits. Count the different cells containing any of them across the whole group.
If their positions occupy exactly three cells altogether, reserve those cells for the three digits.
Keep the existing candidates for those digits and erase all other notes from the three cells. At least one note must be removed to make progress.
Inspect the changed notes. Look for singles, then continue with other eliminations if needed.

A digit with just one possible position is already a Hidden Single. Likewise, a Hidden Pair or Naked Pair may provide an easier deduction. In our prepared examples, those earlier techniques and Naked Triples have already been exhausted.

Check every possible position

Seeing three digits in three chosen cells is not enough. None of the three digits may appear as a candidate in another cell of that group. Even one extra position can break the restriction. Check the entire row, column or block before removing anything.

All three cells must share the group you checked. They need not be adjacent, and they need not have identical notes. Other candidates in the three cells are precisely what the Hidden Triple removes. Occurrences of the chosen digits elsewhere on the grid are allowed; the triple's restriction applies to the selected group.

Do not erase the triple's three digits from its cells, or remove arbitrary notes from other cells. If the three cells already contain only those digits, this Hidden Triple has nothing left to eliminate there. You may be able to use those cells as a naked subset in a shared group.

Hidden and naked subsets

A Hidden Pair confines two digits to two cells; a Hidden Triple confines three digits to three cells. Both remove other digits from the subset's own cells. The triple's digits need not all occur in every one of its cells.

A Naked Triple starts with three cells whose complete candidate lists use only three digits. It removes those digits from other cells in a shared group. After a Hidden Triple cleans up its own cells, those cells have small lists too, but that does not automatically give another elimination in the same group.

Complementary subsets use the remaining empty cells in a group to express the same restriction. With five empty cells, a Hidden Triple has a complementary Naked Pair; with six, a Naked Triple; with seven, a Naked Quadruple. Filled cells are not counted. The second illustration shows the seven-cell case. Choose the view you find easier to recognize.

A pointing triple is different: it concerns one digit in three cells of a block, all in one row or column. See Pointing Candidates for that intersection rule. A Hidden Triple concerns three different digits.

Try it yourself

These prepared positions come from our puzzle library. The links preserve the notes left by earlier deductions. No single, Pointing, Claiming, pair or Naked Triple is available at the start. Find three confined digits and remove other candidates from their cells.

Sudoku #1 — find a Hidden Triple in a column, then look for a single.
Sudoku #2 — find a Hidden Triple in a row, then inspect the changed notes.
Sudoku #3 — find a Hidden Triple in a column, then continue eliminating candidates.

In these exercises, Hint shows the equivalent Naked Quadruple. Its red notes are exactly the ones removed by the Hidden Triple, as in the second illustration. Compare the eliminations, apply the hint once, then request the next hint. Both explanations are valid.

The first two exercises reveal a Hidden Single immediately. The third needs Claiming Candidates and a Naked Pair before a Naked Single appears. These are selected continuations, not a fixed sequence for Hidden Triples.

Hidden Triple is an Expert technique and remains available in Extreme games. Both levels use pencil notes for these hints. All three practice links open directly in pencil mode.