Hidden triples
When three digits can only go in the same three cells of a row, column or box, all other candidates can be cleared from those cells.
What it is
A hidden triple is three digits that, within one row, column or box, can only go in the same three cells. Those cells may have extra candidates noted, which hide the pattern. But because the three digits need three homes, and only these three cells are available, the cells must hold those three digits. Any other candidates in them can be removed.
It is the hidden pair idea scaled up by one. As with naked triples, not every digit needs to appear in all three cells. Each digit can appear in two or three of them. What matters is that none of the three digits appears anywhere else in the unit.
How to spot it
Hidden triples are among the hardest patterns to see, because the cells usually look cluttered. The best approach is to count digits rather than look at cells:
- Pick a row, column or box with plenty of empty cells.
- For each missing digit, note which cells it could go in.
- Look for three digits whose possible cells, put together, are just three cells.
Digits that only have two or three possible cells in a unit are the ones to focus on.
A worked example
Look at column 4. It has six empty cells, in rows 1, 2, 5, 6, 8 and 9.
When you check where each missing digit can go, you find:
- 2 can only go in row 1 or row 5.
- 5 can only go in row 1 or row 8.
- 7 can only go in row 5 or row 8.
So the digits 2, 5 and 7 are confined to three cells: row 1, column 4; row 5, column 4; and row 8, column 4. That is a hidden triple.
Your notes for those cells were:
- Row 1, column 4: 1, 2, 5 and 9.
- Row 5, column 4: 2, 7 and 8.
- Row 8, column 4: 3, 5 and 7.
Remove everything except 2, 5 and 7. Row 1, column 4 becomes 2 and 5. Row 5, column 4 becomes 2 and 7. Row 8, column 4 becomes 5 and 7.
The 1, 9, 8 and 3 you removed now have fewer places to go in column 4. One of them may be left with a single possible cell, which would be a hidden single. The cleaned-up cells may also help with eliminations in their rows and boxes.
Why it works
Column 4 needs a 2, a 5 and a 7, and they can only go in these three cells. If one of those cells took a 9, say, there would be just two cells left for three digits, which is impossible. So none of the three cells can hold anything other than 2, 5 or 7.
A common mistake
The usual mistake is removing the wrong candidates. With a hidden triple, you clear extra digits from the three triple cells. You do not remove 2, 5 or 7 from anywhere else, because by definition they do not appear anywhere else in the unit.
Another slip is missing a cell. If 7 could also go in row 6, column 4, the three digits now spread over four cells and the triple is gone. Check every empty cell in the unit before making removals.
When you will need it
Hidden triples belong to the Hard toolkit on sudoku.ie, with naked triples, quads and the X-wing. They turn up on Thursday and Friday puzzles, and on Saturday’s Expert. They are rarer than naked triples, but when the grid seems stuck and the notes look crowded, they are well worth hunting for.