When three digits in a house can only go in the same three cells, those three cells hold those three digits, so every other candidate in them can be crossed out. No digit has to be able to go in all three.
How to spot it
A hidden pair is two digits that can only go in the same two cells. The obvious way to grow it is three digits that can each go in the same three cells, and that does happen. It is only one shape of several, though, and a reader who looks only for it walks past the rest.
The rule is about the cells the three digits use between them. Take three digits a house is still missing and mark every cell where any of them can go. If that comes to three cells, the digits own those cells. A digit can have two of the three cells, or all three. What matters is the total.
Looking at the cells will not show it. Each can carry four or five pencil marks, and nothing about it looks short. So you count from the digits' side. Pick a house, find the digits with only two or three homes, and ask whether three of them share three cells between them.
Read row nine digit by digit
Row nine has seven empty cells and is missing seven digits. Take them one at a time and list where each can go. The 2 can only go in the first cell or the fourth. The 4 can only go in the second or the fourth. The 9 can only go in the first or the second. The other four, 1, 3, 6 and 7, have four homes or more each. No digit is down to one home, and no two share the same two, so neither a single nor a hidden pair is here.
Count the cells they use between them
Put the three digits' homes together and they come to three cells: the first, the second and the fourth. Not one of the digits can go in all three. Each has two homes, and a different two each time. That is still a hidden triple, and each of its cells shows four candidates, so nothing about them looks special.
Try it both ways round
If the first cell is the 2, the fourth must be the 4 and the second the 9. If the first cell is the 9, the second must be the 4 and the fourth the 2. Now try anything else in the first cell, say its 6. The 9 would need the second cell and the 2 the fourth, and the 4 would have nowhere left to go. So the three cells take the 2, the 4 and the 9, and nothing else fits in them.
Cross out everything else in the three cells
The three cells are spoken for, so their other marks are dead. Six go: the 6 and the 7 from the first cell, the same two from the second, and the 1 and the 3 from the fourth. The other four cells of the row are left alone. That is the opposite of a naked triple, where the strike lands outside the pattern.
Read what the strike left behind
The 6 that came out of the second cell of row nine was one of only two places column two had for a 6. The other is the second cell of row three. Now it is the only one, so it is a 6: a hidden single in the column. Before the strike this grid had no single of either kind anywhere. The triple is what moved it.
Any mix of two and three
The three digits in row nine each had two homes. Other triples mix a digit with three homes and digits with two, or let all three digits use all three cells. The strike is the same every time, because the reasoning never looked at any one digit. It only counted cells across the three.
Look at the three cells after the strike. They read 2 and 9, 4 and 9, and 2 and 4: three cells with three digits between them, and no cell holding all three. That is a naked triple, in exactly the shape the last lesson opened with. A hidden triple, once struck, always leaves a naked one behind.
Three digits, four cells
Box seven, at the bottom left, has a set that looks just like row nine's, and it uses the same three digits. The 2 can go in the first cell of row seven or the first cell of row nine. The 4 can go in the third cell of row seven or the second cell of row nine. The 9 can go in the first or the second cell of row nine. Three digits, two homes each.
Count the cells, though, and there are four. Three digits cannot fill four cells, so one of them will hold something else, and these homes claim nothing. Strike the other marks from all four cells anyway and the 6 and the 7 leave them. That leaves the first cell of row eight as the only home in the box for both the 6 and the 7, and one cell cannot hold two digits. On this grid the spare cell is the third cell of row seven, and it is the 6.
Before you strike, list every cell the three digits can use and count them. The count is the whole test.
2, 4 and 9 have three cells between them. Those cells lose everything else.
2, 4 and 9 spread over four cells. Nothing can be struck.
Two readings of one lock
Every hidden triple has a naked set on the other side of its house. The three digits cannot go anywhere else, so the house's other empty cells hold only the digits that are left. Row nine has seven empty cells. The triple takes three, and the other four hold 1, 3, 6 and 7 between them. That is a naked quad, and it strikes the same six marks. Four cells with three or four marks each are harder to see than three digits with two homes each, which is why the hidden reading is the one to use here.
In a house with six empty cells the two sides are the same size, so a naked triple and a hidden triple always come together. The grid from the naked triple lesson shows it. Its row five had a naked triple in the first three cells: 5 and 9, 5 and 7, 7 and 9. The other three cells, the fourth, the seventh and the last, are the only homes in the row for 1, 4 and 8. That is a hidden triple, and it strikes the same marks: the 9 from the fourth cell, and the 5 and the 7 from the last. One lock, read from either end.
Three cells, three digits between them. The rest of the row loses 5, 7 and 9.
Three digits, three cells between them. Those cells lose everything else. The same three marks go.
Four digits and four cells
A hidden quad is the same rule once more: four digits in a house whose homes come to only four cells between them. Every other candidate leaves those four cells. The digits can have two, three or four homes each, in any mix, exactly as with the triple. There is nothing new to learn for it.
It also has a naked set on the other side, and that side is often smaller. In a house with seven empty cells the other side of a hidden quad is a naked triple, and with six it is a naked pair. A hidden quad is only worth hunting by name when a house has eight or nine empty cells.
Row nine has one after the strike. Its fifth, sixth, eighth and last cells are now the only homes for 1, 3, 6 and 7. It removes nothing, because those four cells never showed anything else. Before the strike the 6 and the 7 could still go in the first two cells, and the 1 and the 3 in the fourth, so the quad was not there yet. The triple made it.
Count the cells across all the digits together. How many homes any one digit has does not decide it. Two digits in two cells make a pair, three in three make a triple, four in four make a quad. One cell too many and there is nothing to strike.
Where this sits on the path
Like the hidden pair, the triple places nothing. It removes candidates, and the placing is left to the singles. On the grid above the strike placed nothing by itself. It made the 6 in column two a single.
When the three cells share a box as well as a line, check both houses. The question below is about a triple that does.
With the quad folded in, this closes the subsets: pairs, triples and quads, naked and hidden, are one idea read from two ends. Next on the path is the X-wing, which locks a single digit across two rows and two columns.
Box one has eight empty cells. Which candidates survive in the first cell of row three?
Take the digits with the fewest homes in the box, and count the cells they use between them.
Recap
- Look for
- Three digits in the same house whose homes come to only three cells between them. Each digit can have two of those cells or all three.
- Scope
- Any house: box, row or column. In a row or column the three cells can sit far apart.
- Result
- Every other candidate comes out of those three cells. The rest of the house is left untouched.
- Tell-tale
- A house with several digits down to two or three homes. Pick three and count the cells they use.
- If it fails
- If the three digits use four cells between them, they claim nothing. Try a different set of three, or add the digit that shares the fourth cell and look for a quad. If the house has only a few empty cells, read it from the cells' side instead, as a naked triple does.