Hidden Pair




The Hidden Pair technique extends the Hidden Single technique in such a way, that it works with pairs of numbers instead of singles.

To recap, this is a Hidden Single.

The only valid cell for a 5 is the one at the top right (within the center box).

This is what a Hidden Pair would look like.

The 5s and 6s cannot be in any of the colored cells. So they must be in the other two cells.

We do not know, though, in what exact cells those two numbers are each.

All we can do here is notate the candidates.

This is generally very common for non-basic techniques. With them, we oftentimes don't get a solution directly but only information on what candidates can be eliminated. In this case we eliminate all candidates except 5 and 6 from those two cells.

Some later clue will eventually give us the info, which one of those two outcomes it will be. (But for now, we do not know.)

Also columns and rows

This, of course, also works with rows and columns.

Here the 6 and 8 cannot be in the left three cells, so they must go in the other two empty ones within that row.

What is not a Hidden Pair

It is extremely important that you make sure that there are indeed only two cells where two candidates may go.

This, for example, is not a Hidden Pair.

The two (5,6)-pairs rule out the first two columns' cells from being 5s and 6s.

So we are able to pin down the 5 and 6 to three cells, and yet, pinning down two numbers to three cells is far less useful than pinning down two numbers to two cells.

Any of those three cells can still be any other number, while with an actual (Hidden) Pair, other numbers are completely ruled out from the two cells.

See what a difference this one missing 6 makes?

The strength of a found pair is not only the knowledge that the numbers are in those cells, but also that all other numbers cannot be there and are therefore pinned down to fewer cells themselves.

We will explorer that a bit more in Groups (pairs, triples, etc.)

What if there are too few empty cells?

In case you find something like this, where two numbers have to be divided among one cell, there is obviously an error in the Sudoku. So when something like this occurs, go back and see if you have made any mistakes with previous steps.

When and how to look for Hidden Pairs

Hidden Pairs can be relatively easy to spot if you have the right approach.

What you can do is combine the search for Hidden Singles with the search for Hidden Pairs (and even bigger Hidden Groups)

What you generally prefer, when looking for Hidden Groups (Singles, Pairs, Triples, etc.), is to

  • prioritize regions with only few empty cells and (if this does not yield something)
  • prioritize regions where many cells can be ruled out at once.

In this example, we might notice that a single cell within the center row can rule out three cells in the left box at once.

Our thought process could be something like this:

Let us see if there is anything in the left box. Maybe we will find a Hidden Single.

There are many numbers in the center row that interact with that box, so let us look at those. Nice, the 7 already rules out three cells. Only two more cells ruled out and we have a Single.

Can we rule out the other cells somehow, too? Let us look at all regions that see that box, i.e. columns 1-3 and rows 4 and 6.

The columns don't yield anything, but there is another 7 in the fourth row, which reduces the number of unblocked cells in the box by one. There are now only two unblocked cells...

... which is one cell too many for a Hidden Single, but there is still a possibility for a Hidden Pair. So we quickly have a look at the rows where we found the 7s and look for other numbers, that are not already in the box.

9 is already in the box, so we ignore that. The next number we look at is the 5. If all rows with a 7 seeing the box also contain some other number that is not in the box, we have found a pair.

And bingo, the 5 is in both those rows. So we now have two numbers (5 and 7) that must be in the same two cells.

So we have successfully found a Hidden Pair, and can notate it in the grid.

For more info on how such a pair helps us, have a look at Groups (pairs, triples, etc.).

When viewed from cell candidates' perspective

If you have already entered several cell candidates, or in the case of the Solver, where all cell candidates are always visible, Hidden Pairs can also be spotted.

Spotting Hidden Groups via cell candidates can be much harder than via fixed numbers, but with harder Sudokus there is no way around it.

In this example we can see that there are only two cells within the row that have the cell candidates 6 and 9.

This is a Hidden Pair, and so we can rule out all other candidates from these two cells.