Naked Pair




The Naked Pair technique extends the Naked Single technique, just like the Hidden Pair technique extends the Hidden Single technique.

To recap: this is a Naked Single.

The only valid cell for a 5 is the one in the center within the center box.

This is what a Naked Pair would look like.

Since the two empty cells see all numbers but 5 and 6, 5 and 6 can only be in those 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.

That is always the case for non-basic techniques. With them, we never get a solution directly but only information on what candidates can be eliminated. In this case we can eliminate all candidates 5 and 6 from all regions that these cells share. (see next section)

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.)

How does finding a Naked Pair help us?

The Naked Pair tells us something about the regions that it is in.

This pair is in two regions.

Both the cells of the pair are in the box (which does not help us, because there are no candidates that we can eliminate)...

... but they are also both in this row.

In the row we can now eliminate all candidates of that pair in all the other cells.

This candidate elimination is the actual result of the Naked Pair technique.

Keep in mind, that there is almost no difference between Hidden Pairs and Naked Pairs. They are both just pairs. "Naked" and "Hidden" only refers to how those pairs are found. A Naked Pair is used to eliminate candidates in all other cells of that region. A Hidden Pair is made into a Naked Pair, by eliminating candidates in the pair's cells.

Pairs are simply Groups, which are explained on the next page.

Combination of regions

Finding Naked Pairs could, of course, also need a combination of regions (rows, columns, boxes).

With Naked Singles we can always use a combination of one box, one row and one column.

See how the intersection of the column, row and box is exactly one cell? That one cell sees all numbers except 9, so it must be a 9

For Naked Pairs it is still possible that all cells of that pair see all the same cells, like in this example.

Notice how we can only have an intersection between box and column here, because the intersection of a row and a column is only one cell (which is not enough to hold a pair).

It is not necessary, though, that each cell of the pair sees the exact same cells.

All that matters is that they see the same numbers, regardless of where those numbers are.

So this example also reveals a Naked Pair.

Both red cells see the numbers 1, 2, 3, 4, 5, 6 and 7. That leaves only the numbers 8 and 9 for the two red cells.

The result of that Naked Pair is, again, that we can eliminate the Naked Pair's candidates from all other cells in that region...

... resulting in another Pair: (3,4).

By the way, that (3,4)-Pair is a Hidden Pair.

When and how to look for Naked Pairs

We look for Naked Pairs after we cannot find any singles. We can, of course, also look for them while looking for singles.

Spotting Naked Pairs is not that difficult, because they are so clearly visible when notating cell candidates (which we do anyway, or at least should do).

If you see two cells in the same region with the same two candidates, you have spotted a Naked Pair.

Here is an example.

At this point we unfortunately don't see any more singles.

So after realizing that there are no singles left, we start to notate cell candidates. We can, of course, also do this while looking for singles.

As always, we look at the most restricted regions to try to spot something.

In this example, we find:

  • two columns with three empty cells each
  • one row with only two empty cells
  • one row with three empty cells
  • one box with three empty cells (at the top, not colored)

Of course, all those Naked Groups are not useful, because the rest of each region is already completely filled with fixed numbers.

What we have now, though, is a good starting point for finding further Hidden Pairs.

Since there is nothing immediately visible, we look further for cells that have only very few candidates.

And voilà, there are some that we still have not notated.

There is now a Naked Pair in the grid. Can you spot it?

It is this one.

Since we know that these two numbers (5 and 6) must be in these two cells, we know that all the other cells in that row cannot be those two numbers.

After finding the Naked Pair, our grid now looks like this.

Why easy?

Earlier I said that it is easy to find Naked Pairs, and yet here we are having half the board filled with cell candidates...

While it is true that it can be cumbersome to find Naked Pairs, there is really nothing very complicated going on here. All we do is

  • check the regions with few empty cells,
  • notate their cells' candidates,
  • look for pairs. Those pairs are very obvious, if you don't overdo it with pencil marks. A good rule of thumb is to only pencil mark those cells that do have two or three candidates.

Sure, in the process of this example we notate far too much. But those notations don't hurt, either. They will most probably be helpful down the road.