Hidden Pairs: A Deep Dive

Hidden pairs are the inverse of naked pairs and they are easier to miss. The whole technique flips the question: instead of asking what cells have, you ask what digits have to go where. Most beginners cannot find them not because the technique is hard but because the scanning style is unfamiliar.

Once the scanning becomes automatic, hidden pairs surface a few times in every hard puzzle. They unlock placements that naked pairs cannot reach.

Why the cells get cleaned

The unit has to contain all nine digits. If X and Y can only live in cells A and B, then A and B together hold X and Y. The other seven digits live elsewhere in the unit. The cells A and B therefore cannot contain anything except X and Y, even if their listed candidates currently include other digits.

That last sentence is the whole technique. The listed candidates in A and B will often look richer than {X, Y}. The point of spotting the hidden pair is to strip them down.

How to scan for one

Pick a unit, preferably a box. Walk each digit one at a time. For each digit, count how many cells inside the unit still list it as a candidate. Most digits will have many. Some will have only two or three. Those few are your candidates for being half of a hidden pair.

When you find two digits that each have exactly the same two candidate cells in the unit, you have a hidden pair. Strip the other candidates from those two cells. Then re-scan for singles, which often appear immediately.

r1c1 {1,3,7,8}   r1c2 {1,5,9}     r1c3 {3,5}
r2c1 {1,7,8,9}    r2c2 {7,8,9}     r2c3 {1,5}
r3c1 {2,3}       r3c2 {2,6,9}     r3c3 {3,5,6,9}

Scan digit 7. Its candidate cells in this box are r1c1, r2c1, r2c2. Three places. Move on. Scan digit 8. Its cells are r1c1, r2c1, r2c2. Same three. Move on. Scan digit 9. Cells r1c2, r2c1, r2c2, r3c2, r3c3. Five places, no help.

Now scan digit 6. Cells r3c2, r3c3. Two places. Scan digit 9 again, then any digit with only two candidate locations. After a few digits you spot that, say, 6 and 9 both restrict to {r3c2, r3c3}. That is a hidden pair. Now r3c2 must be 6 or 9, even though it currently shows {2, 6, 9}. The 2 can be erased. Same for r3c3.

Why they are called hidden

Naked pairs announce themselves visually. Two cells with the same two-candidate set stand out on the page. Hidden pairs do not. The two cells that form a hidden pair often have busier candidate lists than their neighbours. You have to count digit positions to find them, which is mentally different from scanning cells.

When to look

Hidden pairs are most common in boxes that are about half full. Once a box has six placements, candidates are sparse and pairs consolidate naturally. Once a box has only two placements, candidates are too rich for hidden pairs to emerge.

Rows and columns are leaner hunting grounds, but they do happen. The longer the unit's empty span, the rarer the pair.

The mistake to avoid

After finding a hidden pair, the temptation is to do nothing because the digits are already accounted for. That misses the point. The pair tells you that the two cells contain ONLY X and Y. The non-X, non-Y candidates in those cells are wrong. Erase them and re-scan.

For practice, try a hard puzzle and once you are stuck on pencil-marked singles, scan digit by digit for a hidden pair. The technique is faster than X-wing and almost always cracks the stuck moment open. The formal page is at hidden pairs.