Play this puzzle first. It is the hard daily from Friday 11 September, rebuilt from its date.
The starting grid
Daily #3 as served: 26 clues, 55 empty cells.
This one is unusual for a hard grid. It opens with thirteen naked singles in a row, down the left-hand side of the grid, where the clues are dense enough that cell after cell has only one candidate left. Two hidden singles follow. At that point 40 cells are still empty and nothing single is left.
Move 16: the hidden pair
Move 16 of 55. 6 and 9 fit only in R5C5 and R5C8 in row 5, so those two cells hold nothing else. That leaves R5C2 = 4. Red: the two cells of the pair, which lose their other candidates.
Take row 5 and ask, for each missing digit, where it can go. The 6 has only two possible cells, R5C5 and R5C8. So does the 9: the same two cells. Two digits, two cells — those two cells must hold the 6 and the 9 between them, in some order.
That means nothing else can go there. R5C5 had five candidates, 4, 5, 6, 7 and 9, and drops to 6 and 9. R5C8 had 6, 8 and 9, and drops to 6 and 9. The pair was invisible as written because both cells were crowded; it only shows up when you count by digit instead of by cell. That is what makes it hidden.
Removing the 4 from R5C5 matters straight away: R5C2 is now the only cell in row 5 that can take a 4. From there the grid runs out on singles again, with one more hidden single along the way.
How you could have found it
- Pick the row, column or box with the most empty cells that has stalled, and for each missing digit write down which cells can take it.
- Look for two digits with the same two cells. In row 5 here, 6 and 9 both had exactly R5C5 and R5C8.
- Then clean those two cells. The eliminations happen inside the pair, which is the opposite of a naked pair.
The whole path at a glance
| Technique | Moves |
|---|---|
| Naked single | 51 |
| Hidden single | 3 |
| Hidden pair | 1 |
| Total moves | 55 |
Fifty-four singles and one hidden pair. Pairs are uncommon in hard grids here — about two in a hundred need one — which made this a good puzzle to write up. The hidden pairs guide has a second example and explains how to tell the two kinds of pair apart.