Hidden Singles

It helps to hold two ideas side by side. A naked single is about a cell: you look at one empty square, count its candidates, and find only one number can go there. Everything else is ruled out, so that number drops in. A hidden single flips the question. Instead of asking what a cell can hold, you ask where a digit can go within a whole unit — and discover there is exactly one square left for it.

The word hidden matters. The cell that receives the digit might still, on paper, accept two or three other numbers. If you were only reading that cell's candidate list, nothing would stand out. The digit is hidden among its neighbours. But once you look at the entire row (or column, or box) and see that every other empty cell is blocked for that digit, the single home reveals itself. The other candidates in that cell become irrelevant — the digit has nowhere else to be, so it must go there.

This is why hidden singles feel almost like a trick the first few times. You place a number in a cell that clearly could have taken others, and it is still certain. The certainty comes from the unit, not the cell.

The technique is cross-hatching, and you can do it without writing a single pencil mark. Pick a digit — say 7 — and pick a box. Ask a narrow question: within this box, where can a 7 legally go? Then let the sevens already on the board do the ruling out. Every row that already contains a 7 sends a line across the box, striking out its three cells in that row. Every column with a 7 strikes out its column. Whatever cells survive are the only homes 7 has in that box.

Often several cells survive and you learn nothing yet. But now and then the lines converge and only one empty cell is left standing. That is a hidden single, and the 7 goes in. Work box by box, digit by digit, and you sweep up placements in a rhythm.

The same sweep works along a row or a column. Take a row that is already fairly full, note which digit is missing, and check each empty cell against the columns and boxes crossing it. If every gap but one is blocked for that digit, you have found its single home. As ever, hunt the most constrained unit — the row, column, or box that is nearly complete — because that is where the lines are dense enough to pin a digit down.

Look at the grid above. We are working row 5, and the digit we care about is 7. Row 5 has several empty cells, so at first glance 7 could seem to have a choice of homes. Watch what the rest of the board does to those options.

The empty cells of row 5, apart from the gold one, are outlined in blue. Take each in turn. Wherever a 7 already sits in the same column as one of those cells, that column reaches up or down and blocks it — those clashes are shaded red and struck through. Wherever a 7 already sits in the same box, the box blocks it too, again shown in red. One by one the blue cells fall away: this one shares a column with a 7, that one shares a box with a 7, the next shares another column. Every candidate home for 7 in row 5 is eliminated — except one.

The survivor is r5c6, shaded gold. No 7 appears in its column, and no 7 appears in its box, so nothing rules it out. Since 7 has to appear somewhere in row 5, and every other cell is blocked, r5c6 is the only place left. The 7 goes there. Notice that r5c6 may still list other candidates of its own — but they no longer matter. The row has spoken.

It is worth being clear about why this is a deduction and not a guess. Every unit in Sudoku must contain each of the digits 1 through 9 exactly once — that is the whole rule. So row 5 must contain a 7. There is no version of the finished grid where it does not. The only question is which cell holds it.

By cross-hatching we showed that every empty cell in row 5 except r5c6 already has a 7 barring the way, whether from its column or its box. Those cells are genuinely impossible — placing a 7 there would repeat a 7 somewhere else. That leaves r5c6 as the sole remaining candidate, and since the row cannot do without a 7, the digit is forced into it. No trial, no backtracking, no assumption. The logic runs one direction only, and it does not leak.

This is the quiet strength of hidden singles: each one is a fully proven step. String enough of them together and an easy puzzle unwinds on its own, with never a moment where you have to hope you chose right.

When you sit down to a puzzle, reach for hidden singles before anything fancier. Sweep the low-count digits first — the ones that already appear five or six times, because their lines cut across the most boxes and pin things down fastest. For each such digit, run through the boxes and see where it is forced. You will often clear a good number of cells this way before touching pencil marks at all.

A common slip is to stop reading a cell because it looks busy. You glance at a square, see it could take a 3, a 7, or an 8, and move on — missing that the 7 has nowhere else to live in that box. Train yourself to ask the unit's question, not the cell's. The busy cell is exactly where hidden singles like to hide.

Practise the sweep until it becomes automatic: pick a digit, pick a nearly full unit, let the existing copies strike out the impossible cells, and see what is left. Do it calmly and in order rather than darting about the grid. Most of the time you find nothing and move on — but the finds come often enough, and cost so little, that no other technique earns its keep so cheaply.