Naked and Hidden Singles

Naked and hidden singles are the two ways a sudoku puzzle hands you a digit for free, and together they place more cells than every other technique combined. A candidate is a digit still legal in a cell. A unit is one row, one column or one box. Singles come from counting candidates and nothing else.

What the pattern is

A naked single is a cell with exactly one candidate left. The other eight digits are blocked by a solved peer in the same row, column or box, so the survivor has to go in. You read it straight off the cell.

A hidden single is a digit with exactly one legal cell in a unit. That cell may still carry several candidates, which is why the single is hidden: you find it by scanning a unit for a digit, not by scanning a cell for digits.

Both give you a placement rather than an elimination, which makes them the cheapest moves on the board.

When to look for it

• Right after every placement. A new digit strips candidates from up to twenty peers and usually creates a single next door.
• In units that already hold five or more digits.
• Whenever you feel stuck. Most "impossible" positions hide a single nobody scanned for.

On an easy grid hidden singles tend to appear first, because a box with six givens blocks a digit from nearly every empty cell.

How to apply it, step by step

1. Pick a unit, starting with the fullest box.
2. List the digits still missing from it.
3. For each missing digit, mark the cells of the unit where it is still legal.
4. If exactly one cell survives, write the digit in. That is a hidden single.
5. Now look at cells rather than digits: any cell down to one candidate is a naked single.
6. After each placement, strike that digit from the row, column and box of the filled cell, then go back to step 1.

A worked example

A naked single first. Watch D6: the board shows one candidate, and the counting explains why.

123456789

ABCDEFGHJ

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Step 0 of 5

1. Easier steps are used up. Look at D6: it shows a single candidate.

2. 1

   D6 is empty. Check which digits its row, column and box already hold.

3. 2

   Row D already has 8, 4, 7, 9 and 2 (D1, D3, D4, D7 and D8).

4. 3

   Column 6 adds 5 (B6) and 6 (J6).

5. 4

   Box 5 adds 3 (E5). Eight digits are taken, so 1 is the only candidate left for D6.

6. 5

   1 placed in D6.

   1 removed from C6, D5 and E6, which share a row, column or box with D6.

Easier steps are used up. Look at D6: it shows a single candidate.

Row D already holds 8, 4, 7, 9 and 2. Column 6 adds 5 and 6, and box 5 adds 3. That is eight digits, so 1 is the only one left and D6 must be 1. No single house rules out all eight, which is why you check the row, the column and the box together.

Now a hidden single. Watch the 6 in box 1.

123456789

ABCDEFGHJ

3

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Step 0 of 5

1. Look at box 1: it still needs a 6.

2. 1

   Box 1 has eight empty cells and still needs a 6. The other digits are dimmed.

3. 2

   The 6 in B5 rules out row B, and the 6 in C9 rules out row C.

4. 3

   The 6 in H2 rules out column 2, and the 6 in E3 rules out column 3. Only A1 is left.

5. 4

   A1 still has three candidates (3, 6 and 8), so it is not a naked single. It is the only cell in box 1 that can take the 6.

6. 5

   6 placed in A1.

Look at box 1: it still needs a 6.

The 6s in B5 and C9 block rows B and C, and the 6s in H2 and E3 block columns 2 and 3. Only A1 is left in box 1 for a 6, so A1 = 6.

A1 still shows three candidates (3, 6 and 8), so there was no naked single to read there. Only the box-wide count for the digit 6 exposed it.

Common mistakes

• Scanning cells only. If you never scan a unit for a missing digit you will miss every hidden single and stall on a puzzle that is genuinely easy.
• Stale pencil marks. A candidate you forgot to erase can hide a real naked single or invent a false one. Update peers immediately after every placement.
• Stopping the sweep early. One placement often triggers a chain of five more. Finish the cascade before you look for anything cleverer.
• Reaching for a pattern too soon. Pairs, wings and fish only pay off once the grid has no singles left in it.

Where you meet it

Singles alone solve easy classic Sudoku, and they carry most Medium puzzles too. Every harder technique in the game exists for one reason: to create the next single. So the sweep above is the move you will repeat hundreds of times on any grid, at any level.