BUG+1 in Sudoku

Specialist S10 Uniqueness

BUG+1 applies when exactly one unsolved cell has three candidates and all the others have two, with a specific count pattern across the whole board. One candidate appears three times in that cell's row, column and box. In a puzzle known to have one solution, place that candidate.

A candidate is a digit that can still go in an empty cell. Cells see each other when they share a row, column or 3×3 box. A strong link means a digit has exactly two possible cells in one such group: exactly one must contain it.

How to spot it

  • Find exactly one unsolved cell with three candidates. Every other unsolved cell must have exactly two.
  • Count the possible cells for each of its three candidates, separately in its row, column and box. One digit must have three places in each of these groups; the other two must have two places in each.
  • Check the entire board: every other missing digit must have exactly two possible cells in each row, column and box. Digits already placed in a group have no candidate cells there.
  • Use this deduction only when the puzzle is known to have exactly one solution. A three-candidate cell among pairs is not enough on its own.

Why it works

Imagine removing the extra candidate from the three-candidate cell without filling the cell. Every unsolved cell would now have two candidates, and every missing digit would have two possible cells in each row, column and box. This is a Bi-Value Universal Grave, or BUG.

Any solution using only those candidates would have a partner: switch every previously unsolved cell to its other candidate. Every row, column and box would still contain each digit once. So this all-pairs position cannot have exactly one solution.

The original puzzle has exactly one solution. If it did not use the extra candidate, it would also solve the all-pairs position, giving a second solution. Therefore the three-candidate cell must contain the extra candidate.

Place that digit. BUG+1 gives a placement, rather than a separate candidate-removal step.

A real example from Sudokuno

The board below is a real Sudokuno puzzle, shown exactly as it stands just before BUG+1 applies. Everything up to this point was solved with ordinary techniques.

6 7 9 4 8 2 5 1 3
3 8 1 6 7 5 9 2 4
4 5 2 9 3 1 8 7 6
2 8 3
7 6 4 1
1 3 4 2 6 8 7 5 9
7
4 2 3 8
8 4 3 7 2 9 1 6 5
2
8
3 4 9 7
7
4 6 3 8 2

There is nothing to click away here: this technique places a digit rather than removing candidates. The answer is 5 in R6C5 - count the occurrences on the board above and see why the other two candidates cannot be it.

Example: BUG - Sudokuno - not yet published in the puzzle picker

  • Houses: row 6, column 5 and box 5
  • Pivot: R6C5
  • Its three candidates: 1, 5 and 9
  • Occurrences to count: R4C5, R6C3, R6C4, R6C5 and R8C5
  • Places 5 in R6C5
  • Step 43 of 54 in the solve

Key: the three-candidate cell occurrences to count

How the hint helps you

Sudokuno's hint reveals the structure in three steps, without giving away the digit straight away.

  1. Hint 1 The cell with three candidates is highlighted. Every other unsolved cell has two.
  2. Hint 2 Its row, column and box are highlighted, along with occurrences to count. Count each candidate separately: one has three places in each group, the other two have two. Check the count pattern on the rest of the board too.
  3. Hint 3 The cell is marked for placement. Removing its extra candidate would leave the verified BUG pattern, incompatible with the puzzle's single solution.

The whole-board count check and the single-solution requirement both matter. Do not apply BUG+1 just because the board is mostly pairs.

Ready to try it yourself?

You will get a real Sudokuno puzzle at Specialist level S10, where BUG+1 is needed to finish it.

Practise BUG+1

What Sudokuno covers

Sudokuno covers strict BUG+1 on the standard grid: one cell with three candidates, all other unsolved cells with two, the full count pattern verified, and exactly one solution.

Where to go next

  • Simple Colouring Simple Colouring follows one digit through a network of strong links. Colour linked cells alternately: either every cell of colour A holds the digit and none of B does, or the reverse. Those two possibilities can rule out candidates.
  • Rectangle Elimination Rectangle Elimination tests whether a candidate would leave a box with no place for a digit. Assuming the target contains the digit forces another cell to contain it too. If together they rule out every place in the opposite box, remove the candidate from the target.
  • W-Wing A W-Wing connects two cells holding the same candidate pair through a strong link on one digit. This proves that at least one of the pair cells holds the other digit. Remove that other digit from any cell seeing both pair cells.

All Sudoku techniques