BUG+1 in Sudoku

Specialist S10 Uniqueness

BUG stands for Bi-Value Universal Grave: a position where every unsolved cell has exactly two candidates left and every digit has exactly two places left in every row, column and box. A position like that always has an even number of solutions, never one, so a proper Sudoku can never reach it. BUG+1 is that position with one exception - a single cell holding three candidates. One of those three is the only thing standing between the puzzle and the grave, and it can be placed straight away.

How to spot it

  • Late in a solve, check whether every unsolved cell is down to two candidates. That on its own is the first signal, and it is worth checking as soon as the board looks unusually tidy.
  • Find the exceptions. BUG+1 needs exactly one cell with three candidates. Two such cells, or one cell with four, is a different position and this technique says nothing about it.
  • Now count. For each of the three candidates, count how many places that digit still has in the cell's row, then its column, then its box. Two of them will have two places everywhere. One will have three places in all three units.
  • Check the rest of the board too. Every other digit, in every other unit, must still have exactly zero or two places. One irregular unit somewhere else and the position is not a grave, however tidy the cell you were looking at appears.

Why it works

In a genuine grave, every unsolved cell has two candidates and every digit has two places in every unit. Take any solution to such a position and swap every cell to its other candidate: because each digit occupies exactly two of the available places in each unit, the swap is again a valid solution.

So solutions to a grave always come in pairs. There are never exactly one of them - there are zero, two, four, and so on.

This Sudoku has exactly one solution. So the grave must be unreachable: it must have zero solutions, not two.

Pick either of the two ordinary candidates in the three-candidate cell and the third one disappears, leaving precisely that grave. That cannot be where the answer is. The exceptional candidate - the one with three places rather than two - is the only remaining possibility, so it is the answer.

Unlike the other Specialist techniques, this one ends in a placement rather than an elimination. Nothing is removed anywhere.

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 Sudokuno marks the one cell with three candidates, without saying which of the three matters. Every other unsolved cell on the board has exactly two.
  2. Hint 2 The occurrences appear: every place each of the three candidates still has in that cell's row, column and box. Two of the candidates have two places in each unit. One has three.
  3. Hint 3 The exceptional candidate is marked. Either of the other two would leave the deadly all-pairs position, which a puzzle with one solution cannot reach, so this digit is placed.

The check that matters is the count, not the tidiness. A board full of pairs is only a grave if every digit really does have two places in every unit - and the technique only licences anything at all because the puzzle is known to have a single solution.

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 implements strict BUG+1 on the standard grid: exactly one cell with three candidates, every other unsolved cell with two, and a verified grave everywhere else. BUG+2 and higher, BUG-Lite, Reverse BUG and Reverse BUG-Lite, MUG, generalised grave structures, variant grids such as Sudoku-X, and chain or AIC formulations are all outside S9.

Where to go next

  • Simple Colouring Simple Colouring looks at one digit at a time. When a row, column or box has exactly two places left for that digit, the two cells are strongly linked: exactly one of them holds it. Join such pairs together and you get a network that can be coloured alternately in two colours, and then exactly one of the colours is the true one. That is enough to remove candidates, even though you do not know which colour is which.
  • Rectangle Elimination Rectangle Elimination follows one candidate around four boxes. Start with a hinge H whose row or column has exactly one other place for the digit, S. A possible target T lies on the perpendicular line through H. If T held the digit, H would be off and the strong side would force S on. When T and S then see every remaining place for that digit in the diagonally opposite box, the box would have nowhere to put it. T therefore cannot hold the digit.
  • W-Wing A W-Wing starts with two cells that have exactly two candidates each, and the same two: say 7 and 9. Somewhere else on the board, one of those digits - say 7 - has exactly two places left in a single row, column or box, and each of the two cells sees one of those two places. That is enough to prove that at least one of the two cells holds 9, so any cell that sees both of them cannot.

All Sudoku techniques