Sudoku Written by Brian Hamilton and Karan Hamilton.
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Near the end of a hard puzzle you can reach a board where almost every open cell is down to two candidates. If exactly one cell holds three, there is a shortcut that names that cell’s answer at a glance. It is called BUG+1, after the Bivalue Universal Grave, and it rests on a promise about the puzzle rather than on the rules: a sudoku with a single solution can never end in a BUG.
We ran a BUG detector over the first 500 puzzles of our Evil sudoku bank, wherever the 18 techniques of our sudoku helper run out. It fired at 4 of 808 stalls and finished exactly one puzzle the 18 could not, which is the worked example below. Then comes the catch most guides mention only in passing: on a puzzle with two solutions the rule gives wrong answers, and we measured how often.
What is a BUG in sudoku?
A BUG is a state of the pencil marks with two properties at once:
- every unsolved cell has exactly two candidates, and
- in every row, column and box, each digit that is still a candidate there appears in exactly two cells.
“Bivalue” means two candidates, “universal” means every cell on the board, and “grave” nods to the deadly patterns of the uniqueness family: sets of candidates a one-solution puzzle can never end in. (Nothing to do with software bugs.) You never play a pure BUG. You play BUG+1, where the board is one candidate away from a BUG and that candidate must be the answer.
A candidate is a digit still possible in a cell, written as a small pencil mark. A cell with exactly two is bivalue. A house is any row, column or 3×3 box. BUG reasoning is only as good as your marks: one candidate erased by mistake can make a board look like a BUG when it isn’t, so every mark has to be there and right.
Why can’t a sudoku end in a BUG?
Suppose a BUG had a solution. In each house, each digit has exactly two cells, and the solution uses one of them. Now switch every unsolved cell to its other candidate. In every house, each digit moves from one of its two cells to the other, so it still appears exactly once. The switched grid is a second solution. Solutions of a BUG therefore come in pairs: a BUG has no solution or an even number of them, never exactly one.
That is the deadly-rectangle argument behind unique rectangles, stretched from four cells to the whole board. Guides often shorten it to “a BUG has two solutions”, but on a real puzzle it usually has none: with the extra 4 deleted from the example board below, a solver finds no solution at all. What the argument rules out is a solution that avoids the extra candidate, because that solution would come with a twin.
How does BUG+1 work?
Every unsolved cell has two candidates except one, which has three. Remove the right one of those three and the board would be a BUG. A puzzle with one solution can’t be in a BUG, so that candidate can’t be false: it is the cell’s answer.
Here it is on a real board, puzzle 185 of our Evil bank, at the first point where the helper’s 18 techniques have nothing left. R5C3, the green cell, is the only one with three candidates; the blue 4s are the other 4s in its row, column and box.
Take the 4 out of R5C3 and every house balances, which is a BUG. So R5C3 is 4. The placement leaves four naked singles behind, and more naked singles finish the puzzle.
How do you spot the extra digit?
You don’t have to count the whole board. If no hidden single is left and every cell of a house has two candidates, that house already balances: k open cells need k different digits, each digit needs at least two spots or it would be a hidden single, and k bivalue cells hold only 2k marks between them. So only the three houses of the odd cell can be off, and counting there gives you the digit.
- Clear the singles first. BUG only applies when no naked or hidden single is waiting.
- Find the cells with more than two candidates. Exactly one, holding three: carry on. None: see the last section. Two or more: at best it’s a BUG+n, below.
- Count that cell’s three digits in its row. One should appear three times and the other two twice.
- Check the column and the box, then place it. All three houses must name the same digit. If anything disagrees, it isn’t a BUG+1.
On the example board, row 5 alone shows the 4 three times, and column 3 and box 4 agree.
What are BUG+2 and BUG+n?
With two or more extra candidates, setting all of them aside leaves a BUG, so at least one of the extras is true. It won’t say which, so it rarely places a digit, but it can eliminate whatever would clash with every extra. If the extras are all the same digit, any cell that sees all of them loses that digit. If they all sit in one cell, that cell’s other candidates go.
In our 500 Evil puzzles a BUG+n was the move once, on puzzle 224 at its fourth stall. Reaching that stall took one forcing chain and two XY-chains on top of the 18.
How often does BUG come up?
Rarely, and only near the end. These are our counts:
| Measure | Evil | Extreme |
|---|---|---|
| Puzzles the helper’s 18 techniques finish | 332 of 500 | 1 of 500 |
| With BUG added | 333 of 500 | 1 of 500 |
| First stalls where a BUG applies | 1 of 168 (0.6%) | 0 of 499 |
| All stalls where a BUG applies | 4 of 808 (0.5%) | 0 of 499 |
| BUG moves used | 4 (three BUG+1, one BUG+2) | 0 |
| BUG+1 positions while the 18 still had a move | 31, in 31 puzzles | 0 |
The first 500 puzzles of each bank. The helper’s 18 techniques run simplest first until they stall; at a stall, BUG is tried before X-chains, XY-chains and forcing chains, so a different order would credit it differently. Measured in October 2026.
BUG lives at the end of a solve. The four stalls where it was used had 21 to 36 cells open, and the 31 BUG+1 positions a median of 22, against a median of 44 at a typical first stall. At those 31 the helper took a move of its own instead: an almost locked set 15 times, an XY-Wing 9 times, a W-Wing 5 times. If you see the BUG+1 first, take it: on a one-solution puzzle it is just as valid.
Our Extreme sudoku puzzles never get near one. The 18 stall at the very first position in 499 of 500, with 57.5 cells open and fewer than one two-candidate cell on average.
At all four stalls where we used a BUG, XY-chains were available too: 28 to 41 of them. On puzzle 185 the shortest runs through five cells, R3C3, R6C3, R6C4, R8C4 and R8C8, and applied instead of the BUG it lets the helper finish the puzzle just the same. If you would rather not lean on uniqueness, the chain is there.
Is BUG+1 safe on any puzzle?
No. BUG+1 follows from the promise that the puzzle has exactly one solution, not from the rules, and when the promise breaks, the rule breaks with it. We tested that. Deleting one clue at a time from the same 500 Evil puzzles gave 656 puzzles with exactly two solutions. Solved with every helper technique except Unique Rectangle (the one of the 18 that also assumes uniqueness), 22 of them reached a BUG+1 position, and in 21 of the 22 the digit the rule names is in neither solution. Delete the 2 at R4C4 from puzzle 295, for one: the rule says R2C2 is 8, and the two solutions have a 5 and a 6 there.
So use it only where the promise holds. Every one of the 5,000 puzzles in our Evil bank, and in our Extreme bank, has been counted to exactly one solution. Don’t use it on a puzzle whose source makes no such promise, and never to decide whether a puzzle is unique, since it assumes the answer. The helper’s 18 don’t include BUG, so it will never suggest one.
Stuck with nearly every cell down to two candidates?
Count the cells with more than two candidates.
- One, holding three: check for a BUG+1 as above.
- None, and no single anywhere: something went wrong earlier. By the counting argument every house balances, so this is a full BUG, which a one-solution puzzle can’t reach with correct pencil marks. A candidate was erased that shouldn’t have been, or a digit went into the wrong cell; go back over your last few moves.
- Two or more: look for a BUG+n elimination. If there is none, a board this full of bivalue cells is exactly what XY-chains run on, with X-cycles and forcing chains after that. Our guide to sudoku techniques puts them all in order.