Sudoku Stuck With Only Pairs Left? 3 Ways Out

When a sudoku is stuck with nothing but pairs, so that almost every empty cell has exactly two candidates, three techniques usually break it. If exactly one cell has three candidates and the rest have two, use BUG+1: that cell takes the digit that appears three times in its row, column, and box. If four cells in a rectangle across two boxes share the same pair, use the unique rectangle. Otherwise, look for an XY-Wing among the two-candidate cells. Both of the first two rely on the puzzle having a single solution, which every proper sudoku does.

A board full of pairs feels like the end of the road, but it is the opposite: two-candidate cells are the easiest cells to reason about, because each one is a yes-or-no question. Both examples below come from real puzzles that the 13 standard techniques, up to simple colouring, could not finish. That is exactly the kind of position where people get stuck with pairs.

First, rule out the easy fixes

Before reaching for the patterns below, make sure the stall is real. Rescan for hidden singles digit by digit, since a pencil mark you forgot to erase can make a solved cell look like a pair. Then check every group for two cells that hold the same pair. If another cell in that group still lists either digit, that is a naked pair you have not applied yet. Our stuck checklist has the full order.

1. BUG+1: one cell with three candidates

BUG stands for bivalue universal grave. Picture a board where every empty cell has exactly two candidates and every candidate appears exactly twice in each row, column, and box. That board always has at least two solutions: you can flip every cell to its other candidate and everything still checks out. A proper puzzle has one solution, so it can never reach that state.

Now picture the board one candidate away from it. Every cell has two candidates except one cell, which has three. The only way to avoid the deadly two-solution state is for that cell to take its extra digit: the one that appears three times in its row, column, and box instead of twice.

• 5 clue
• 5 already solved
• 5 new digit
• 5 pattern candidate
• 5 candidate to remove

To use BUG+1 on paper, you do not need to check every group. Confirm that only one cell has three candidates, then count its three digits in that cell's row. The digit with three appearances is the answer. Checking the column and box as well is a good safeguard against a stale pencil mark.

2. Unique rectangle: four cells, one pair

Take four cells that form a rectangle across exactly two rows, two columns, and two boxes. If all four could only be 3 and 5, you could swap the 3s and 5s and get a second valid solution. A proper puzzle does not allow that, so at least one of the four cells must hold something other than 3 or 5.

That gives you a clean elimination whenever three corners have exactly the pair and the fourth has the pair plus extra candidates. The fourth corner cannot be either digit of the pair.

The rectangle has to span exactly two boxes. Four corners spread over four boxes do not form a deadly pattern: each box would hold just one corner, and changing its digit would leave that box with a duplicate. For more on why two-solution patterns matter, see can a sudoku have more than one solution.

3. XY-Wing: three pairs that work together

If neither pattern applies, the two-candidate cells are still your best lead, and the XY-Wing is the pattern to try. Pick one cell with candidates X and Y as a pivot. Look for two cells it can see, one with X and Z, one with Y and Z. One of those two pincers must end up as Z, so any cell that sees both pincers loses Z. There are worked XY-Wing examples in the stuck checklist and in how to solve sudoku when there are two options.

When the XY-Wing is not there either, the next tools are chains: an XY-chain is an XY-Wing with more links, and a forcing chain follows both candidates of one pair until they agree or one collapses. Both are logic, not guessing. Our guide to solving sudoku without guessing shows a three-move forcing chain on a real board.

A note on uniqueness techniques

BUG+1 and the unique rectangle only work if the puzzle has exactly one solution. Every puzzle from a proper source does, but a badly made puzzle might not, and then these techniques can remove the wrong candidate. If you solve puzzles from an unknown generator, treat them as a last resort. All puzzles in Sudoku Royale have one solution, and all of them can be finished without either technique: the app's puzzles stop at simple colouring and the Jellyfish, so you will not need BUG+1 there. For practice with the patterns that do come up, the Dojo has an XY-Wing drill and a Naked Pair drill.