When a sudoku cell has two options left, do not pick one and hope. Look for a second cell it works with. Two cells in the same group with the same two options form a naked pair. Three two-option cells that see each other in a chain can form an XY-Wing. A digit with two possible places can be chained with others through simple colouring. And if all else fails, follow each option a few moves ahead: when both lead to the same digit somewhere else, that digit is proved. All four are covered below, each with a real position.
"Two options" can mean two different things, and both are useful:
- A cell with two candidates (a bivalue cell): R4C2 can only be 6 or 9.
- A digit with two places in a row, column, or box (a conjugate pair): in column 2, the 4 can only go in R3C2 or R6C2.
Either way, one of the two is true and the other false. Every technique on this page uses that fact. (R4C2 means row 4, column 2. The grids are labelled on the edges.)
1. Find a matching cell: the naked pair
If another cell in the same row, column, or box has exactly the same two options, the two cells share those two digits between them. You still do not know which is which, but no other cell in that group can have either digit.
- 5 clue
- 5 already solved
- 5 new digit
- 5 pattern candidate
- 5 candidate to remove
The naked pairs guide has more examples. The mirror image, two digits that each have only the same two places in a group, is covered in the hidden pairs guide.
2. Follow both options: when they agree, you have your digit
Take the two-option cell and ask what each option forces, one easy step at a time. If both paths put the same digit in the same cell, that digit is certain, whichever option turns out to be right. This is not guessing: you have checked every possibility, and there were only two.
Keep these checks short. Two to five forced moves per option is easy to hold in your head or on scrap paper. If a path runs longer, stop and look for a pattern instead: a long chain of assumptions is where mistakes creep in. Our page on solving sudoku without guessing explains where the line between a check and a guess sits.
3. Link three two-option cells: the XY-Wing
The XY-Wing is the named, reusable version of step 2. You need a pivot cell with options X and Y and two pincer cells it can see, one with X and Z, one with Y and Z. Whatever the pivot is, one of the pincers ends up as Z. So any cell that sees both pincers cannot be Z.
To find XY-Wings, look only at the two-option cells. Pick one as the pivot and check the two-option cells it can see for the shape XZ and YZ. It is the hardest step in 72% of the Master puzzles in Sudoku Royale, so it is worth the practice.
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Download Sudoku Royale: Free on iOS4. Chain the digits with two places: simple colouring
Now the other kind of two options: a digit that has only two places in a group. Link those places into a chain with simple colouring, colouring them in two alternating colours. In each link one end is the digit and the other is not, so across the whole chain one colour is true and the other false.
A three-link chain like this one is also called a turbot fish. Longer ones are the reason simple colouring is the signature technique of the hardest level in our library: 95% of Extreme puzzles need it.
When two options really are a coin flip
There is one case where two options are genuinely interchangeable: four cells at the corners of a rectangle, in two boxes, all with the same two options. If the puzzle had to choose between them, it would have two solutions. A proper puzzle cannot, which is the idea behind the unique rectangle technique. If you meet this at the very end of a puzzle with nothing else to place, the puzzle is broken. Both answers are correct, so fill in either one. See what to do when you are stuck with only pairs for the unique rectangle and its cousin, BUG+1.
Practising with two-option cells
In Sudoku Royale's Dojo, the Naked Pair, Hidden Pair, XY-Wing, and Simple Colouring drills each serve a run of boards where that pattern is the next move, with a short lesson first. In Classic mode, if you stall, the lightbulb hint names the technique the board needs without giving you the digit.
Frequently Asked Questions
How do you solve sudoku when there are two options?
Look for a partner cell. Two cells in one group with the same two options are a naked pair. Three linked two-option cells can form an XY-Wing. A digit with two places can be chained by simple colouring. Or follow each option a few forced moves ahead: if both lead to the same digit in the same cell, that digit is certain.
What do you do when a sudoku has multiple candidates in every cell?
Make sure your pencil marks are complete and up to date, then rescan for hidden singles one digit at a time. Next look for pointing and claiming, then pairs. Cells with many candidates usually shrink quickly once one of those patterns removes a few.
Is it OK to guess between two numbers in sudoku?
It is never necessary in a proper puzzle. Instead of guessing, follow both options a few moves: if one leads to a contradiction, the other is proved, and if both force the same digit somewhere, that digit is proved. Both are logic.
What is a bivalue cell in sudoku?
A cell with exactly two candidates left. Bivalue cells are the building blocks of naked pairs, the XY-Wing, and XY-chains.