Empty Rectangle in Sudoku: How the Box Pattern Works

An Empty Rectangle is a box where every candidate for one digit lies on a single row and a single column of that box, forming a cross, so the other four cells of the box (the empty rectangle) have none. Combined with a strong link on the same digit that touches the cross's column (or row) from outside the box, it removes the digit from one target cell.

To spot one, pick a digit and look for a box where its candidates fit on one row plus one column of the box, with at least one candidate off the crossing point on each arm. Then look along the cross's lines for a row or column where the digit has exactly two spots, one of them in line with the cross.

How the Empty Rectangle works

This position comes from a real puzzle in our library. Focus on the digit 1.

• Box 2 has three cells that can be 1: R1C4, R1C6 and R3C6. Every one of them lies on row 1 or column 6. The four cells R2C4, R2C5, R3C4 and R3C5 have no 1, which is the empty rectangle that gives the pattern its name.
• Row 9 has exactly two cells that can be 1: R9C6 and R9C8. That is the strong link. R9C6 sits in column 6, the column arm of the cross.

One of R9C6 and R9C8 is the 1 of row 9. Check each case against the target, R1C8.

• If R9C8 is 1, then R1C8 is in the same column, so it is not 1.
• If R9C6 is 1, column 6 has its 1, so R1C6 and R3C6 are not 1. The only spot left for box 2's 1 is R1C4. That puts the 1 of row 1 in R1C4, so R1C8 is not 1.

Both cases clear R1C8, so 1 comes out of it. The cross is what makes the second case work: whatever happens in box 2, its 1 lands on row 1 or column 6, so blocking one arm pushes the digit onto the other.

Puzzle 201897 from the Sudoku Royale library. Givens (row by row, dots are blanks): ............2.953....57.9....6.28.7.2.7.63.9..4....3..431..78..7.8................ The candidates shown are what is left after singles, pointing and claiming, naked and hidden pairs, triples and quads, X-Wing and Swordfish, with nothing else applied.

How to spot an Empty Rectangle

1. Pick a digit and go box by box.
2. Test for a cross. Can you choose one row and one column of the box that together cover every candidate for the digit? If all candidates sit in a single line, you have pointing instead, which you should already have used. A useful cross has candidates on both arms.
3. Follow the column arm out of the box. Look in that column, outside the box, for a cell whose row holds the digit in exactly two places. That row is your strong link.
4. Find the target. Go to the other end of that strong link. The target is where its column meets the row arm of the cross.
5. Repeat with the arms swapped. The same pattern works with a column strong link that starts on the row arm of the cross.

A box with only two candidates on different rows and columns also forms a cross, in two ways. That small version often gives the same elimination as a 2-String Kite, so it pays to know both patterns.

Empty Rectangle vs pointing

Pointing uses a box where a digit sits on one line, and clears that line outside the box. The Empty Rectangle uses a box where the digit sits on two crossing lines. On its own that proves nothing, because the digit could be on either arm. The strong link from outside closes one arm and the logic then runs like pointing on the other.

Related patterns

• 2-String Kite: a row link and a column link joined in a box.
• Skyscraper: two parallel strong links joined along a line.
• X-Chain: longer single-digit chains that include all of these as short cases.
• Pointing pairs and box-line reduction: the simpler box patterns to clear first.

For where these fit in the solving order, see our list of sudoku techniques in learning order and the overview of advanced sudoku strategies.