A digit's candidates in a box form a pattern that doesn't fill both a full row and column, combined with an outside conjugate pair for an elimination.
Empty Rectangle is a single-digit chain technique. Inside a box, if a digit's candidates are restricted to specific rows and columns such that at least one row-column intersection within the box does NOT contain that digit, an "empty rectangle" is formed. This empty area, combined with a conjugate pair outside the box, creates a two-branch chain of inference that eliminates the digit from a target cell.
The technique sits alongside Skyscraper and Two-String Kite as a single-digit chain pattern. Once you understand conjugate pairs and how box constraints interact with line constraints, Empty Rectangle becomes a natural extension of that logic.
Pick a digit. Find a box where that digit's candidates are confined to specific rows and columns, with at least one row-column intersection inside the box empty of that digit. The candidates might form an L-shape, a line along one row plus one cell in another column, or similar restricted patterns. The key requirement: they occupy exactly 2 rows and 2 columns (or fewer), with at least one intersection cell missing.
Next, find a conjugate pair for that digit outside the box. One cell of the conjugate pair must share a row or column with the box's candidate pattern.
The elimination works through two-case logic:
Case 1: The conjugate pair places the digit at its first endpoint. This forces the digit into a specific position within the box (through the row or column constraint), which in turn eliminates the digit from the target cell via the box's column or row.
Case 2: The conjugate pair places the digit at its second endpoint. This directly eliminates the digit from the target cell (they share a row or column).
Since both cases eliminate the digit from the target cell, it cannot contain that digit. This is the core of the chain inference.
Example 1: Empty Rectangle
Look at digit 3 in box 3 (top-right). The candidates form a pattern that doesn't fill every row and column within the box. R1C8 and R3C7 are the key pattern cells, combined with an outside conjugate pair.
Case 1: if the conjugate pair places 3 at the endpoint sharing a line with box 3, the box forces 3 into R1C8, eliminating 3 from R4C8 via column 8. Case 2: if the pair places 3 at the other endpoint, it directly eliminates 3 from R4C8. Either way, R4C8 cannot hold digit 3.
Eliminate digit 3 from R4C8. R4C8 reduces to {2}, a Naked Single.
Example 2: Empty Rectangle
Look at digit 5 in box 5 (rows 4-6, cols 4-6). Candidates for 5 form an L-shape: R4C4 and R4C5 along row 4, plus R5C6 in column 6. Outside the box, R4C9 and R7C9 form a conjugate pair for digit 5 in column 9 (the only two cells in that column with candidate 5).
If R4C9 = 5: row 4 of box 5 is cleared, so box 5 forces R5C6 = 5. Then R7C6 sees R5C6 in column 6, eliminating 5. If R7C9 = 5: R7C6 shares row 7 with R7C9, eliminating 5. Either way, R7C6 cannot hold digit 5.
Digit 5 eliminated from R7C6. R7C6 reduces to {3, 9}.
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