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.
Look inside a box for a digit whose candidates all lie on one row and one column of that box, with their crossing cell free of the digit. Then find a row or column outside the box where the same digit has exactly two positions. The two branches together eliminate the digit from one cell.
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: the candidates are confined to a single row and a single column within the box (a cross/plus shape, of which an L-shape is a special case), meeting at the empty intersection cell.
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}.
Because four cells in the box hold no candidate for the digit, and those four cells form a rectangle. When the digit's candidates are confined to one row and one column of the box, the two leftover rows and two leftover columns cross in four cells, all empty of that digit.
Any shape confined to one row and one column of the box, with the crossing cell empty of the digit. An L of two cells along the row plus one down the column is the common case, and the widest form fills both arms around that empty crossing. Nothing may sit off the row or column.
Once the box pattern is set, look along the row and the column that the candidates occupy. You need a line outside the box in which the digit has exactly two candidate cells, and one of those two cells must share that row or column with the box pattern.
At the cell that both branches reach. One endpoint of the outside pair sees the target directly along a shared row or column; the other endpoint forces the digit into a specific cell of the box, which then sees the target along the box's other line. Both routes remove the digit.
Both are single-digit chains, but the Kite needs two conjugate pairs while the Empty Rectangle needs only one. In an Empty Rectangle the box pattern replaces the second pair: the digit may have three or more candidate cells in that box, as long as they all lie on one row and one column.
After Skyscraper and Two-String Kite have failed on the same digit. Empty Rectangle sits in the Extreme tier with those two listed as prerequisites, and Simple Coloring is the natural next step. All three are single-digit chain patterns built on conjugate pairs.
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