From Sudopedia, the free Sudoku reference guide

From Sudopedia

The Empty Rectangle is a single-digit solving technique which uses the absence of candidates to perform an elimination.

An alternative term is hinge, which was actually coined before the term Empty Rectangle, but to a smaller audience. The acronym ER is used by many players.

How it works

Take a look at this diagram:

.-------.-------.-------.
| . . . | . . . | . . . |
| - A - | - - - | - B - |
| . . . | . . . | . . . |
:-------+-------+-------:
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
:-------+-------+-------:
| . . . | . . . | - X - |
| . * . | . . . | X . X |
| . . . | . . . | - X - |
'-------'-------'-------'

Row 2 has a strong linked pair of candidates in cells A and B. Call that candidate x. Box 9 has all candidates x confined to 1 boxcol and 1 boxrow. The 4 cells without candidate x form the Empty Rectangle (marked with a dash). When x is true in A, x is false in r8c2. When x is false in A, it is true in B, and hence the remaining candidates x in box 9 are confined to row 8, causing x to be false in r8c2. Thus x can never be true in r8c2, and so can be eliminated from r8c2.

This technique is a special case of Grouped Turbot Fish.

An extended form of the empty rectangle

The pattern shown in the diagram below is an extension of the ER technique which can result in 1 or 2 additional candidate cell eliminations. In order to use this extended technique there must be an ER pattern and at least one additional conjugate pair with one cell which is a peer either of the cell A in the ER pattern or the two cells in row 8 box 7. This example is the former with two conjugate pairs CE and CD. Note that the ER pattern in the diagram is the same as in the original example.

.-------.-------.-------.
| . . . | . . . | . . . |
| - A - | - - - | - B - |
| . . . | . . . | . . . |
:-------+-------+-------:
| - - E | . . . | . . . |
| - C - | - D - | - - - |
| - - - | . . . | . . . |
:-------+-------+-------:
| . . . | . . . | - X - |
| . * * | . * . | X . X |
| . . . | . . . | - X - |
'-------'-------'-------'

How it works. If B is X then X is in row 8 of box 7 and r8c235 cannot be X. If B is not X, it's conjugate A is X and C which is a peer of A must be not X. Therfore both of C's conjugates D and E must also be X and therefore r8c235 cannot be X. Note that this extension technique can also be used with the 2-string kite

Here is a second example using the same ER pattern. In this case a single additional conjugate pair CD is in column 5 and C is the peer cell.

.-------.-------.-------.
| . . . | . - . | . . . |
| - A - | - - - | - B - |
| . . . | . - . | . . . |
:-------+-------+-------:
| . . . | . - . | . . . |
| . * . | . D . | . . . |
| . . . | . - . | . . . |
:-------+-------+-------:
| . . . | . - . | - X - |
| . * . | . C . | X . X |
| . . . | . - . | - X - |
'-------'-------'-------'

See Also

• 2-String Kite