Mirrored from Sudopedia, the Free Sudoku Reference Guide

# Empty Rectangle

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.

DOUBLE EMPTY RECTANGLE

A special case of the simple ER shown above is the double ER in which each of the conjugate pairs is aligned with the non-ER row(column) of an ER box. This can result in additional cell eliminations as illustrated in the partially worked puzzle shown below. Here the double ER consists of the 9 conjugate pair in row 1 and the ER boxes 4 and 5. If r1c2=9 then r5c3 must also be 9 and if r1c5=9 then r5c6 must also be 9. Thus either r5c3 or r5c6 must be 9 and 9 can be eliminated from r5c58.

Double ER Example 1

```.-------------------+-------------------+---------------------.
|    4  289     3   |    7   289     6  |   28      1      5  |
|    6    1    89   |    3     5   289  |  278     79      4  |
|   27 2789     5   |    4     1   289  |    3      6     89  |
|-------------------+-------------------+---------------------|
|    8   79     6   |    5   279     3  |    4    279      1  |
|    3    4    79   |    1  2789  289   |    5    279      6  |
|    1    5     2   |    6    79     4  |   78      3     89  |
|-------------------+-------------------+---------------------|
|    5    6     1   |    2     4     7  |    9      8      3  |
|  279   278    4   |   89     3     1  |    6      5     27  |
|  279    3    78   |   89     6     5  |    1      4     27  |
'-------------------+-------------------+---------------------'
```

The original puzzle is:

003000000 010050000 000400060 800003001 040000500 002600000 000007980 000001000 000000040

In the second Double ER example, the 2 ER boxes are not in the same row(column). It consists of the 4 conjugate pair in row 6 and ER boxes 3 and 7. If r6c9=4 then r2c8=4, and if r6c3=4, then r9c2=4 and also r2c1=4. Therefore either r2c1 or r2c8 must be 4 and it can be eliminated from the four cells. r2c3569. This example requires correction: The second Double ER example in Sudopdia is incorrect, there cannot be a resolved 1 in H7 as indicated in the grid, and there is a candidate 4 in A7 which does not permit box c to hold an ER. The original puzzle is: 003000000 010050000 000400060 800003001 040000500 002600000 000007980 000001000 000000040 The problem is in box j. The original string shown above shows a given 4 in J8, rather than the given 9 in the numeric grid of the example, also the numeric grid in the example has a resolved 1 in H7 immediately next to a resolved 1 in H6 in box j. When these problem are corrected as above, box c holds an ER in 4, but box g holds only a single 4 in H3. Using the 4-strong link in row F this forces F9 to 4 and removes 4’s from F3, E7, E9, D7, D8, D9, C9, B9, and A9. This forces A7 to 4 and removes 4’s from A5, A3, and A2, this forces B1 to 4 which removes 4’s from B3, C3, B5, B6, D1 and E1.

Double ER Example 2

```.-------------------+-------------------+---------------------.
|    6  347   347   |    8  12347    9  |   13      5   12347 |
|  348    1  3457   | 2356 23467   3456 |    9     27    2347 |
|    9    2  3467   | 1347  1347    345 |    8      6    1347 |
|-------------------+-------------------+---------------------|
| 2348 3489    16   |  356   346  3456  |   13   1247  123479 |
|  234  349    16   |    7   346     8  |  135    124  123459 |
|    7    5    34   |    1     9     2  |    6      8     34  |
|-------------------+-------------------+---------------------|
|  238   38     9   |    4  2368     7  |   15      1    1568 |
|    5    6  2478   |    9    28     1  |   27      3     78  |
|    1  378  2378   |  236     5    36  |   27      4    678  |
'-------------------+-------------------+---------------------'
```

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 9. 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 9 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 - |
'-------'-------'-------'
```