Let's start with a definition of the Empty Rectangle (ER) itself: If one candidate is restricted to exactely one row and one column within a box, the remaining cells form an ER. Look at the two examples below: On the left candidate 9 in box 5 is restricted to row 4 and column 6. The ER is r5c45 and r6c45. On the right candidate 1 in box 5 is restricted to row 4 and column 5. The ER is r5c46 and r6c46.
If an ER has been found, it can be checked for eliminations: Find a conjugate pair (a row or column containing only two candidates) where one of the candidates is in the row (column) forming the ER. If the column (row) of the ER contains a candidate, that can see the second candidate of the conjugate pair, it can be eliminated.
An ER pattern that follows the description above can always be seen as a
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or as a Grouped Nice Loop .
On the left: Row 4 and column 6 form an ER in box 5. r48c2 form a conjugate pair. One side of the conjugate pair is in the ER row (r4c2), the other can see a candidate in the ER column (r8c6). 9 can be eliminated from r8c6.
To see why this works, let us start with candidate 9 in r8c2: If r8c2 is 9, r8c6 cannot be 9 (same row). If r8c2 is not 9, r4c2 has to be 9 (conjugate pair/strong link), which means that neither r4c5 nor r4c6 can be 9. This leaves r6c6 as last candidate 9 in box 5, it must be true and r8c6 must be false again. Since both possibilities in r8c2 lead to the same result (r8c6<>9), 9 can be eliminated from r8c6.
The equivalent Finned Mutant X-Wing: 9 c2b5 r4c6 fr8c2 => r8c6<>9.
The equivalent Grouped Nice Loop: r8c6 -9- r8c2 =9= r4c2 -9- r4c56 =9= r6c6 -9- r8c6 => r8c6<>9.
In the example on the right row 4 and column 5 form the ER, r7c59 is the conjugate pair, and r4c9 can be eliminated.
If the ER contains only two candidates the Empty Rectangle is not clearly defined. The eliminations are of course still valid. The move can be replicated as Turbot Fish or as X-Chain . Some sudoku players don't consider ERs with only two candidates valid, therefore HoDoKu supports them only optionally.
Note: Since Turbot Fish is before ER in the standard solving order, ERs with two candidates are always found as Turbot Fishes, even if they are enabled. Please move Turbot Fish behind ER (or disable it) to find the ERs below.
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