Almost Unique Rectangles

An Almost Unique Rectangle or AUR is a set of four cells populated in such a way that if one or two of them does not have a specific candidate 'x', then those four cells allow a Uniqueness Test.

Example 1

As one can see, the 4 cells in r13c25 would form a deadly pattern for digits 1 and 2 if it were not for the extra candidates for digit 8 and 9.

If the candidates for digit 8 were not present, it would form a Uniqueness Test 2, allowing us to eliminate all other candidates for digit 8 in row 1.

If the candidates for digit 9 were not present, it would form a Uniqueness Test 2, allowing us to eliminate all other candidates for digit 9 in row 3.

In its present form, we can make no direct eliminations, but can use this pattern in a nice loop, using the strong link between digit 8 in r1c25 and digit 9 in r3c25.

The result is this Nice Loop fragment:

 [r1c2|r1c5]=8|9=[r3c2|r3c5]

Example 2

To show that this also works with 1 extra candidate at both sides:

If the candidate for digit 8 were not present, it would form a Uniqueness Test 1, allowing us to eliminate all candidates for digit 9 for all peers of r3c2.

If the candidate for digit 9 were not present, it would form a Uniqueness Test 1, allowing us to eliminate all candidates for digit 8 for all peers of r1c5.

In its present form, we can make no direct eliminations, but can use this pattern in a nice loop, using the strong link between digit 8 in r1c5 and digit 9 in r3c2.

The result is this Nice Loop fragment:

 [r1c5]=8|9=[r3c2]

External Links

• almost-unique-rectangles-description-and-use-in-nice-loops on the new Sudoku Player's forum.