¹Definition: Unique Rectangles (URs) are similar to B.U.G.+1 patterns except on a smaller scale: instead of occupying the entire board, URs occupy four cells in a rectangular pattern. These cells all contain the same two digits, and 1-3 of the cells are BVCs. The four cells must occupy exactly two rows, two columns, and two blocks.

²Look For: A BVC where both of its digits occur in another cell along its row and column, and one of these cells is in its block and the other lies outside its block, and these two cells see another cell that also contains the same two digits (forming a rectangle of cells).

³Consequence: This varies based on the type of UR. In our example of a type 1, the BVC digits can be erased from the sole non-BVC cell. See below for a breakdown of consequences by type.

⁴Why it Works: In all UR types, the erasures can be made because if the digits were valid candidates in the cell it would result in an ambiguous puzzle. In our example without the 1 in G2 the four cells could resolve to 7-8-8-7 or to 8-7-7-8. Therefore, the 7/8 is not valid in one of these cells, and the only one which has a different possibility is G2; having another digit present.

⁵So far we have only considered type 1 URs. But URs come in 7 types. Before going into detail, here is a table of types for quick reference.

Type	#BVCs	Arrangement	Consequence
1	3	any	Erase BVC digits from non-BVC cell
2	2	non-BVCs in same lane w/same extra digit	Erase extra digit in cells outside of UR that see all UR's non-BVC cells
3	2	non-BVCS in same lane w/diff. extra digits	Imagine virtual single cell made up of non-BVC extra digits, use it w/other non-UR cells in lane to make Naked pair, triple, or quad.
4	2	non-BVCS in same lane w/diff. extra digits	Erase the BVC digit that is not a locked-candidate from both non-BVC cells
5	1-2	non-BVCs on diagonal w/same extra digit	Erase extra digit in cells outside of UR that see all UR's non-BVC cells
6	2	non-BVCs on diagonal w/diff. extra digits	Erase BVC digit from both non-BVC cells if the digit forms an X-Wing within the UR (i.e. occurs nowhere else along the URs 2 rows, 2 cols, and block
7	1-2	any	If the UR cell diagonal from a UR BVC has a BVC digit that occurs in neither of its lanes (outside the UR cells) the the other BVC digit can be erased from its cell

⁶Type 2 & Type 5: These types are essentially the same. There are 2-3 non-BVCs and they all have the same extra digit. Type 2 has its non-BVC cells lined up on the same lane, while Type 5's are diagonal from each other. Type 2 has exactly two non-BVC cells, while Type 5 can have two or three non-BVC cells. Everything else is the same between them.

⁷Type 2: Let's consider the following Type 2 example. We have digits 2/9 in the required UR arrangement. The non-BVCs are in the same lane: column C, with the same extra digit: 5. So, in order to stave off the non-unique pattern of a UR with all 2/9 BVCs, we know that either C1 or C3 must resolve to a 5. This means that any other cells in their shared house (column C and block 1) cannot be a 5. So we can erase the 5's from those cells.

⁸Type 5: Here is a Type 5 example. We have digits 6/8 in the required UR arrangement. The non-BVCs are diagonal from each other, and in fact, we have three of them, all with the same extra digit: 4. So, in order to stave off the non-unique pattern of a UR with all 6/8 BVCs, we know that one of these three non-BVCs must resolve to a 4. This means that any other cell that can can see all the non-BVC cells can have its 4 erased.

⁹Type 4 & Type 6: These types are similar. In each case a BVC digit does not appear outside the UR cells along a non-BVC lane. The other BVC digit can be erased from both non-BVCs. The difference between them is that Type 4 has its non-BVCs in the same lane, and only requires one BVC digit to be missing from the non-BVC lane, while the Type 6 has diagonal non-BVCs and requires that one of the BVC digits is missing from the two rows and the two columns of the UR.

¹⁰Type 4: Let's consider the following Type 4 example. We have digits 3/9 in the required UR arrangement. Our non-BVC cells are in row 9, with differing extra digits. We notice that the 3 does not appear outside the UR cells along row 9, and so is a locked candidate for the non-BVC cells (marked as hard-candidates in Sue). So, we know one of these cells must resolve to a 3. But if one of them could resolve to a 9, then we would have the ambiguous situation again: 3-9-9-3 or 9-3-3-9. So, 9 can be erased from both of the non-BVC cells.

¹¹Type 6: Here is a Type 6 example. We have digits 1/2 in the required UR arrangement. Our non-BVCs are in a diagonal with different digits: a 1 in D3 and a 6 in E8. We notice that the BVC digit 2 does not appear in any lane of the UR, outside the UR cells (i.e. along row 3, row 8, column D, or column E). (Digit 2 forms an X-Wing.) Therefore, digit 2 can be erased from the non-BVC cells, Leaving the remaining 2's as hidden singles, which can immediately be entered as Big Numbers.

¹²Type 7 (aka "Hidden Rectangle"): In this Type we can have 2-3 non-BVCs (as with a Type 5, but here the extra digits are not all the same across cells). Here is a Type 7 example. We have digits 4/7 in the required UR arrangement. We have three non-BVCs. We choose the one that is diagonal from the BVC: A9. We look to see if the 4 or 7 is missing from both of A9's lanes (row 9 and column A) outside of the UR cells. We see that this applies to digit 4. Therefore we can erase the other BVC digit (7) from A9.

¹³Type 3: In this type of UR we have two non-BVC in the same lane, with multiple extra digits which may differ between them. This time we do something radically different: we disregard the BVC digits, then take the extra digits of the non-BVCs and pretend that they are all in one cell of the lane. We will refer to this as a virtual cell. We can then use this virtual cell together with the non-UR cells in the same lane to form a Nake pair, triple, or quad.

¹⁴Take a look at our Type 3 example. We have digits 1/5 in the required UR arrangement. The non-BVCs align in column H, and have the following extra digits: 4, 6, and 9 (we ignore duplication, since we can never have two of the same digit in a cell; even a virtual cell). Now, we need to imagine a virtual cell, replacing the non-BVCs, containing just the extra digits: Something like this.

¹⁵From this "thought experiment" we see that the virtual cell forms a Naked Triple of digits 4,6, and 9 with the cells H1 and H2. We can therefore do what we do with any real Naked Triple: erase these digits from other cells in the lane.

¹⁶Your Turn!
Take a look at this puzzle. Find the UR pattern, determine its type, and then erase the candidate(s) that the Unique Rectangle eliminates.

Hint