Unique Rectangle Type 4
A Unique Rectangle Type 4 is a uniqueness technique, like the Unique Rectangle of Type 1. Four cells at the corners of a rectangle in two rows, two columns and two boxes all hold the candidates A and B; two neighbouring corners, the floor, hold nothing else, and the two opposite them, the roof, hold more. If A has no other candidate cell in a row, column or box that the two roof corners share, one of them must be A. The other cannot then be B, or all four corners would hold only A and B, a deadly pattern with two solutions. So B is removed from both roof corners. It is an Expert technique in Bare Sudoku.
When to use it
When two neighbouring corners of a rectangle are down to the same pair and the opposite two corners still hold more candidates, so that Type 1 does not apply. Type 4 is the most common of the further Unique Rectangle types in Bare Sudoku puzzles. It needs complete notes for the four corners and for one digit of the pair in the roof's unit.
How to spot it
Find two cells in the same row or column with exactly the same two candidates, A and B, and complete the rectangle with two cells that also hold A and B, in exactly two boxes in total. If the two roof cells still have other candidates, look at the row, column or box they share: if A has no candidate there outside the two roof cells, remove B from both. Then try the same with B as the confined digit, which would remove A.
Example
| c1 | c2 | c3 | c4 | c5 | c6 | c7 | c8 | c9 | |
|---|---|---|---|---|---|---|---|---|---|
| r1 | 2 | 59 | 8 | 57 | 1 | 3 | 4 | 6 | 79 |
| r2 | 4 | 1 | 36 | 78 | 268 | 29 | 5 | 289 | 3789 |
| r3 | 7 | 59 | 36 | 4 | 2568 | 259 | 1 | 289 | 389 |
| r4 | 1 | 8 | 5 | 6 | 9 | 4 | 7 | 3 | 2 |
| r5 | 6 | 4 | 7 | 3 | 23 | 25 | 9 | 58 | 1 |
| r6 | 3 | 2 | 9 | 1 | 58 | 7 | 6 | 458 | 48 |
| r7 | 9 | 7 | 4 | 35 | 35 | 8 | 2 | 1 | 6 |
| r8 | 8 | 6 | 2 | 9 | 4 | 1 | 3 | 7 | 5 |
| r9 | 5 | 3 | 1 | 2 | 7 | 6 | 8 | 49 | 49 |
Bold digits are given, blue digits were placed while solving, small digits are candidates. Highlighted cells form the pattern, crossed-out candidates are removed, and a circled candidate is the digit to place.
- r5c4, r5c5, r7c4, r7c5 are the corners of a rectangle in two rows, two columns and two boxes, 5 and 8. All four hold the candidates 3 and 5.
- r7c4, r7c5 hold nothing but 3 and 5. The other two corners, r5c4, r5c5, hold 3 and 5 together with other candidates.
- In row 5, 3 has no candidate cell other than r5c4, r5c5. So one of these two corners must be 3.
- If the other one were 5, the four corners would hold only 3 and 5, and swapping the two digits would give a second solution. Every Bare Sudoku puzzle has exactly one solution, so neither corner can be 5.
- Remove 5 from r5c4, r5c5.
Common mistakes
- Removing A instead of B. The digit that is confined to the two roof cells stays; the other digit of the pair is removed from them.
- Using a rectangle whose corners lie in four boxes, or a unit where A has a third candidate cell. Without the two-box shape there is no deadly pattern, and without the strong link on A one roof corner could still be B.
- Applying it to a puzzle that may have more than one solution. Every Bare Sudoku puzzle is checked for exactly one solution, so the argument holds in the game.
In Bare Sudoku
Needed from level: Expert
In the game, the hint says: Unique Rectangle Type 4
Frequently asked questions
What is a Unique Rectangle Type 4?
A rectangle of four cells in two rows, two columns and two boxes that all hold the candidates A and B. Two neighbouring corners hold only A and B; the other two hold more, and in a unit they share, A is confined to those two corners. Then B is removed from both of them.
Why does Type 4 work?
One of the two roof corners must be A, because A has nowhere else to go in their shared unit. If the other roof corner were B, all four corners would hold only A and B, and the two digits could be swapped for a second solution. A proper Sudoku has one solution, so B is impossible in both roof corners.
How is Type 4 different from Type 1?
Type 1 has three corners with only the pair and removes both digits from the fourth. Type 4 has two corners with only the pair and uses a strong link on one digit between the other two corners; it removes the other digit from those two.
Which Unique Rectangle types does Bare Sudoku find?
Types 1 to 6 and the Hidden Rectangle. Type 1 and Type 4 are the most common in its puzzles; the hint names the type it used.
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