Unique Rectangles II in Sudoku
Unique Rectangles II continues Unique Rectangles. The same four-cell rectangle in two boxes must never be left with only its two digits, because the puzzle would then have two solutions. Cases C, D and E use a strong link or a naked pair to find what that rules out.
A candidate is a digit that can still go in an empty cell. Cells see each other when they share a row, column or 3×3 box. A strong link means a digit has exactly two possible cells in one such group: exactly one must contain it.
How to spot it
- Start as before: four unsolved cells in two rows, two columns and exactly two boxes, all holding the same pair, say 4/6. Cases A and B are on the Unique Rectangles page.
- Case C: two corners hold only 4/6 (the floor). In a row, column or box containing the other two corners (the roof), one pair digit has no place outside the roof. The other pair digit leaves both roof cells.
- Case D: one corner holds only 4/6. At the opposite corner, one pair digit has no place outside the rectangle in that corner's row and column. The other pair digit leaves the opposite corner.
- Case E: the floor holds only 4/6, and the roof's extra digits are exactly two, say 7/8. If another cell in a house with both roof cells holds exactly 7/8, the two make a naked pair: 7 and 8 leave the rest of that house.
Why it works
If the four corners were ever reduced to 4/6, swapping 4 and 6 around the rectangle would give a second solution. A puzzle with one solution cannot get there, so something must stop it.
Case C: one roof cell must be the linked digit. The other roof cell cannot then take the other pair digit, or the rectangle would hold only the pair. Either roof cell could be the linked one, so both lose the other pair digit.
Case D: if the opposite corner took the other pair digit, both of its neighbours in the rectangle would need the linked digit, and the pair-only corner would finish the rectangle. So the opposite corner cannot take it.
Case E: at least one roof cell must use an extra digit, so the roof acts like one cell holding 7/8. Together with the 7/8 cell it locks 7 and 8 into two cells of the house.
A real example from Sudokuno
The board below is a real Sudokuno puzzle, shown exactly as it stands just before Unique Rectangles II applies. Everything up to this point was solved with ordinary techniques.
| 4 | 5 | |||||||
| 2 | 5 | 9 | 3 | 6 | ||||
| 6 | 1 | 5 | 7 | 3 | 2 | |||
| 2 | 1 | 9 | ||||||
| 1 | 8 | 3 | 2 | 5 | 4 | |||
| 2 | ||||||||
| 5 | 9 | 6 | 8 | 2 | ||||
| 5 | ||||||||
| 4 | 8 | 5 | 9 |
Click a candidate to cross it out; click it again to restore it. Remove the 2 candidates ruled out by this technique, then press Check. This exercise leaves your game unchanged.
Sudoku: SNC-00895 — Sudokuno
- Houses: column 5
- Pattern: R4C2, R4C5, R6C2 and R6C5
- Removes 4 from R4C5 and R6C5
- Step 13 of 63 in the solve
Key: floor target cells
How the hint helps you
Sudokuno's hint reveals the structure in three steps, without giving away the digit straight away.
- Hint 1 The hint names Unique Rectangles II. No cells or digits are shown yet.
- Hint 2 The four corners are marked. They lie in two rows, two columns and exactly two boxes, and share one candidate pair.
- Hint 3 The case appears: the floor and roof with the linking house (C), the pair-only corner and the opposite corner (D), or the roof and its naked-pair partner (E).
- Hint 4 The targets are marked. Removing them is what keeps the rectangle from ever holding only its pair.
Every case rests on the puzzle having exactly one solution. Sudokuno only offers these steps on puzzles where that is known.
Ready to try it yourself?
You will get a real Sudokuno puzzle at Specialist level S12, where Unique Rectangles II is needed to finish it.
Practise Unique Rectangles IIWhat Sudokuno covers
Sudokuno covers cases C, D and E. Case E uses a naked pair only; larger subsets, Extended Unique Rectangles and other uniqueness patterns are outside this technique.