What this technique is
Unique rectangles with missing candidates is a small but useful extension of standard unique-rectangle logic. In a classic unique rectangle you find four cells that form a rectangle (two rows × two columns) and each corner contains only two digits, say a and b. That situation can allow a swap of a and b that would produce two solutions unless another restriction eliminates one of the swaps. In the “missing candidates” variant one corner has already had one of those two digits removed — it’s no longer bi-valued in a/b — and that change can still let you eliminate candidates elsewhere.
Why the “missing” corner matters
When one corner no longer has both a and b, the set of placements that could create a duplicate solution is reduced but not always removed. If that missing candidate is, say, a, then the only places left inside the rectangle that could be a are the remaining bi-valued corners. If some outside cell sees all of those remaining “a” possibilities, placing a in that outside cell would allow the rectangle to be filled with b’s in a way that causes a second solution. To prevent that, the outside cell cannot be a — so you can eliminate a there.
When to look for it
- After normal candidate cleanup: look for near-unique-rectangle shapes where three corners are bi-valued in the same pair and the fourth corner is not.
- Pay attention when one corner is reduced to a single of the two digits (or lost one of the two) because it turns a full-UR into this missing-candidate case.
- Check cells that see both of the remaining bi-valued locations for the digit that’s now limited inside the rectangle — those might be eliminations.
Step-by-step practical test you can use
1) Find the rectangle: pick two rows and two columns with four empty corner cells.
2) Identify the two digits of interest, a and b, that are the only candidates in three of the corners (or in all four before one was reduced).
3) Confirm the fourth corner is missing one of those digits (it might contain the other digit or other candidates).
4) Locate cells outside the rectangle that can see (are in the same row, column or block as) every remaining corner that still could contain that missing digit. If a single outside cell sees all those spots, the missing digit can be eliminated from that outside cell.
Worked candidate-grid example
Below is a small, focused example. The rectangle corners are at (row 2, col 3), (2,8), (7,3) and (7,8). The two UR digits are 5 and 9.
- (2,3), (2,8) and (7,8) are all bi-valued {5,9}.
- (7,3) has lost candidate 9 and is only {5} — that’s the missing candidate case.
Because 9 cannot be placed at (7,3), the only remaining places for 9 inside the rectangle are (2,3), (2,8) and (7,8). The two cells in column 8 that could be 9 are (2,8) and (7,8). Any cell that sees both of those (for example a cell in column 8) cannot also be 9, or you would allow an alternate placement that leads back toward a twin solution. In the example the cell at (5,8) includes 9 among its candidates; unique-rectangle-with-missing-candidate logic removes that 9.
How to read the visual example
- Focus cells mark the four rectangle corners so you can spot the shape quickly.
- Candidates list what’s possible in the empty cells we’re discussing.
- The eliminated mark shows the outside cell that loses a candidate because of the UR-missing-candidate deduction.
What to watch out for
- The outside elimination must see all remaining candidate positions for the digit you want to remove. If it only sees some but not all, you cannot make the elimination.
- This technique depends on the uniqueness assumption (a valid Sudoku has one solution). Use it only as a logical deduction, not as brute-force guessing.
- Make sure the rectangle corners are genuinely limited to the digits you name. If some corner contains additional candidates beyond the two digits, the basic UR logic may not apply without further linking techniques.
Practice tips
- Scan for near-UR shapes early in a solve after you’ve created candidate lists. They’re often easy to miss until candidate lists are tidy.
- Check columns and blocks that intersect the rectangle: outside cells that lie in a common column or block with the two remaining candidate locations are good elimination targets.
- When you find one, mark the rectangle corners and the eliminated candidate in your notes. That helps avoid repeated checking.
Quick summary
Unique rectangles with missing candidates is a small, local deduction: three corners limited to the two UR digits and a fourth corner that has already lost one of those digits can still produce eliminations outside the rectangle. Look for cells that see every remaining inside-place for the missing digit — those outside cells lose that candidate. It’s a tidy addition to your UR toolkit and worth scanning for whenever you tidy candidates.