Unique Rectangles

Every other technique on this ladder reasons about the numbers in the grid. The Unique Rectangle reasons about the puzzle itself: a properly made sudoku has exactly one solution, so any arrangement that would give it two cannot be part of the answer. That single fact removes candidates nothing else can touch.

What the pattern is

Four cells sitting at the corners of a rectangle — two rows, two columns — and, crucially, spread over exactly two boxes. If all four corners end up holding only the same two candidates, say 3 and 7, the puzzle is broken:

        c2        c5
      +--------+--------+
 r4   |  37    |  37    |
      |        |        |
 r9   |  37    |  37    |
      +--------+--------+
        box 4     box 5      <- exactly two boxes

  3 7        7 3
  7 3   and  3 7   both solve the grid

Swap the two digits diagonally and every row, column and box still holds each digit exactly once. Two valid solutions. Since a proper puzzle has one, this shape — the deadly pattern — can never appear in the finished grid, and whatever prevents it must be true.

The same argument works for longer even loops of cells that alternate between two units: a six-cell or eight-cell Unique Loop is read exactly like the four-cell rectangle.

The four types

The deadly pattern is only a threat when the four corners are close to holding just the pair. What you eliminate depends on how many corners carry extra candidates.

Type 1 is the common one and the easiest to trust: if that corner also held only 3 and 7, the deadly pattern would be complete, so it must keep one of its extras.

Bivalue Universal Grave (BUG)

A rectangle is a deadly pattern on four cells. The Bivalue Universal Grave is the same argument stretched across the whole grid: if every unsolved cell held exactly two candidates, and every candidate value occurred exactly twice in every row, column and box, the entire grid could be swapped between two solutions at once. A proper puzzle can never actually settle into that state.

In practice you catch it one cell short of that — every unsolved cell bivalue except one, which holds three candidates. That state is called BUG+1. Two of the extra cell's candidates behave exactly like a true BUG: each appears exactly twice in every row, column and box it belongs to. The third appears three times in each of the cell's own units — its row, its column and its box. That third candidate has to be the answer — if the cell took either of the other two instead, the rest of the grid would be left in a genuine BUG state, which a proper puzzle cannot have.

r5c4 candidates: 4, 6, 9   (every other unsolved cell in the grid is bivalue)

digit 6: row 5 x2, column 4 x2, box 5 x2   -> clean, appears twice everywhere
digit 9: row 5 x2, column 4 x2, box 5 x2   -> clean, appears twice everywhere
digit 4: row 5 x3, column 4 x3, box 5 x3   -> the odd one out

r5c4 = 4

Types 2 to 4 generalise the idea the same way the unique rectangle types above do: more cells carry extra candidates, and the argument turns into an elimination across a shared unit instead of a single placement. BUG+1 is harder to spot than a rectangle — most solvers, including automated raters such as Sudoku Explainer, rate it noticeably above the plain unique rectangle.

When to look for it

Uniqueness is a late technique, not because it is hard to see but because it is easy to see everywhere once you start looking. Run the singles, subsets, pointing pairs and fish first; reach for a rectangle when the grid has gone quiet and pairs are repeating across two boxes.

One condition is absolute: the puzzle must be known to have a single solution. Every puzzle on this site does, so the argument is always available here. If you are solving a grid of unknown origin, or one you built yourself, uniqueness can quietly give you a wrong answer.

How to apply it, step by step

1. Scan for cells holding exactly two candidates and note the pair.
2. Look for the same pair in a cell sharing its row, then check the two cells that complete the rectangle.
3. Confirm the rectangle spans exactly two boxes — four boxes means no deadly pattern and no deduction.
4. Count the corners that carry extra candidates: one corner is type 1, two corners are type 2, 3 or 4.
5. Apply the elimination for that type, then re-run the basic techniques — a rectangle usually unlocks a chain of singles.

A worked example

R4C2, R4C5 and R9C5 hold only 3 and 7. R9C2 holds 3, 7 and 9:

        c2         c5
      +---------+---------+
 r4   |  3 7    |  3 7    |
 r9   |  3 7 9  |  3 7    |
      +---------+---------+
       box 4      box 5

  R9C2 cannot be 3 and cannot be 7  ->  R9C2 = 9

If R9C2 were 3 or 7, all four corners would hold nothing but 3 and 7, and the grid would have two solutions. It has one, so R9C2 keeps its 9 and drops the pair — a placement, not just an elimination.

Common mistakes

• Using a rectangle that spans four boxes. The diagonal swap then breaks a box, so there is no deadly pattern and no argument.
• Applying it to a puzzle that may have several solutions. The whole technique rests on the puzzle being proper.
• Forgetting that the other three corners must hold only the pair. One extra candidate anywhere else and the pattern is a different type, or nothing at all.
• Confusing it with Rectangle Elimination, which is ordinary single-digit logic and makes no assumption about the solution being unique.
• Eliminating the extras instead of the pair in type 1. The corner keeps its extras; it loses the two pair digits.

Where you meet it

Unique Rectangles are a diabolical-tier pattern. They start paying off in Expert puzzles and stay useful all the way up, often turning a stalled grid back into a run of singles. They appear in Killer Sudoku too, where the cage sums frequently leave exactly the two-candidate corners the pattern needs.