Unique Rectangle

Three corners sharing a pair would force two solutions.

A Unique Rectangle is a solving move built on a guarantee every published Sudoku makes: the grid has exactly one answer. When three corners of a rectangle that spans two boxes hold the identical two-candidate pair, the fourth corner cannot also be reduced to that same pair — doing so would let the two digits swap freely and hand the puzzle a second solution. The unique rectangle sudoku technique turns that impossibility into a firm elimination, clearing the shared pair from the odd corner out.

Worked example

R1C1, R1C2 and R4C1 are all {1,2}; were R4C2 also {1,2}, the four corners could swap two ways (a deadly pattern), so R4C2 must be 5.121212125
R1C1, R1C2 and R4C1 are all {1,2}; were R4C2 also {1,2}, the four corners could swap two ways (a deadly pattern), so R4C2 must be 5.

Here R1C1, R1C2 and R4C1 are each exactly {1,2}, and together with R4C2 they mark the four corners of a rectangle crossing two boxes, two rows and two columns. R4C2 holds {1,2,5} — the same pair plus a 5. If R4C2 were only {1,2}, the 1s and 2s could be arranged around the rectangle in two mirror-image ways, so the puzzle would have two valid solutions. That is forbidden, so R4C2 cannot be 1 or 2. We eliminate both, leaving R4C2 = 5.

When to use it

Reach for it late in a hard puzzle, once the grid is thick with bivalue cells and your other tactics have stalled. Scan for three cells sharing an identical two-candidate pair that form three corners of a rectangle sitting in exactly two boxes, two rows and two columns. The fourth corner should carry that same pair plus one or more extras — those extras are what survive the elimination, as the 5 does in R4C2.

How to use it

  1. Spot the rectangle. Find four cells in two rows, two columns and exactly two boxes where three share the same two candidates.
  2. Avoid the deadly pattern. If the fourth corner were also just those two digits, the puzzle would have two solutions — which a real puzzle never has.
  3. Clear the fourth corner. Remove both shared digits from the fourth corner, leaving its extra candidates.
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Frequently asked questions

What is a Unique Rectangle in Sudoku?

It is a deduction that leans on a Sudoku having a single solution. When three corners of a two-box rectangle share the same two candidates, the fourth corner is barred from being that identical pair, because a fourth matching corner would let the pair swap two ways and create a second solution. On this page's board, that logic forces R4C2 to shed its 1 and 2 and settle on 5.

How do I spot a Unique Rectangle on the grid?

Look for four cells at the corners of a rectangle that occupies exactly two boxes, two rows and two columns. Three of those corners must be the same bivalue pair — here {1,2} in R1C1, R1C2 and R4C1 — while the fourth carries that pair plus extra candidates, like the {1,2,5} in R4C2. When that shape appears, the shared pair can be eliminated from the fourth corner. The rectangle must span exactly two boxes, because that is the geometry that would let the two digits swap and produce a second solution.

Is using the Unique Rectangle technique cheating?

No. Every valid, properly-published Sudoku has exactly one solution by definition, so reasoning from uniqueness is completely sound rather than a shortcut. The one caveat is that it must not be applied to a puzzle known to permit multiple solutions, where the deadly-pattern assumption breaks down. On a standard single-solution grid like this one, ruling 1 and 2 out of R4C2 is airtight.