Unique Rectangle: uniqueness is an extra premise

Why uniqueness matters

A Unique Rectangle avoids a configuration whose digits could be swapped into another solution. That argument assumes uniqueness; the ordinary row, column and box rules alone do not supply it. We discuss Classic with verified uniqueness and sound prior eliminations.

HoDoKu — Uniqueness · SudokuWiki — Unique Rectangles

Our Type 1 example

Let unsolved, non-given cells A1, A4 and C1 have {2,7}, while C4 has {2,5,7}. The corners occupy two rows, two columns and exactly two boxes. Choosing 2 or 7 at C4 makes a four-corner 2/7 arrangement that can be swapped without changing the digit set in any affected unit.

Any completed solution using that arrangement would therefore have a second, swapped completion. Under the uniqueness premise, C4 must instead be 5. This partial scenario does not itself establish that an entire puzzle is unique.

Four boxes and givens change the argument

A1,A4,D1,D4 occupy four boxes. Swapping those corners changes individual box contents, so the same proof fails. A given corner also cannot be swapped. A rectangle-shaped outline alone is insufficient.

When unsure, use another proof

Do not assume uniqueness for an incompletely copied or multi-solution puzzle. Diagonal or cage rules may also forbid the proposed swap. This Classic argument is not a universal rule for every variant.

In Classic 9×9, use Scan to scan the current entries, inspect justified eliminations and open this lesson. Chain and ALS searches are bounded; UR Type 1 and BUG+1 require verified uniqueness. Reading this lesson never changes your saved game.

Try before looking

References and further reading

Read the source explanations and compare them with our examples. Links are references, not endorsements or claims of affiliation.

• HoDoKu — Uniqueness
• SudokuWiki — Unique Rectangles