Step 20 · Master
What is Unique Rectangle? How to Solve Killer Sudoku
Uses the fact that a proper puzzle has only one solution: four cells forming a rectangle across two boxes can't all end up as the same pair of digits, so you can remove the candidates that would make that happen.
What it is
Uses the rule that a proper sudoku has exactly one solution.
Picture the four corners of a rectangle. They sit in 2 rows, 2 columns, and 2 boxes.
If those corners could only hold 3 and 7, then "top-left 3, top-right 7, bottom-left 7, bottom-right 3" and the swapped "top-left 7, top-right 3, bottom-left 3, bottom-right 7" would both work: every row, column, and box still gets one 3 and one 7. That's two solutions, which can't happen.
So any candidate that would leave the corners with only 3 and 7 can be removed.
Example: three corners can only be 3 or 7, and the fourth can be 3, 7, or 9. If the fourth were 3 or 7, there would be two solutions, so it must be 9.
How to spot it
Look for cells with only the same two candidates (like 3 and 7) on two or three corners of a rectangle. The rectangle must span exactly two boxes.
Example
Dotted outlines are cages, with their total in the top-left corner. Yellow cells are where to look, orange cells are the reason, red digits are candidates you can remove, and the green digit is the answer.
A proper sudoku has exactly one solution. This trick uses that rule. R3C4, R3C5, R4C4, and R4C5 are the four corners of a rectangle. They sit in just 2 rows, 2 columns, and 2 boxes. Imagine the four corners could only hold 2 and 8. Then "R3C4=2, R3C5=8, R4C4=8, R4C5=2" works, and so does the swapped "R3C4=8, R3C5=2, R4C4=2, R4C5=8". Either way, each of those rows, columns, and boxes gets one 2 and one 8, so no rule is broken. No cage contains just one of these corners, so the swap keeps every cage sum the same too. That would mean two solutions, so the four corners can never end up as just 2 and 8.
On this board, the three corners R3C4, R4C4, and R4C5 can only be 2 or 8. If R3C5 were also 2 or 8, all four corners would be just 2 and 8, and the swap above would work. So R3C5 is neither 2 nor 8. Remove 2 and 8 from R3C5.