Every published sudoku has exactly one solution, and the unique rectangle turns that promise into deductions. Four cells forming a rectangle across exactly two boxes, all containing the same two candidates, would be fatal: the two digits could swap around the rectangle and produce a second valid solution. A proper puzzle cannot allow it, so something must break the pattern — and whatever prevents it is forced.
Three corners hold bare 3-9; the fourth (red) must take its extra candidate 1 to avoid a second solution.
Corners r3c4, r3c6 and r7c4 hold only 3-9; the fourth corner r7c6 shows 1-3-9. If r7c6 ever collapsed to 3 or 9, the four corners would form a deadly 3-9 rectangle with two interchangeable solutions — impossible in a proper puzzle. So r7c6 must be 1, and both the 3 and the 9 vanish from it.
Verify the four corners span exactly two boxes and that the bare-pair corners truly contain nothing else. A stray extra candidate in a 'clean' corner voids the whole argument — and never use this on puzzles that are not guaranteed unique.
These grids come from rated games on this site — our solver confirmed that unique rectangle was part of the shortest solving path for each one.
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