Techniques · Expert
Unique rectangle
Four corners and a swap
A unique rectangle, the kind on this page, is four cells in two rows, two columns, and two boxes. The digits in the corners are the same two. Type 1 is the version where three corners hold exactly those two digits, and the fourth holds those two plus something else. The two digits come off the fourth corner.
On this board the digits are 5 and 8. The rectangle is row 1 and row 3, column 6 and column 7. That is two boxes: box 2 holds the column 6 corners, and box 3 holds the column 7 corners. Three corners are clean. Row 1, column 7 can be 5 or 8. Row 3, column 6 can be 5 or 8. Row 3, column 7 can be 5 or 8. The fourth corner is row 1, column 6, and it can be 5, 8, or 9.
Why the fourth corner cannot be 5 or 8
Suppose row 1, column 6 were only 5 or 8, like the other three. Then the rectangle could flip. One solution puts 5 in row 1, column 6 and row 3, column 7, and 8 in row 1, column 7 and row 3, column 6. The other solution swaps them. Each of those two rows, two columns, and two boxes still holds one 5 and one 8. No other cell in the grid can tell the two arrangements apart. The puzzle would have two solutions.
The row does not forbid a 5 in row 1, column 6. The column does not forbid an 8 there either. Both digits are still legal in the houses. What removes them is a fact the pencil marks do not show: this puzzle has one solution. The 5 and the 8 leave row 1, column 6. The 9 stays, and the step places 9 there.
The assumption, said plainly
This is the move I trust least, and I still use it. It is a real deduction. It rests on the promise that the grid was built with a single solution. A newspaper puzzle, and every puzzle this site generates, comes with that promise. The solver checks it before it will take the rectangle. If the check finds a second solution, the rectangle is skipped and the two digits stay.
If you typed a grid in by hand and never confirmed it has one solution, do not strip this corner. You might be deleting the digit that belongs to the other filling. I have done that on a mistyped puzzle and then spent the evening sure the logic was broken. The logic was fine. The puzzle was not a puzzle.
Three clean corners, one busy one
Row 1, column 6 is the corner with an extra 9. The other three corners are only 5 and 8. Step through. Both of those digits leave, and the cell becomes 9.
See the pattern
Look at these four cells.
The four cells on this board
Row 1, column 4 is 2 and row 1, column 8 is 7. Row 1, column 3 is 4. Row 1, column 9 is 3. The top edge of the rectangle is still open: row 1, column 6 can be 5, 8, or 9, and row 1, column 7 can be 5 or 8.
Row 3 is almost full. Row 3, column 1 is 3, row 3, column 2 is 9, row 3, column 3 is 7, and row 3, column 5 is 4. Row 3, column 9 is 2. The two open corners are row 3, column 6 and row 3, column 7, each 5 or 8.
Three corners, exactly 5 and 8. The fourth, row 1, column 6, adds a 9. The 5 leaves and the 8 leaves. The step places 9 in row 1, column 6. The three clean corners keep both digits. Clearing 5 off row 1, column 7 would throw away a corner the argument needs.
Type 1, and the rectangles that are not it
Type 1 is this shape: three corners naked on the two digits, one corner with extras, and the extras are what remain. If two corners are busy, the swap is no longer free and this elimination is the wrong one. Other types move the deletion onto a different cell. This solver does not do those. Two busy corners, and I leave the rectangle alone.
An X-wing is a rectangle of one digit. These corners hold two digits, and the only cell that loses anything is the busy corner. A rectangle stuck inside one box is a different picture too. Here column 6 is in box 2 and column 7 is in box 3. The two boxes are required.
After the swordfish
Unique rectangles are Expert. They come after the swordfish. The next page is BUG+1. The 1,483 archived NYT puzzles never need one. Their hardest steps stop at triples. Expert is where this site will hand you the rectangle.
I look for two rows and two columns whose four crossings use two boxes, then I count candidates. Three corners on the same two digits, and a fourth that still carries them, is Type 1. Then I remember why the digits leave. One solution was promised. That is the whole move.
That jumps a puzzle forward until Unique rectangle is the next move. Or start a full Expert puzzle.