The unique rectangle is the first technique that argues not from the digits but from the puzzle itself. Its raw material is four cells on the corners of a rectangle — lying in two rows, two columns, and, crucially, just two boxes — that could all hold the same two candidates. If they ever did, the puzzle would break.
Here is the uniqueness argument. Picture a finished grid whose four corners hold only those two digits, one diagonal each way. Swap them — every 2 for a 9 and back — and every row, column, and box involved still checks out, because the four cells pair up inside the same two of each. That is two valid solutions from one grid. A proper Sudoku is published with exactly one solution, so the bare rectangle is a deadly pattern no legitimate puzzle can contain.
In its simplest form (type 1), three corners are already reduced to the bare pair while the fourth still carries extra candidates. The deadly pattern is avoided only if that fourth corner takes one of its extras — so the pair’s two digits are struck from it. Note the two-box condition: spread over three or four boxes, the swap would break the boxes, no second solution would exist, and the argument would not hold.
On the board
Three corners of this rectangle — R2C4, R2C7, and R3C7 — are bare {2,9} cells. The fourth, R3C4, shows {2,7,9}. The four cells span rows 2 and 3, columns 4 and 7, and just two boxes. If R3C4 were 2 or 9, all four corners would hold nothing but 2s and 9s, and the finished puzzle could swap the diagonals into a second valid solution. Uniqueness forbids it: the 2 and the 9 are struck from R3C4, and the cell resolves to 7.
This is the exact board the app's interactive Unique Rectangle demo uses — step 29 of its 65-step walkthrough, drawn here with its live pencil marks.
