Unique Rectangle
Short answer: four cells at the corners of a rectangle that spans exactly two boxes, three of them holding only the same two shapes. If the fourth held only those two as well, the pair could be swapped right round the rectangle to give a second complete solution. A proper puzzle has one, so the fourth cell must be one of its other shapes.
What a Unique Rectangle looks like
Every technique before this one argues from the grid. This one argues from the puzzle. It does not ask what the cells can hold — it asks what the setter can have intended, and uses the promise that the puzzle has exactly one answer as if it were another rule of sudoku. That makes some solvers uneasy, and the discomfort is worth taking seriously.
The two boxes matter. If the four corners were spread over four boxes, swapping the pair would break a box constraint and the second solution would not exist — so there would be nothing to avoid. The pattern only threatens uniqueness when each of the two boxes involved holds one corner of each pair.
How to use a Unique Rectangle
- Look for repeated pairs. Scan for cells whose pencil marks are exactly the same two shapes. You need at least three of them before there is anything to look at.
- Check they form a two-box rectangle. Three of them must sit at corners of a rectangle — two rows, two columns — and the four corners must fall in exactly two boxes. Four boxes means no pattern.
- Look at the fourth corner. It must contain both shapes of the pair plus at least one more. If it holds only the pair, the puzzle already has two answers and something has gone wrong earlier in your solve.
- Strike the pair from it. Remove both shapes of the pair from that fourth cell. What is left is what it must be — often a single candidate, which places the cell outright.
This is type 1, the only form that needs no extra machinery. The other types handle a fourth cell with exactly one extra shape, or two cells with extras, and they eliminate elsewhere on the grid rather than in the rectangle. The same argument applied to cells you have already filled in is an avoidable rectangle.
Is it legitimate?
It depends on the puzzle, and that is not a dodge. A Unique Rectangle assumes the grid you were given has exactly one solution — true of any properly set puzzle, and true of every puzzle in form9, but not true of a grid you have half-invented or copied down wrong. It also proves nothing about the position on its own: if you make a mistake earlier, the rectangle happily eliminates the correct shape. Purists avoid it for the same reason they avoid guessing — it reasons about the puzzle rather than the grid. Everyone else uses it, because it is often the shortest way through.
Find Unique Rectangle in your own grid
Paste a puzzle and this page will look for Unique Rectangle in it — the same pattern shown above, in your grid instead of ours. Nothing is uploaded; it runs in your browser.
Frequently asked questions
Why does it need exactly two boxes?
Because that is what makes the swap legal. With one corner of each pair in each of two boxes, exchanging the two shapes around the rectangle leaves every row, column and box still valid — a genuine second solution. Spread the corners over four boxes and the swap breaks a box, so no second solution exists and there is nothing to rule out.
What if all four corners hold only the pair?
Then the puzzle as you have it does have two solutions. On a properly set puzzle that means you have made an error earlier — most often a wrongly placed shape or a candidate you struck by mistake. Go back rather than eliminating anything.
Can I use this on any puzzle?
Only on one you trust to have a single answer. Puzzles from a reputable source, and every puzzle in form9, qualify. A grid you typed in by hand, or one from a generator that does not check uniqueness, does not — and there the technique can strike the true shape.
What are the other types?
Types 2 to 6 handle the cases where more than one corner has extra candidates, or where the extras interact with a nearby unit. They eliminate away from the rectangle rather than inside it and are much harder to spot. Type 1 covers most of the value.