Unique Rectangles in Sudoku: the Deadly Pattern and When It’s Safe

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Published April 14, 2026 · Updated October 14, 2026

Every other technique in our Sudoku Helper follows from the rules: each digit once in every row, column and box. A unique rectangle follows from a promise instead, that the puzzle has exactly one solution. Take puzzle 1011 from our Evil sudoku bank. Nineteen hidden singles and thirteen naked singles fill all but 21 squares, and then the singles stop. R3C9, R4C8 and R4C9 hold only 6 and 7; R3C8, the fourth corner of their rectangle, holds 6, 7 and 4. The promise says R3C8 is 4.

A rule that leans on a promise raises two questions, and we measured both on our own puzzles. How often do you need one? Never below Evil, and on 17 of the first 500 Evil puzzles. And what does the same hint do to a puzzle that has two solutions? It can wipe out both, as a real board below shows.

What is a unique rectangle, and what is a deadly pattern?

A unique rectangle, UR for short and sometimes called a deadly rectangle, is four open squares at the corners of a rectangle: two rows, two columns and exactly two boxes. Three corners hold the same two candidates and nothing else; they are the floor. The fourth, the roof, holds those two digits and at least one more. The rule: strike the two digits from the roof. Its answer is one of its extras.

The rule exists to dodge a deadly pattern: all four corners down to the same two digits. Those could be swapped round the rectangle without breaking any rule, so a puzzle that reached one would have two solutions, and a puzzle with one solution never does.

That is Type 1, the basic form, and the only one our helper looks for. The other types start from the same four squares with the extra candidates spread differently; they come further down.

Why can’t the roof take either digit?

Start with the floor on puzzle 1011. R3C9 shares column 9 with R4C9, and R4C8 shares row 4 with it. Whichever of 6 and 7 goes into R4C9, the other must go into both R3C9 and R4C8. So the floor can be filled in two ways only: R3C9 and R4C8 both 6 with R4C9 7, or the other way round.

Now the roof. R3C8 shares row 3 with R3C9 and column 8 with R4C8, so it can never take their digit. The only one of the pair it could take is R4C9’s, and then the four corners would hold 6 and 7 crosswise with nothing to fix which way round. Swap every 6 and 7 among them and each of rows 3 and 4, columns 8 and 9 and boxes 3 and 6 still has one of each: a second solution. A puzzle with one solution can’t allow that, so the roof is neither 6 nor 7.

The rectangle never says which way round the floor goes; the singles after it do (6 in R3C9 and R4C8). Every mark in the first picture is one the digits alone allow, since nothing but singles came before. Load this position into the helper: its first Next Hint pencils in the marks, and the second names this rectangle and says two techniques are available. The other is an Almost Locked Set.

Why must the four squares be in exactly two boxes?

Because the swap has to keep every box legal. Rows and columns look after themselves: each of the rectangle’s rows and columns has one 6 and one 7 among its corners, before the swap and after it. A box only does if it holds two of the corners. On puzzle 1011 box 3 holds R3C8 and R3C9 and box 6 holds R4C8 and R4C9, so each keeps its 6 and its 7. Spread the corners over four boxes and each box holds one corner; swap its 6 for a 7 and the box has two 7s. That swap isn’t a second solution, so there is nothing to avoid and nothing to remove.

The quick test: the two rows lie in the same band of three, or the two columns in the same stack. The helper’s run over all 5,000 Evil puzzles produced 204 unique rectangles: 105 had their rows in one band and 99, like puzzle 1011’s, their columns in one stack.

Clues matter in the same way. A clue can’t be swapped, so four corners with a clue among them are no deadly pattern; our helper only considers four open squares. Some solvers stretch the idea to corners they have filled in themselves, calling it an avoidable rectangle. It isn’t among the helper’s eighteen.

What happens on a puzzle with two solutions?

The rule is only as good as the promise. To see what it does without one, we took the first 500 Evil puzzles and deleted one clue at a time. That makes 13,041 new puzzles, and 656 of them have exactly two solutions, the same set the BUG guide tested. In 101 of the 656 the two solutions differ in just four squares: a deadly pattern made real.

Puzzle 1011 gives one. Delete the 6 from R2C3 and the solutions differ only in R2C1, R2C3, R6C1 and R6C3, where 4 and 6 change places. Until October 2026 our helper took the promise on trust; after 32 moves it offered a unique rectangle on exactly those four squares.

The hint leaves R2C1 with 7, a digit neither solution has there, and from that moment the puzzle has no solution at all. Sixteen moves later the helper reached R4C8 with no candidates left.

Across all 656, the helper played a unique rectangle on 38. On 20 it did no harm: the digits it struck were in neither solution. On the other 18 the rectangle sat on exactly the four squares where the solutions differ and struck the two digits both solutions put in the roof; 17 of those went on to a square with no candidates. It never chose one solution over the other. When it went wrong, it removed both. BUG+1 fails the same way: on the same 656 puzzles it reached 22 positions and named a digit in neither solution at 21.

So the rule is safe exactly where the promise holds. Each of the 5,000 puzzles in our Evil bank and the 5,000 in our Extreme bank has exactly one solution; we count them. Our Easy, Medium, Hard and Expert puzzles come from a generator that removes a digit only while singles can still finish the grid (with locked candidates on Hard, and naked pairs as well on Expert), which leaves one solution by construction. A puzzle from a source that makes no promise, or one copied with a clue missing, can have two. Our helper now counts the solutions of a puzzle you type in, paste or link to, stopping at two, and on one with more than one it leaves unique rectangles out and says why. If a unique rectangle ever leads to a square with no candidates, either a mark was wrong before it or the puzzle has more than one solution.

What are unique rectangle Types 2 to 6?

When two corners carry extra candidates, the rectangle can still pay out, in other ways. The numbering isn’t standard. Types 2 to 4 mostly agree from one site to the next; 5 and 6 don’t, and SudokuWiki’s Type 5 is a different pattern from HoDoKu’s. With the floor two squares in one row or column and the two roofs in the other:

• Type 2: both roofs have the same single extra digit. One roof must be that digit, so it goes from every square that sees both roofs.
• Type 3: the roofs’ extras act as one square. If they are 1 and 2, say, and another square in the roofs’ row holds only 1 and 2, the three squares work like a naked pair: 1 and 2 go from the rest of the row. With more extras, or more squares, it works like a naked triple instead.
• Type 4: one of the rectangle’s digits can go nowhere else in a row, column or box the roofs share, so one roof holds it. Then the other digit goes from both roofs.
• Types 5 and 6 (in the numbering of the HoDoKu solver) put the roofs on a diagonal. Type 5 is Type 2 with diagonal roofs, or with the extra in three corners. Type 6 strikes a rectangle digit from both diagonal roofs when that digit can go nowhere else in the rectangle’s two rows, or its two columns.

We measured Type 2, because it is cheap to check. Slotted into the helper’s list just after Type 1, it was the move on 5 of the first 500 Evil puzzles, each time in place of an Almost Locked Set, and it finished no puzzle there that the eighteen couldn’t. Across all 5,000 it was the move on 53, three of them where the eighteen had nothing, and the puzzles finished rose from 3,655 to 3,659. On the helper’s way through those puzzles, at the 4,417 positions with no single left, a Type 1 rectangle was on the board at 139 (3.1%) and a Type 2 at 61 (1.4%). Puzzle 278 is one of the five:

At that point the helper’s only move is an Almost Locked Set; the Type 2 is far easier to see. (The 9 turns out to be in R4C2.) We haven’t measured Types 3 to 6, and every other count in this article is for Type 1.

Where do unique rectangles turn up, and how do you find one?

Look late in a puzzle. The rectangles our helper played came with a median of 35 squares still open, and the one on puzzle 1011 with 21: four squares can only line up once plenty of them are down to two candidates. Then:

1. Find two squares in one box, in the same row or column, holding the same two candidates and nothing else. On puzzle 1011, R4C8 and R4C9 hold 6 and 7 in box 6. Two like that are a naked pair already.
2. Follow their columns (or rows) into the other boxes of the same stack (or band). Any row there where both squares are open and hold both digits gives the other two corners: row 3, R3C8 and R3C9.
3. Read the two new corners. One holds just the pair and the other more: strike the pair from that one. Both with extras: check Types 2 to 4. Both with just the pair: a deadly pattern on the board, so a mark has gone missing, a digit is wrong, or the puzzle has two solutions.

Don’t expect the roof always to turn into a single. Of the 204 rectangles, 108 had a roof with one extra, which became a naked single at once; 82 had two extras and 14 had three. To check a position of your own, enter it in the helper: Unique Rectangle appears in its Available Techniques menu whenever its pencil marks hold a Type 1.

How rare are unique rectangles?

To size it up, our helper solved puzzles from each of our levels, always choosing the simplest of its eighteen techniques, and we counted every puzzle where a unique rectangle was ever the choice.

• Easy to Expert: never. None of the 800 puzzles, 200 per level, called for one. On Expert a rectangle showed up at 25 of the 277 positions where the singles had run out; a simpler move was always available alongside it.
• Evil: about one puzzle in 26. 17 of the first 500 needed one, and 195 of all 5,000. Where no single was left, one was available at just 3.1% of positions, less often than any other technique on the helper’s list beyond the singles.

Our helper’s eighteen techniques, simplest first, as its Next Hint takes them. “Hardest step”: nothing further down its list was needed; only the Almost Locked Set is. The first 500 lines, from a bank stored in score order, are harder than the rest. With five more techniques added before the Almost Locked Set (finned fish, 3D Medusa and others), 19 of the first 500 need a rectangle; the order decides which move gets the credit. Measured October 2026.

What if you’d rather not rely on uniqueness, or can’t find a rectangle?

You rarely have to. On puzzle 1011, the Almost Locked Set the helper lists beside the rectangle takes the 7 from R3C8 (and from R8C7) instead; play that, and one hidden single and 20 naked singles finish the puzzle with no promise needed.

If there is no rectangle, don’t keep hunting: at about 97 Evil positions in 100 with no single, there isn’t one. Try the commoner moves first, hidden pairs, X-Wings and wings such as the XY-Wing, then Almost Locked Sets and chains: XY-chains, X-cycles and, failing everything, forcing chains. The guide to sudoku techniques lays out the order to learn them in.