Chains, Colouring & Uniqueness

Unique Rectangle

A Unique Rectangle uses the fact that a proper Sudoku has exactly one solution. When three corners of a rectangle — two rows, two columns, exactly two boxes — hold only the same pair {a, b} and the fourth corner holds that pair plus other candidates, the fourth corner cannot be a or b: either would let the two digits swap around the rectangle and give the puzzle a second solution. So a and b are removed from the fourth corner, and only its extra candidates remain.

What you need to see on the grid

Look for a Unique Rectangle late in a hard Classic puzzle, when the grid is full of cells with two candidates and the simpler patterns have run dry. It only works when the puzzle has exactly one solution — as every published Skudo puzzle does — so never use it on a puzzle whose uniqueness you cannot trust. Four terms carry the method:

The rectangle: Four unsolved cells where two rows cross two columns. For a Unique Rectangle they must lie in exactly two boxes: both rows in the same band of three boxes, as in the example below, or both columns in the same stack. Spread over four boxes, the same shape proves nothing.
The deadly pattern: Such a rectangle whose four corners hold only the same two digits. It can be filled in two ways that differ only by swapping those digits, and no other cell notices the difference — so a puzzle that reached it would have two solutions.
The extra corner (Type 1): In Type 1, three corners hold exactly {a, b} and one corner holds a, b and at least one more candidate; that corner is the only place the elimination lands. Type 1 is the only type the app's hint engine applies. The literature also describes Types 2 to 6 and hidden rectangles, in which several corners carry extras; they are not covered here.
The uniqueness assumption: The rows, columns and boxes alone do not force the elimination: it holds because the puzzle has a single solution, and every published Skudo puzzle has one. A grid typed in from an unverified source gives no such guarantee, and in Killer the argument fails anyway: swapping the pair can change a cage total, so the swapped grid need not be a solution.

How to apply it, step by step

Step 1 shows a position that can never occur; steps 2 to 5 follow one worked example. Every diagram shows the whole grid: the cells a step needs are filled in, the others are left blank and hatched. A digit missing from a drawn cell is one its row, column or box already holds — even where that part of the grid is blank — or one an earlier step removed, and no simpler technique changes any cell that is drawn.

Step 1See why a deadly pattern can never appear

Look at rows 2 and 3. Every cell is settled except four — R2C2, R2C7, R3C2 and R3C7, the corners of a rectangle — and each of them can only be 4 or 7. They can be filled in two ways: 4 and 7 along row 2 with 7 and 4 below, or the other way round. Both work. Whichever you choose, each of the two rows, the two columns (2 and 7) and the two boxes (1 and 3) gets one 4 and one 7, and no other cell on the grid has to change.

So a puzzle that reached this position would have two solutions. A proper puzzle has exactly one — every published Skudo puzzle does — so it can never get here. That is the whole idea of a Unique Rectangle: a candidate that would lead to this picture must be false.

A position that cannot occur: rows 2 and 3 are complete except the four corners, and each corner is 4 or 7. Swapping 4 and 7 around the rectangle gives a second valid filling.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2	1		8	5	6	9		2	3
R3	9		6	3	1	2		5	8
R4
R5
R6
R7
R8
R9

Digit already on the grid
Corner of the rectangle
Candidates 4 and 7, the digits studied
Not needed for this step

Filled in: rows 2 and 3. The other 63 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.

Step 2Find three corners with one pair and a fourth with extras

Now a position of the kind you meet in a real game. Scan for cells with exactly two candidates and look for two that hold the same pair in one row: R2C2 and R2C7 both hold {4, 7}. Then read the same two columns in another row. Row 3 gives R3C2 = {4, 7}, the third corner, and R3C7 = {1, 4, 7}: the pair plus one extra candidate, 1.

Count the boxes before going further. R2C2 and R2C7 sit in different boxes (1 and 3), so the second row must come from the same band of boxes, rows 1 to 3, to keep all four corners inside those two boxes. Row 1 cannot help: R1C2 is already a 3. Row 3 can, and the four corners lie in exactly two boxes. Had the first two corners shared a box, the second row could come from any band instead.

Technique: Unique Rectangle

Rows 1 to 3 in full. R2C2, R2C7 and R3C2 hold only {4, 7}; R3C7 holds {1, 4, 7}. The four corners lie in boxes 1 and 3.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1	2	3	5			8		9
R2	1		8	5	6	9		2	3
R3	9		6			2		5	8
R4
R5
R6
R7
R8
R9

Digit already on the grid
The rectangle's two rows
Corner of the rectangle
Candidates 4 and 7, the digits studied
Not needed for this step

Filled in: rows 1, 2 and 3. The other 54 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.

Step 3Ask what happens if the extra corner takes the pair

Suppose R3C7 were 4. Row 3 then makes R3C2 a 7, box 1 makes R2C2 a 4, and row 2 makes R2C7 a 7. All four corners now hold only 4 and 7 — the picture of step 1 — and the two digits could be swapped for a second solution. Suppose instead R3C7 were 7: R3C2 becomes 4, R2C2 7, R2C7 4, and the same swap works.

Either way the puzzle would have two solutions. It has one, so R3C7 is neither 4 nor 7. Notice that nothing in the rows, columns or boxes forbids those digits there: only the guarantee of a single solution does. That is why this technique is safe only on a puzzle known to have exactly one solution.

Technique: Unique Rectangle

Boxes 1 and 3, which hold the rectangle. If R3C7 took 4 or 7, the four corners would hold only 4 and 7 and could be swapped.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1	2	3	5					9
R2	1		8					2	3
R3	9		6					5	8
R4
R5
R6
R7
R8
R9

Digit already on the grid
The rectangle's two boxes
Corner of the rectangle
Candidates 4 and 7, the digits studied
Not needed for this step

Filled in: boxes 1 and 3. The other 63 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.

Step 4Remove the pair from the extra corner only

Take both 4 and 7 out of R3C7: either one would rebuild the deadly pattern. Its extra candidate, 1, stays — it is what the corner has left. The three corners holding just {4, 7} keep both digits: each of them still has to be one or the other.

The note under the diagram says it plainly: this removal assumes the puzzle has exactly one solution. Every published Skudo puzzle does, which is why the move is safe here.

Technique: Unique Rectangle

Rows 2 and 3, the rectangle's two rows. 4 and 7 leave R3C7; the three corners holding only {4, 7} keep both.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2	1		8	5	6	9		2	3
R3	9		6			2		5	8
R4
R5
R6
R7
R8
R9

Digit already on the grid
The rectangle's two rows
Corner of the rectangle
Candidates 4 and 7, the digits studied
Candidate removed
Not needed for this step

Filled in: rows 2 and 3. The other 63 cells are left blank: they are not needed for this step. Removed: 4 from R3C7; 7 from R3C7. This removal assumes the puzzle has exactly one solution.

Step 5Collect what the elimination unlocks

R3C7 had three candidates and has lost two, so only 1 is left: a Naked Single, R3C7 = 1. When the extra corner carries a single extra digit, as here, a Unique Rectangle hands you a placement straight away.

With two or more extras the corner keeps all of them, and its shorter list often completes a pair or a single in its row, column or box — look there next.

Technique: Naked Single

Row 3 after the Unique Rectangle. R3C7 lists only 1.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3	9		6			2	1	5	8
R4
R5
R6
R7
R8
R9

Digit already on the grid
The cell that must be 1
Candidate 1, the digit studied
Digit placed
Not needed for this step

Filled in: row 3. The other 72 cells are left blank: they are not needed for this step. Placed: R3C7 = 1.

Common mistakes

A rectangle spread over four boxes

R2C2, R2C8, R6C2 and R6C8 form a rectangle too, but each corner sits in a different box. Swapping the pair there means a box loses its 4 and gains a 7 it may already hold elsewhere, so the swapped grid need not be a solution and nothing is proved. Count the boxes first: exactly two, every time.

Clearing the wrong corner

The pair leaves only the corner that carries extras. Taking 4 or 7 out of one of the three corners that hold just the pair strips that cell of a digit it may need, and the grid becomes unsolvable a few moves later. And when two or more corners carry extras, Type 1 does not apply at all: one of those corners may still be 4 or 7 without forming a deadly pattern.

Using it without a uniqueness guarantee

The whole argument assumes one solution. On a puzzle from an unverified source, on a grid where a wrong digit has crept in, or in Killer, where cage sums can make the swapped grid illegal, the "deadly" corners may be perfectly possible and the elimination can remove the true digit. Use it only on a puzzle known to have a single solution, and on a grid you know is right.

Keep going

Naked PairTwo cells locked to the same two digits clear those digits from every unit they share.
XY-WingThree two-candidate cells force a digit into one of two pincers — cells seeing both lose it.
XY-ChainA run of bivalue cells pins one digit to one end of the chain or the other — every cell that sees both ends loses it.

Learn the move, not just the digit

In a Classic game, the hint in the app shows the next cell you can fill and explains why. When a Unique Rectangle cleared the way, the explanation names it, or another technique of the same level used on the way, as the reason the candidates were ruled out. It does not draw the rectangle for you, and Killer hints never use it.

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