Hidden Rectangle

A Hidden Rectangle is a uniqueness technique in which the deadly pattern is hidden among other candidates. Four cells at the corners of a rectangle in two rows, two columns and two boxes all hold the candidates A and B, and one corner holds nothing else. In the row and in the column of the opposite corner, A has no candidate cell beyond the corners of the rectangle. If that opposite corner were B, both corners next to it would have to be A, and the first corner would have to be B: the four corners would hold only A and B, a deadly pattern with two solutions. So B is removed from the opposite corner. It is an Expert technique in Bare Sudoku.

When to use it

When a rectangle's corners all hold the same two candidates, one of them holds nothing else, and the other three still have extra candidates, so that Type 1 and Type 4 do not apply. Hidden Rectangles are about as common as Type 4 in Bare Sudoku puzzles. They need complete notes for one digit of the pair in two lines.

How to spot it

Start from a cell with exactly two candidates, A and B. Look for a rectangle through it, in exactly two boxes, whose other three corners also hold A and B. Take the corner opposite your starting cell and check its row and its column: if A has no candidate there outside the rectangle, that corner loses B. Then try the same with the roles of A and B swapped.

Example

Hidden Rectangle
c1c2c3c4c5c6c7c8c9
r1542896791678783
r29683467567812
r3768128263495
r46839125261847
r56857194692318
r6214387569
r74765239291382818
r8195623837274
r9328714956

Bold digits are given, blue digits were placed while solving, small digits are candidates. Highlighted cells form the pattern, crossed-out candidates are removed, and a circled candidate is the digit to place.

  1. r1c5, r1c7, r2c5, r2c7 are the corners of a rectangle in two rows, two columns and two boxes, 2 and 3. All four hold 6 and 7. r2c5 holds nothing else; the opposite corner, r1c7, also holds 8.
  2. In row 1, the only candidate cells for 6 are r1c7 and r1c5. In column 7, they are r1c7 and r2c7.
  3. Suppose r1c7 were 7. Then r1c5 and r2c7 would both have to be 6, and r2c5, which sees both of them, would have to be 7. The four corners would hold only 6 and 7, and swapping them would give a second solution.
  4. Every Bare Sudoku puzzle has exactly one solution, so r1c7 is not 7. Remove 7 from r1c7.

Common mistakes

In Bare Sudoku

Needed from level: Expert
In the game, the hint says: Hidden Rectangle

Play Sudoku

Frequently asked questions

What is a Hidden Rectangle in Sudoku?

A rectangle in two rows, two columns and two boxes whose four corners all hold the candidates A and B, one of them nothing else. If A is confined to the rectangle in both the row and the column of the opposite corner, that corner cannot be B, because the four corners would otherwise hold only A and B and the puzzle would have two solutions.

Why is it called hidden?

Because the deadly pattern is covered by extra candidates in three of the corners. In the plain Unique Rectangle you see three corners with just the pair; here you must look at where the digit A can go in two lines to uncover the pattern.

How is a Hidden Rectangle different from Type 4?

Type 4 needs two neighbouring corners with only the pair and one strong link in the unit of the other two. A Hidden Rectangle needs just one corner with only the pair and two strong links, in the row and the column of the opposite corner.

Can the two-candidate corner be the one that loses B?

No. The strong links run through the row and the column of the corner with extra candidates, and that corner is where B is removed. The corner with only the pair is the one that would be forced to B in the deadly pattern.

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