A number appears in two blocks, twice, both in the same row or column
This technique allows us to rule out possible numbers from being in a cell. This
may result in a cell only having one possible number in it. To make this easier
to understand, the possible numbers that can be in each cell have been filled in.
This technique works around that fact that if there are only two places where
a given number can be (in a row, column or block), it has to be in one of them. Also,
if it is in one of them, it can't be in the other one, and vice versa. This is called
a conjugate, or conjugate pair.
Find a number in any 3x3 block that appears only twice in that block, and both
of the numbers appear in the same column. Now if you can find another block that:
Is in the same row of three blocks
Has the same number in it twice, and only twice
Both numbers are in the same column
The numbers appear in the same two rows as in the first block
Then you can remove that number from all the other cells in that row (except for
the cells in the two blocks that the numbers appear twice in). This works on the
premise that there are only two places in two blocks where a number can be,
and if the number is in one of those it can't be in the other. However, because there
are only two places where it can be in each block, it cannot be anywhere else in
the matching row.
A
B
C
D
X
X
X
X
X
X
Example 1: Block/Block Interaction
For example, imagine that the only two places in the top left, and top middle blocks
that we could put a 1, are labelled A, B, C and D. Because A and C are in the same row,
if A has the number 1 in, then C cannot be 1. Likewise, if C has the number 1 in,
then A cannot. Whichever cell has the number 1 in, it's either going to be A or C,
never both.
Furthermore, if we put 1 into A, because there are only two places in that
block where the number can go, B cannot be 1 (and vice versa). Because A and C are in
the same row, if A is 1, then C cannot be 1. But we have to have a 1 in that cell
somewhere, and if it's not C, it has to be D.
This means that 1 is either in A and D, or C and B. Whichever way round it is,
we will have a 1 in row 1 (A or C) and row 3 (B or D), so we can't have a 1 in
any of the cells marked X.
This also works in columns, and the blocks don't even need to be next to each other.
A
B
C
D
X
X
X
X
X
X
Example 2: Block/Block Interaction by Column
A
B
X
X
X
X
X
X
C
D
Example 3: Block/Block Interactions
Here is a real-world example:
3
7
4 5
1
8
6
2
9
4 5
2 4 8 9
4 5 8 9
2 8 9
7
3
2 9
6
4 5 8
1
1
2 5 6
5 6 8 9
4
2 5
5 9
3 5
3 5 7
5 7 8
7
4
3
8
1 6
2
5
1 6
9
8 9
1
6 8 9
5
7 9
4
3
2
6 7
5 6 9
5 6
2
3 6 9
1 3 6 7
6 7
8
1 6 7
4
6 9
5
7
4
2
8
6 9
3
1
2 4 9
4 9
3
1
6
5
2 8 9
7 8 9
2 7 8 9
2 6
8
1
7
9
3
2 5 6
4
5 6
Example 4: Block/Block Interaction Cells in block 1,cells in block 2,remove
numbers from these cells
Looking in the top left block, the number 5 appears twice, in [r1,c3] and [r3,c3].
In the top middle block the number 5 appears twice, in [r1,c5] and [r3,c5]. In both
blocks, the numbers appear in the same column. Between both blocks, the numbers
both appear in the same row.
