Twin A

- Code: Select all
` 7 2 . | 3 . . | . . 8 `

3 . . | . 8 5 | . . 2

. . . | 2 6 . | . . 3

-------+-------+------

. . . | . 5 . | 2 3 .

. . 3 | 6 2 . | . . 1

. . 2 | . 3 . | . . .

-------+-------+------

2 3 . | 5 9 6 | . . .

. . 9 | . . 2 | 3 . .

. 6 . | 8 . 3 | . 2 9

Twin B

- Code: Select all
` . . 6 | . . . | . . 2 `

. . . | . . . | . . 4

. . . | . 3 . | . 7 5

-------+-------+------

. . . | . 5 . | . . .

5 . . | 3 . . | 2 . 9

. . . | . . 9 | 5 4 .

-------+-------+------

. . 4 | 1 8 3 | . . .

. 3 . | . . . | 4 . 8

8 9 . | . . 4 | . . .

Thanks to Pat for posting the above two grids.

We notice that digit 6 corresponds to digit 3 in column 4, column 5 and column 6 of twin A and twin B respectively. We expect is a linkage in the pair of linked puzzles which by definition must have a single unique solution. Furthermore, this is the only linkage we can find, that is with 3 digits in one grid correspond to 3 digits in another grid. Hence this must be the starting point for tackling the puzzle.

We can write the correspondence in this way: 6 --> 3.

As a result of this correspondence, column 4, column 5, column 6, row 3, row 5 and row 7 are all "fixed" in position. Fixing a row means that all the digits in the same row must remain within the row. However, a digit can shift to another position in the same row but in a different column within the same box. Similarly, fixing a column means that all the digits in the same column must remain within the column. However, a digit can shift to another position in the same column but in a different row within the same box. As column 4, column 5 and column 6 intersect with row 7 in both grids, this means that all the digits in the first row of box 8 in both grids are fixed in position. This is because all the digits occupy the points of intersections of column 4, column 5 and column 6 with row 7. Hence we have the following correspondence for the three digits in the first row of box 8:

Twin A --> twin B

5 --> 1

9 --> 8

6 --> 3

As row 7 is fixed, it means that only row 8 and row 9 can interchange with each other within the block of rows. On examining row 8 and row 9 in both grids, we find 6 in row 9 of twin A and 3 in row 8 of twin B. This implies that row 8 and row 9 of twin A can be interchanged with each other to produce twin B. This can be shown by interchanging the row 8 and row 9 to produce the new grid for twin A:

Twin A

- Code: Select all
` 7 2 . | 3 . . | . . 8 `

3 . . | . 8 5 | . . 2

. . . | 2 6 . | . . 3

-------+-------+------

. . . | . 5 . | 2 3 .

. . 3 | 6 2 . | . . 1

. . 2 | . 3 . | . . .

-------+-------+------

2 3 . | 5 9 6 | . . .

. 6 . | 8 . 3 | . 2 9

. . 9 | . . 2 | 3 . .

Twin B

- Code: Select all
` . . 6 | . . . | . . 2 `

. . . | . . . | . . 4

. . . | . 3 . | . 7 5

-------+-------+------

. . . | . 5 . | . . .

5 . . | 3 . . | 2 . 9

. . . | . . 9 | 5 4 .

-------+-------+------

. . 4 | 1 8 3 | . . .

. 3 . | . . . | 4 . 8

8 9 . | . . 4 | . . .

After exchanging row 8 with row 9 in twin A, we find the following correspondence in row 8 of both grids tallying with what we have discovered so far:

Twin A --> twin B

6 --> 3

9 --> 8

Such tally with earlier findings implies that exchanging the rows is a correct move. Hence we can fix the position of row 8. AS row 7 and row 8 are fixed, it implies that the row 9 is now fixed in twin A.

As 2 and 4 occupy the points of intersection of column 6 and row 9 in in twin A and twin B respectively, it implies that they are fixed in position. It also implies that 2 in twin A is corresponding to 4 in twin B.

Twin A --> twin B

2 --> 4

On examining column 9 of both grids, we find that the following correspondence tallies with what we have discovered so far:

Twin A --> twin B

2 --> 4

9 --> 8

This implies that column 9 in twin A is fixed in position. Since 2 and 4 correspond to each other in row 2 of twin A and twin B respectively, it implies that row 2 is fixed in position. Since row 2 and row 3 are fixed in position, row 1 is also fixed in position. In other words, there is no exchange of rows in the first three rows of twin A to produce twin B. Since the following fixed rows -- row 1, row 2, row 3, row 5 and row 8 intersect with the fixed column 9, we get the correspondence for the following digits which occupy the points of intersection.

Twin A --> twin B

8 --> 2

2 --> 4

3 --> 5

1 --> 9

9 --> 8

On examining row 4 of both grids, we find that 5 corresponds to 5. This is illogical as we already discovered the following correspondence between 3 and 5.

Twin A --> twin B

3 --> 5

As 2 occupies the point of intersection of fixed row 5 and fixed column 5, it is fixed in position. Row 4 and row 6, however, are not fixed in position. The discrepancy 5 --> 5 implies that row 4 can be exchanged with with row 6 in twin A. Interchanging row 4 and row 6 in twin A, we get the following equivalent grids:

Twin A

- Code: Select all
` 7 2 . | 3 . . | . . 8 `

3 . . | . 8 5 | . . 2

. . . | 2 6 . | . . 3

-------+-------+------

. . 2 | . 3 . | . . .

. . 3 | 6 2 . | . . 1

. . | . 5 . | 2 3 .

-------+-------+------

2 3 . | 5 9 6 | . . .

. 6 . | 8 . 3 | . 2 9

. . 9 | . . 2 | 3 . .

Twin B

- Code: Select all
` . . 6 | . . . | . . 2 `

. . . | . . . | . . 4

. . . | . 3 . | . 7 5

-------+-------+------

. . . | . 5 . | . . .

5 . . | 3 . . | 2 . 9

. . . | . . 9 | 5 4 .

-------+-------+------

. . 4 | 1 8 3 | . . .

. 3 . | . . . | 4 . 8

8 9 . | . . 4 | . . .

Now we have the following correspondence between 3 and 5 in row 4 of twin A and twin B respectively.

Twin A --> twin B

3 --> 5

Since this tallies with what we have discovered so far, we implies that it is correct to exchange row 4 and row 6. Hence we can fix the position of row 4 and row 6. Note that 3 and 5 are fixed in position now because they occupy the point of intersection of fixed row 4 and fixed column 5. Looking at row 6 again, we discover the following discrepancy in correspondence:

Twin A --> twin B

2 --> 5 (false)

3 --> 4 (false)

This correspondence does not tally with what we have discovered so far. As column 7 and column 8 of twin A are not fixed in position, we can interchange them to get the right correspondence for the digits in row 6. Swapping column 7 and column 8 in twin A, we get the following equivalent grids:

Twin A

Now we have the following correspondence between 3 and 5 in row 4 of twin A and twin B respectively.

Twin A --> twin B

3 --> 5

Since this tallies with what we have discovered so far, we implies that it is correct to exchange row 4 and row 6. Hence we can fix the position of row 4 and row 6. Note that 3 and 5 are fixed in position now because they occupy the point of intersection of fixed row 4 and fixed column 5. Looking at row 6 again, we discover the following discrepancy in correspondence:

Twin A --> twin B

2 --> 5 (false)

3 --> 4 (false)

This correspondence does not tally with what we have discovered so far. As column 7 and column 8 of twin A are not fixed in position, we can interchange them to get the right correspondence for the digits in row 6. Swapping column 7 and column 8 in twin A, we get the following equivalent grids:

Twin A

- Code: Select all
` 7 2 . | 3 . . | . . 8 `

3 . . | . 8 5 | . . 2

. . . | 2 6 . | . . 3

-------+-------+------

. . 2 | . 3 . | . . .

. . 3 | 6 2 . | . . 1

. . | . 5 . | 3 2 .

-------+-------+------

2 3 . | 5 9 6 | . . .

. 6 . | 8 . 3 | 2 . 9

. . 9 | . . 2 | . 3 .

Twin B

- Code: Select all
` . . 6 | . . . | . . 2 `

. . . | . . . | . . 4

. . . | . 3 . | . 7 5

-------+-------+------

. . . | . 5 . | . . .

5 . . | 3 . . | 2 . 9

. . . | . . 9 | 5 4 .

-------+-------+------

. . 4 | 1 8 3 | . . .

. 3 . | . . . | 4 . 8

8 9 . | . . 4 | . . .

Now the correspondence of the digits in row 6 of both grids tallies with what we have discovered so far:

Twin A --> twin B

2 --> 4

3 --> 5

By now all the rows and columns except the first three columns in twin A have been fixed. Looking at column 2 of both grids, we notice that 6 in twin A maps correctly to 3 in twin B.

Twin A --> twin B

6 --> 3

With this correct mapping or correspondence, we can fix column 2. Looking at column 1 of both grids, we cannot map any digit in that column for both grids. However we can map the digits in column 1 of twin A and column 3 of twin B. Swapping the column 1 and column 3 of twin A, we get the following equivalent grids.

Twin A

- Code: Select all
` . 2 7 | 3 . . | . . 8 `

. . 3 | . 8 5 | . . 2

. . . | 2 6 . | . . 3

-------+-------+------

2 . . | . 3 . | . . .

3 . . | 6 2 . | . . 1

. . | . 5 . | 3 2 .

-------+-------+------

. 3 2 | 5 9 6 | . . .

. 6 . | 8 . 3 | 2 . 9

9 . . | . . 2 | . 3 .

Twin B

- Code: Select all
` . . 6 | . . . | . . 2 `

. . . | . . . | . . 4

. . . | . 3 . | . 7 5

-------+-------+------

. . . | . 5 . | . . .

5 . . | 3 . . | 2 . 9

. . . | . . 9 | 5 4 .

-------+-------+------

. . 4 | 1 8 3 | . . .

. 3 . | . . . | 4 . 8

8 9 . | . . 4 | . . .

Now looking at column 1 of both grids, we can find the following correspondence which tallies with what we have discovered so far:

Twin A --> twin B

3 --> 5

9 --> 8

With such correspondence that tallies with the previous findings, we can fix column 1 of twin A. Since column 1 and column 2 are fixed in twin A, this implies that column 3 of twin A is also fixed. Now looking at column 3 of both grids, we can map the following digits in twin A and twin B:

Twin A --> twin B

7 --> 6

2 --> 4

Note that the mapping of 7 in twin A to 6 in twin B is new. Digit 7 in twin A must be equivalent to digit 6 in twin B because column 3 is fixed in position. So far We have mapped all the digits except digit 4 in twin A:

Twin A --> twin B

1 --> 9

2 --> 4

3 --> 5

4 --> ?

5 --> 1

6 --> 3

7 --> 6

8 --> 2

9 --> 8

The only digit left in twin B that has not been mapped to other digits is 7. Hence, 4 in twin A must map to 7 in twin B.

Twin A --> twin B

4 --> 7

With all the equivalent digits found, we can substitute them in twin A and twin B as shown in the grids below:

Twin A

- Code: Select all
` . 2 7 | 3 . . | . . 8 `

. . 3 | . 8 5 | . . 2

. . . | 2 6 . | . 4 3

-------+-------+------

2 . . | . 3 . | . . .

3 . . | 6 2 . | 8 . 1

. . . | . 5 1 | 3 2 .

-------+-------+------

. 3 2 | 5 9 6 | . . .

. 6 . | 8 . 3 | 2 . 9

9 1 . | . . 2 | . 3 .

Twin B

- Code: Select all
` . 4 6 | 5 . . | . . 2 `

. . 5 | . 2 1 | . . 4

. . . | 4 3 . | . 7 5

-------+-------+------

4 . . | . 5 . | . . .

5 . . | 3 4 . | 2 . 9

. . . | . 1 9 | 5 4 .

-------+-------+------

. 5 4 | 1 8 3 | . . .

. 3 . | 2 . 5 | 4 . 8

8 9 . | . . 4 | . 5 .

Using the usual sudoku strategy, we finally obtain the solutions for the twin puzzles as shown below:

Twin A

- Code: Select all
` 6 2 7 | 3 4 9 | 5 1 8 `

4 9 3 | 1 8 5 | 7 6 2

1 8 5 | 2 6 7 | 9 4 3

-------+-------+------

2 5 1 | 7 3 8 | 4 9 6

3 7 9 | 6 2 4 | 8 5 1

8 4 6 | 9 5 1 | 3 2 7

-------+-------+------

7 3 2 | 5 9 6 | 1 8 4

5 6 4 | 8 1 3 | 2 7 9

9 1 8 | 4 7 2 | 6 3 5

Twin B

- Code: Select all
` 3 4 6 | 5 7 8 | 1 9 2 `

7 8 5 | 9 2 1 | 6 3 4

9 2 1 | 4 3 6 | 8 7 5

-------+-------+------

4 1 9 | 6 5 2 | 7 8 3

5 6 8 | 3 4 7 | 2 1 9

2 7 3 | 8 1 9 | 5 4 6

-------+-------+------

6 5 4 | 1 8 3 | 9 2 7

1 3 7 | 2 9 5 | 4 6 8

8 9 2 | 7 6 4 | 3 5 1