A number that only appears as a candidate twice in a row, column or block is called
a conjugate pair. In the X-Wing technique, a square is
formed from these conjugate cells. By finding two conjugate pairs in two columns, where
both cells are on the same rows as each other, it is possible to remove that number from
the other cells in those two rows. Two conjugates in two columns, spread over two rows.
Swordfish is the next step up from that. You must find three conjugate pairs in three
columns, where the cells are spread over three rows. There is one extra condition, which
is that each row must only have two cells each on it.
It works both ways round, if you find the conjugate pairs in three rows, then you can
remove the canidate number from the columns that the conjugate cells appear in.
Because each of the cells involved is part of a conjugate pair, the value we're
looking for must appear in one of those cells. Becuase there are three conjugate pairs,
and they're all interlinked, it is possible to say for certain where numbers cannot appear.
A
F
B
C
D
E
Example 1: Swordfish by columns Swordfish cells,
remove number from these cells
In this example, cells A and B are the only cells in column 2 where we could place
and 8. Cells C and D are the only places in column 5 where an 8 could go, and E and
F are the only places in column 8. These are the three conjugate pairs. They also
only appear on three rows, row 2, row 5 and row 8.
AB is one conjugate, CD is another and EF is the third. Looking at this combination,
there are only two ways round that it could be. Starting fron conjugate AB, if we put the
8 in A, it cannot be in B or F. EF is a conjugate, so if it's not in F, it must be in E,
which means it cannot go in D. CD being a conjugate, means that the number has to go in C.
Let's do that the other way round. If we start by saying the 8 is in B, then it can't be
in A or C, so it must be in F and D, and not in E. Either of these combinations means that
there would be an 8 once in each row and column in the pattern, and beecause they're
conjugates, one of them has to be correct.
Because the conjugates are in columns, we can remove the number 8 from the rows that
the conjugate cells appear in.
A
B
C
D
E
F
Example 2: Swordfish by rows Swordfish cells,
remove number from these cells
It works by rows as well. AB is the conjugate in row 2, CD is the conjugate in row 4
and EF is the conjugate in row 6. Therefore the number can go in either cells A, D and F,
or cells B, C, and E. This implies that the number cannot go anywhere else in columns
1, 5 and 9, so the candidate can be removed from the marked cells.
7
5 8
6
1
5 8 9
2
5 8 9
4
3
2
3
9
6
5 8
4
5 8
7
1
5 8
1
4
7
5 9
3
5 6 8 9
2
6 8
5 8
2
5 8
6
3
9
4
1
7
3
6
7
1
4
5
9
2
8
1
4
9
2 8
7
2 8
3
6
5
2 5
7
1
3
5 6 8
4
2 9
6 9
5 8
4
9
3
7
5 8
2 6
5 8
1
2 6
2 5 6
8
2 6
2 5 9
5 9
1
4
3
7
Example 3: Swordfish found in columns Swordfish cells,
remove number from these cells
In the puzzle above, the number 8 only appears in columns 1, 3, and 4 twice. It appears
in column 1 in [r3,c1] and [r4,c1], in column 3 in [r4,c3] and [r9,c3], and in column 4
in [r3,c4] and [r9,c4]. These cells also only appear in three rows, rows 3, 4 and 9.
Because the conjugate pairs are in columns, we can remove the number 8 from all the other
cells in those three rows. This allows 8 to be removed from [r3,c7] and [r3,c9].
