Hi Kites,

This is a long answer to some short questions ... sorry about that!

[Just before posting this, I noticed em's reply. As usual,

she has found a much more succinct explanation than mine!

But here is my long version anyway. And I'll get em back

some other time, for calling me Old Nick! ]

Here is the same puzzle with just the candidate 1's showing:

- Code: Select all
` `

. . . | 1 1 . | . 1 .

. . . | . . . | . . .

. . . | 1 . . | 1 . .

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

. 1 1 | . . . | 1 1 .

. . 1 | . . . | 1 . .

. . . | . . . | . . .

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

. . 1 | 1 . . | . . .

. . . | . . . | . . .

. 1 . | . 1 . | . . .

Let's do a little swordfish hunting.

Let's start with the columns. We're looking

for columns with 2 or 3 1's. (And we can

ignore the number of 1's in each row, for now).

Well, several columns meet that condition.

We need to find 3 of those columns, such that the 1's

all fall on 3 common rows.

Let's start with column 2. Column 2 and column 3

can't be part of the same swordfish: the 1's share

a total of 4 rows. Similarly for columns 2 and 4.

But next we look at columns 2 and 5. Aha! 3 commmon

rows. So we are almost there ... we need a 3rd column.

Column 7 doesn't work ... but column 8 does.

All the 1's in columns 2, 5, and 8 are confined to

3 rows: 1, 4, and 9. We have found a swordfish.

Now we can apply the swordfish rule: we can remove

all the 1's from each cell of those 3 rows, except

for the swordfish cells, with this result:

- Code: Select all
` `

. . . | . 1 . | . 1 .

. . . | . . . | . . .

. . . | 1 . . | 1 . .

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

. 1 . | . . . | . 1 .

. . 1 | . . . | 1 . .

. . . | . . . | . . .

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

. . 1 | 1 . . | . . .

. . . | . . . | . . .

. 1 . | . 1 . | . . .

We were looking for swordfish based on columns ...

but we could have checked the rows instead.

Let's go back to the start:

- Code: Select all
` `

. . . | 1 1 . | . 1 .

. . . | . . . | . . .

. . . | 1 . . | 1 . .

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

. 1 1 | . . . | 1 1 .

. . 1 | . . . | 1 . .

. . . | . . . | . . .

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

. . 1 | 1 . . | . . .

. . . | . . . | . . .

. 1 . | . 1 . | . . .

We're looking for rows with 2 or 3 1's.

(We can ignore the number of 1's in the columns.)

There are several, but

we need to find 3 of those rows, such that the 1's

are all in 3 common columns.

Let's start by considering row 1. Imagine sweeping all the 1's in row

1 down through the puzzle, as a way to find other rows with

1's in common columns. Here we strike out. For example, looking

at the 1's in rows 1 and 3, they share 4 columns. And similarly

for the 1's in row 1 and any other row.

We have better luck with row 3. The 1's in rows 3 and 5 share 3 columns.

And then we sweep further down the puzzle and find row 7.

All the 1's in columns 3, 5, and 7 are confined to

3 columns: 3, 4, and 7. We have found a swordfish.

Now we can apply the swordfish rule: we can remove

all the 1's from each cell of those 3 columns, except

for the swordfish cells, with this result:

- Code: Select all
` `

. . . | . 1 . | . 1 .

. . . | . . . | . . .

. . . | 1 . . | 1 . .

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

. 1 . | . . . | . 1 .

. . 1 | . . . | 1 . .

. . . | . . . | . . .

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

. . 1 | 1 . . | . . .

. . . | . . . | . . .

. 1 . | . 1 . | . . .

It's the same result as before. (But finally let me end by saying

this won't always happen: finding a swordfish based on columns

does not necessarily mean you will find one based on rows.)