First of all, this is no new technique. It falls under the catagory of grouped strong links. However, I think this one has it's worth in that it is much easier to spot and to deal with than grouped x-cycles.

Some of this have already been posted in the "skyscraper & 2-string kite" thread, but I thought it deserved a thread on it's own, besides I have thought of a few improvements

Here goes:

definition: An Empty Rectangle (ER) is when four cells in a box does not contain a specific candidate, and these four cells form a rectangle.

so that with this candidate-distribution: (theoretical example)

- Code: Select all
`. 1 . | . . . | . . .`

1 1 1 | . . . | . . .

. 1 . | . . . | . . .

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

1 . 1 | . 1 . | . . .

. . . | 1 . . | . . .

. . 1 | 1 . . | . . .

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

1 . . | . . 1 | . . .

. 1 . | . . 1 | . . .

. 1 . | 1 1 1 | . . .

you would have these empty rectangles:

- Code: Select all
`X 1 X | . . . | . . .`

1 1 1 | . . . | . . .

X 1 X | . . . | . . .

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

1 . 1 | . 1 . | . . .

X X . | 1 X X | . . .

X X 1 | 1 X X | . . .

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

1 . . | X X 1 | . . .

X 1 X | X X 1 | . . .

X 1 X | 1 1 1 | . . .

Now with every Empty Rectangle, you will get two Empty Rectangle Lines! (ERL). These are the two straight lines you can draw inside the box with the ER without touching the ER itself:

Example:

- Code: Select all
`X | X | . . . | . . .`

--|-------ERL | . . .

X | X | . . . | . . .

--|---+-------+------

. | . | . . . | . . .

. | . | . . . | . . .

.ERL. | . . . | . . .

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

. . . | . . . | . . .

. . . | . . . | . . .

. . . | . . . | . . .

Now luckily we don't have to concider these ERL's at all! Our only concern is the point where they intersect, and this we call the Empty Rectangle Intersection. (ERI)

Here are all the ERI's marked with a "+" in the above example (without the candidates)

- Code: Select all
`X . X | . . . | . . .`

. + . | . . . | . . .

X . X | . . . | . . .

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

. . + | + . . | . . .

X X . | . X X | . . .

X X . | . X X | . . .

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

. + . | X X . | . . .

X . X | X X . | . . .

X . X | . . + | . . .

And now for the use of this thing:

If you have a strong link and an ERI that shares a line (row or column) then you can eliminate a candidate at the point where the other end of the strong link and the ERI intersect.

pics says more than words, so here are some examples:

- Code: Select all
`. . . | . . . | X . X`

. . . | . . . | X . X

. a . | . . . | . + .

--|------------------

. | . | . . . | . . .

. | . | . . . | . . .

. | . | . . . | . . .

--|------------------

. b . | . . . | . * .

. . . | . . . | . . .

. . . | . . . | . . .

. . . | . . . | X X .

. a . | . . . | . . +

. | . | . . . | X X .

--|------------------

. | . | . . . | . . .

. | . | . . . | . . .

. | . | . . . | . . .

--|------------------

. b . | . . . | . . *

. . . | . . . | . . .

. . . | . . . | . . .

. a . | . . . | + . .

. | . | . . . | . X X

. | . | . . . | . X X

--|------------------

. | . | . . . | . . .

. | . | . . . | . . .

. | . | . . . | . . .

--|------------------

. b . | . . . | * . .

. . . | . . . | . . .

. . . | . . . | . . .

a-b mark the strong links

X shows the Empty Rectangle (ER)

+ marks the Empty Rectangle Intersection (ERI)

* shows the candidate that can be eliminated

A few treats about this techinque:

*It does not matter what the distribution of the candidates in the box are! We are only looking where there are NO candidates!

*It is VERY easy to spot!

*Can eliminate candidates that escapes the easier methods, and would require extensive knowledge of grouped x-cycles to find otherwise.

*Absorbs one of the three basic patterns in the Turbot Fish. (The one that is not Skyscraper or 2-string kite)

hope you like it!

havard