## When will an Empty Rectangle work?

Post the puzzle or solving technique that's causing you trouble and someone will help

### When will an Empty Rectangle work?

1....6.89..67..1.3......5.6..964....67.2...3.8....76..36...1..5.4.96......2.7..6.
At this stage in this puzzle Boxes 3 & 8 have ERs for candidate 4, but Boxes 2 & 9 do not. However, all four of these boxes do meet the requirement to be corners for Kites or Skyscrapers in candidate 4
(as in not being limited solely to a single Row or Column.) I had been assuming that for an ER elimination to work or be reliable the four corner boxes had to all have ERs in that candidate, but in this case the 4ER in Box8 does work with the 4 Conjugate Pair in Row1 to eliminate 4 from r9c7.
So is this an anomily, or would it generally be true that an ER elimination can be relied on if you have an ER which can see into one end of a CP, even if the other boxes involved are only good for Kites
and do not have an ER for your candidate?
Yogi
2017 Supporter

Posts: 135
Joined: 05 December 2015
Location: New Zealand

### Re: When will an Empty Rectangle work?

Code: Select all
`*--------------------------------------------------------------------------------*| 1       235     357      |b345     235     6        |a247     8       9        || 2459    2589    6        | 7       2589    24589    | 1       24      3        || 2479    2389    378      | 1348    12389   23489    | 5       247     6        ||--------------------------+--------------------------+--------------------------|| 25      1235    9        | 6       4       358      | 278     1257    1278     || 6       7       145      | 2       1589    589      | 489     3       148      || 8       1235    1345     | 135     1359    7        | 6       12459   124      ||--------------------------+--------------------------+--------------------------|| 3       6       78       |c48     *28     *1        | 24789   2479    5        || 57      4       1578     | 9      *6      *2358     | 2378    127     1278     || 59      1589    2        |c3458    7      d3458     |e38-4    6       148      |*--------------------------------------------------------------------------------*`

Just to answer the question in the title of your post, an Empty rectangle "works" when the ER box has the ER candidate confined to exactly one row and one column in that box.

My solver's output says : Empty Rectangle in 4's: Base Cells r1c4 r1c7 + Empty Rectangle Box 8 + Target Cell r9c7.

The 4 in r1c7 sees the 4 in the target cell directly, so if it was True, the 4 in r9c7 would be eliminated.

In the case of the other 4 in Row 1, if it was True, the 4's in r79c4 would be false, and the 4 in r9c6 would then be True, eliminating the 4 in r9c7.

Since there are only 2 4's in Row 1, and either of them being True would eliminate the 4 in r9c7, it can definitely be eliminated.

Note that there could have been 4's in r8c4 and r9c5 and the ER would still have worked. The Empty Rectangle of 4's in Box 8 are the cells r78c56, which I have marked with *'s in the diagram.

Hope this helps, Leren
Last edited by Leren on Tue Feb 07, 2017 12:16 am, edited 1 time in total.
Leren

Posts: 3368
Joined: 03 June 2012

### Re: When will an Empty Rectangle work?

There are several interpretations of this elimination.
Leren explained the ER logic.

Basically ER's always can be expressed as grouped kites or skyscrapers too.
In this case a grouped skyscraper this way:

Alternatively you have the grouped skyscraper
4r1c7=r1c4-r7c4=r7c78
and the grouped kite
4r1c7=r1c4-r23c6=r9c6
eleven

Posts: 1999
Joined: 10 February 2008

### Re: When will an Empty Rectangle work?

I agree that there is a large overlap between ER's, Kites and Skyscrapers, but I don't think the overlap is 100 on ER's%.

Here is a puzzle that has an ER move, but no Kite or Skyscraper equivalent that I can see.

2.4..3587.3......2..8..6.............2...4.39....9...659........4.5...1.61.3.9..5

Leren
Leren

Posts: 3368
Joined: 03 June 2012

### Re: When will an Empty Rectangle work?

Please mark it, i can only see a kite in 2 (r7c5=r8c6-r4c6=r4c7).
Ah, probably i call a kite, what is not a genuine one (with strong links in lines and the weak one in a box).
So i better should have said "2 (grouped) strong links" instead of skyscraper &kite.
eleven

Posts: 1999
Joined: 10 February 2008

### Re: When will an Empty Rectangle work?

Code: Select all
`*--------------------------------------------------------------*| 2     6     4      | 9     1     3      | 5     8     7      || 79    3     1      | 47    8     5      | 46    469   2      || 79    5     8      | 247   27    6      | 3     49    1      ||--------------------+--------------------+--------------------|| 4     7     9      | 6     3    *12     |*12    5     8      || 1     2     6      | 8     5     4      | 7     3     9      || 3     8     5      | 27    9     127    | 124   24    6      ||--------------------+--------------------+--------------------|| 5     9     3      | 1    *27    8      | 6-2   267   4      || 8     4     27     | 5     6    *27     | 9     1     3      || 6     1     27     | 3     4     9      | 8     27    5      |*--------------------------------------------------------------*`

Empty Rectangle in 2's: Base Cells r4c6 r4c7 + Empty Rectangle Box 8 + Target Cell r7c7 => - 2 r7c7

There was also a grouped Skyscraper in 1's in Yogi's puzzle, for which there were no corresponding Kite or ER moves.

Leren
Leren

Posts: 3368
Joined: 03 June 2012

### Re: When will an Empty Rectangle work?

i'm not sure how an ER overlaps directly with a 2-string kite or a skyscraper or sashimi x-wing or finned-xwings

i can see them using different sectors and replicating an identical elimination { is this the point of the topic??}

an ER only functions with a bivavled row/col whom has 1 endpoint visible to an ER intersection.

a 2 string kite uses a box as a weak link intersection where the intersection of the row/col with in said box is always void of a candidate.

a skyscraper uses 2, rows/cols that share a weak link across an intersecting col/row

graphics for clarity
notes:
/ = no candidate
-x exclusion location
x may contain candidate

Code: Select all
`ER (max) --------------------------------  .  /  .  |  .  .  .  |  .  .  .  .  x  .  |  . -x  .  |  .  .  .  .  /  .  |  .  .  .  |  .  .  .   --------------------------------  .  /  .  |  /  x  /  |  .  .  .  .  x  .  |  x  x  x  |  .  .  .  .  /  .  |  /  x  /  |  .  .  .   --------------------------------  .  /  .  |  .  .  .  |  .  .  .  .  /  .  |  .  .  .  |  .  .  .  .  /  .  |  .  .  .  |  .  .  .--------------------------------`

Code: Select all
`ER (min) --------------------------------  .  /  .  |  .  .  .  |  .  .  .  .  x  .  |  . -x  .  |  .  .  .  .  /  .  |  .  .  .  |  .  .  .   --------------------------------  .  /  .  |  /  x  /  |  .  .  .  .  x  .  |  x  /  /  |  .  .  .  .  /  .  |  /  /  /  |  .  .  .   --------------------------------  .  /  .  |  .  .  .  |  .  .  .  .  /  .  |  .  .  .  |  .  .  .  .  /  .  |  .  .  .  |  .  .  .--------------------------------`

Code: Select all
`grouped 2 string kite {max} --------------------------------  .  .  .  |  .  /  .  |  .  .  .  . -x  .  |  .  x  .  |  .  .  .  .  .  .  |  .  /  .  |  .  .  .--------------------------------  .  .  .  |  .  x  .  |  .  .  .  /  x  /  |  x  /  x  |  /  /  /  .  .  .  |  .  x  .  |  .  .  .--------------------------------  .  .  .  |  .  /  .  |  .  .  .  .  .  .  |  .  /  .  |  .  .  .  .  .  .  |  .  /  .  |  .  .  .--------------------------------`

Code: Select all
`2 string kite {min} --------------------------------  .  .  .  |  .  /  .  |  .  .  .  . -x  .  |  .  x  .  |  .  .  .  .  .  .  |  .  /  .  |  .  .  .--------------------------------  .  .  .  |  .  x  .  |  .  .  .  /  x  /  |  x  /  /  |  /  /  /  .  .  .  |  .  /  .  |  .  .  .--------------------------------  .  .  .  |  .  /  .  |  .  .  .  .  .  .  |  .  /  .  |  .  .  .  .  .  .  |  .  /  .  |  .  .  .--------------------------------`

Code: Select all
`Skyscraper  { ie 2 sashimi x-wings} --------------------------------  .  /  .  |  .  /  .  |  .  .  .  .  x  .  |  -x / -x  |  .  .  .  -x / -x  |  .  x  .  |  .  .  .--------------------------------  .  /  .  |  .  /  .  |  .  .  .  .  x  .  |  .  x  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .--------------------------------  .  /  .  |  .  /  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .--------------------------------`

Code: Select all
`Finned X-wing (max)--------------------------------  .  /  .  |  .  x  .  |  .  .  .  .  x  .  |  -x x -x  |  .  .  .  .  /  .  |  .  x  .  |  .  .  .--------------------------------  .  /  .  |  .  /  .  |  .  .  .  .  x  .  |  .  x  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .--------------------------------  .  /  .  |  .  /  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .--------------------------------`

Code: Select all
`Finned X-wing (min)--------------------------------  .  /  .  |  .  /  .  |  .  .  .  .  x  .  |  -x x -x  |  .  .  .  .  /  .  |  .  x  .  |  .  .  .--------------------------------  .  /  .  |  .  /  .  |  .  .  .  .  x  .  |  .  x  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .--------------------------------  .  /  .  |  .  /  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .--------------------------------`

Code: Select all
`Sashimi X-wing {max} --------------------------------  .  /  .  |  .  /  .  |  .  .  .  .  x  .  |  -x / -x  |  .  .  .  .  /  .  |  .  x  .  |  .  .  .--------------------------------  .  /  .  |  .  /  .  |  .  .  .  .  x  .  |  .  x  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .--------------------------------  .  /  .  |  .  /  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .--------------------------------`

Code: Select all
`Sashimi X-wing {min} --------------------------------  .  /  .  |  .  x  .  |  .  .  .  .  x  .  |  -x / -x  |  .  .  .  .  /  .  |  .  x  .  |  .  .  .--------------------------------  .  /  .  |  .  /  .  |  .  .  .  .  x  .  |  .  x  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .--------------------------------  .  /  .  |  .  /  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .  .  /  .  |  .  /  .  |  .  .  .--------------------------------`

There was also a grouped Skyscraper in 1's in Yogi's puzzle, for which there were no corresponding Kite or ER moves.

i presume you mean this?
Finned X-Wing: 1 r49 c29 fr4c8 => r56c9<>1

the 4 elimination alternatives with out going to complex fish size above 2
Finned X-Wing: 4 r17 c47 fr7c8 => r9c7<>4
Empty Rectangle: 4 in b8 (r1c47) => r9c7<>4
Some do, some teach, the rest look it up.

StrmCkr

Posts: 942
Joined: 05 September 2006

### When will an Empty Rectangle work?

Thanx guys. You've given me a great deal to work through and ponder upon.
Unfortunately I'm about to head away on holiday, but I will give you my thoughts once I've got some!

Catch up later.
- Yogi
Yogi
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Joined: 05 December 2015
Location: New Zealand

### When will an Empty Rectangle work?

Just to wrap up this particular enquiry, I was mainly trying to identify the scenarios in which a reliable ER elimination could be made, after I found a case which did not fit my previous belief.
I’m finding that it is true that ERs, Kites and Skyscrapers all depend on the same raw materials (Conjugate Pairs interacting through Boxes of a certain type) and they can and often do overlap to find the same elimination. However, it is also clear that they don’t always overlap (as in Leren’s example) so each can be tried in turn by the P & P solver as a different method to try to move things along.
Something else I've found is that if you have fewer than 34 known numbers in your puzzle the odds are against your finding a useful Kite or Skyscraper. However, it is still generally better to look for them first, because that exercise will identify the boxes and candidates that could be useful when you move on to ERs, if you want or need to.
Yogi
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Joined: 05 December 2015
Location: New Zealand