## Help with empty rectangles

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

### Help with empty rectangles

I have got the hang of most of the sudoku solving techniques but am stuck with empty rectangles and can't finish my book until I get my head around it. Does anyone know of a simple explanation of this method. I have found a couple which were obviously intended for rocket scientists.
delboy007

Posts: 5
Joined: 14 May 2016

### Re: Help with empty rectangles

Try here and here for a general explanation at some good teaching sites.

Here is an example of the Empty Rectangle rule in action.

Code: Select all
`*--------------------------------------------------------------*| 5     9     8      | 6     4     3      | 17    17    2      || 12    12    3      | 7     5     9      | 6     4     8      || 6     7     4      | 1     2     8      | 5     9     3      ||--------------------+--------------------+--------------------|| 4     5     7      | 2    c169  d16     | 8     3    e9-1    || 9     18    6      |*3    c18   *7      | 4     2     5      || 18    3     2      |*4    c189  *5      | 17    6     179    ||--------------------+--------------------+--------------------|| 27    26    5      | 9    b16    4      | 3     8    a17     || 3     4     1      | 8     7     2      | 9     5     6      || 78    68    9      | 5     3     16     | 2     17    4      |*--------------------------------------------------------------*`

There are 2 1's in Row 7 at r7c5 and r7c9 so one of them must be True. If Row 7c9 is True r4c9 is obviously False.

If r7c5 is True r456c5 are all False, so r4c6 is True and r4c9 is False.

Since it is False for both r7c5 and r7c9, one of which must be True, it must be False and can be eliminated.

The trick to understanding Empty rectangles for me is to focus on what is in the ER box, rather than what is not in the ER box (Box 5 in this example).

The so called Empty Rectangle in Box 5 is the four cells marked with a * r56c46 (they have no 1's). However, what this really means is that the 1's in Box 5 are constrained to exactly one row (Row 4) and one column (Column 5).

So if r7c5 is True all the 1's in Box 5 Column 5 are False, so you are left with a single row of 1's (actually only a single 1 in this puzzle at r4c6) in Box 5 that makes r4c9 False. (Note that there could be a 5th 1 in Box 5 at r4c4 and the move would still work, with 2 1's at r4c46, one of which must be True).

So, summarizing the ER box requirements, there must be a minimum of 2 instances of the ER candidate, and a maximum of 5 instances. You must be able to cover all such instances with exactly one vertical (column) line and one horizontal (Row)
line. You need both cover lines for the box to be suitable, covering all instances with just one line will no be suitable. And, yes, when you can do this, there will be a rectangle of cells in the box that will not contain the candidate (plus possibly other empty cells), so the two ways of thinking are equivalent.

For me, thinking about it this way makes it easier to understand - hopefully you don't think this explanation is rocket science.

Leren
Last edited by Leren on Sat May 28, 2016 6:58 am, edited 4 times in total.
Leren

Posts: 3958
Joined: 03 June 2012

### Re: Help with empty rectangles

Welcome Mr delboy Bond, we've been expecting you.

In a nutshell the requirement is four cells in a box that form a rectangle none of which can hold a digit.
Code: Select all
`*-------*   *-------*  | \ 3 \ |   | \ 3 \ || 3 3 3 |   | 3 \ 3 |  \ = 3 is missing | \ 3 \ |   | \ 3 \ |*-------*   *-------*`

On the left, if the digit can't be true in the middle row it must be true in the middle column and vice versa (a strong link)

On the right, the common cell is empty as well so additionally if the digit must be true in the row, it can't be true in the column and vice versa (a weak link) so either type of link can be used.

The links Leren have cited give examples. (On posting I see he's expanded his original reply, but never mind.)

DPB
David P Bird
2010 Supporter

Posts: 1043
Joined: 16 September 2008
Location: Middle England

### Re: Help with empty rectangles

Sorry. I have sat down a few times allowing time for my brain to cool and I just can't begin get my heard around this. I am sure rocket sciencetry must be easier than this.
delboy007

Posts: 5
Joined: 14 May 2016

### Re: Help with empty rectangles

Are you familiar with AIC (and other chains)? Empty Rectangles are perhaps the simplest of the extensions to chain techniques, and are useless on their own. Unless you are already well versed in chains they are not going to make any sense.

JasonLion
2017 Supporter

Posts: 641
Joined: 25 October 2007
Location: Silver Spring, MD, USA

### Re: Help with empty rectangles

I am ok with remote pairs but not chains (I assume they are different). I will take a look at chains then. Can anyone recommend an idiots guide to chains?
delboy007

Posts: 5
Joined: 14 May 2016

### Re: Help with empty rectangles

The teaching sites I usually recommend can be found here and here.

While I'm at it I'll try to explain how the empty rectangle move I posted above works as a chain as simply as possible - hopefully no Rocket Science. I've added a label e to r4c9 to make the explanation easier.

Suppose the 1 in cell a is False (ie not True). Then the 1 in cell b would have to be True (because there are only 2 1's in Row 7 and one of them must be True). In chain parlance this is called a Strong link. In general terms a Strong link exists between X and Y if you assume X is False and that forces Y to be True).

OK, so cell b is True. Well, that means that the 3 cells marked c in Column 5 Box 5 are all False, since there can only be one True 1 in Column 5. This inference is called a Weak link. A Weak link exists between X and Y if you assume X is True and that forces Y to be False.

Now here is the clever "Empty Rectangle" bit : Since all the cells marked c are False in Box 5, cell b must be True. In chain parlance a Strong link exists between cells marked c and d in Box 5.

OK, so cell d is True. Well, since there can only be one 1 in Row 4, all the other 1's in that row must be False, including the 1 in cell e.

So, we have followed the chain via cells a-b-c-d-e and shown that if you assume that cell a is False then cell e must be False.

Well, what does that prove. Strictly speaking it proves nothing ! (Even though many sites will claim that it proves that the 1 in cell e must be False and can be eliminated).

What is usually left out is the other half of the proof, although it's so obvious, its the reason that it usually doesn't get a mention.

Here is the other half : Assume the 1 in cell a is True. Then since there can only be 1 True 1 in Column 9 all the other 1's in that column must be False, including the 1 in cell e. A Weak link exists between the 1's in cells a and e.

Now we have gotten somewhere. Why is that ? Well, the 1 in cell a is either True or False. But it doesn't matter which is the case, in both cases the 1 in cell e is False, so we can eliminate the 1 in cell e without at this stage knowing whether the 1 in cell a is True or False.

Hopefully there was not too much Rocket Science in this explanation.

Leren
Last edited by Leren on Sat May 28, 2016 7:37 am, edited 6 times in total.
Leren

Posts: 3958
Joined: 03 June 2012

### Re: Help with empty rectangles

thanks I will give that a go
delboy007

Posts: 5
Joined: 14 May 2016

### Re: Help with empty rectangles

thanks. I will look into that. So is a remote pair a kind of chain ?
delboy007

Posts: 5
Joined: 14 May 2016

### Re: Help with empty rectangles

Yes Remote Pairs is kind of chain, you can look up the descriptions on the same 2 sites as before.

Leren
Leren

Posts: 3958
Joined: 03 June 2012