Strategies for Popular Number Puzzles

Empty Rectangles

As of October 2023 Empty Rectangles has been retired in favour of the simpler pattern Rectangle Elimination
Empty Rectangles can be re-expressed as Grouped X-Cycles. Examples are given below. In order to see the Empty Rectangle the tick-box on the solver for Rectangle Elimination must be off. Counter-examples would be appreciated if discovered.

Empty Rectangles in Boxes
Empty Rectangles in Boxes
Empty Rectangles constitute a clever but obscure strategy for Sudoku. An Empty Rectangle occurs within a box where four cells form a rectangle which does NOT contain a certain candidate.

In the first diagram is a spread of candidate 8 around the board but each box must be considered separately when imagining the rectangle, so I have coloured the boxes differently. To be able to apply this strategy we must have at least two candidates left in the box and four or more empty cells.

The red crosses represent the Empty Rectangles. Each of these theoretical rectangles is independent of the others.
Empty Rectangle Lines
Empty Rectangle Lines

Whatever the formation of the Empty Rectangle (ER), we will have a vertical and horizontal gap where we can imagine two Empty Rectangle Lines (ERLs). These are two lines we can draw inside the box without touching the ER itself.

I have picked box two in this example.
Empty Rectangle Intersections
Empty Rectangle Intersections
We can quickly forget about these lines once we've spotted where they cross - a place known as the Empty Rectangle Intersection (ERI).

In this diagram the ERIs have been marked with a brown cell.
Example ERI arrangement
Example ERI arrangement
If you have a strong link and an ERI that shares a line (row or column), then you can eliminate a candidate at that point where the other end of the strong link and the ERI intersect. Let's look at some visual examples of the formation. Two different arrangements are shown in this and the next diagram.

The plus sign now indicates where the Empty Rectangle Intersection (ERI) is placed. In the first example down column 2, two 8s exist and form a strong link (X and Y). One end of the strong link (X at C2) can see the ERI at C8.
Example ERI arrangement
Example ERI arrangement

In the second example the ERI at A7 can see an 8 in A4 (the X) and this is paired with the 8 on the cell marked Y. Y can see the 8 in Z which in turn is visible to the original ERI.

It is proposed in both examples that the 8 in Z can be removed.
Empty Rectangle Example
Empty Rectangle Example : Load Example or : From the Start
Here is a real example based on the centre box. The remaining options for 6 in box 5 is D4, D5, E6 and F6. This gives us an Empty Rectangle shown by the four yellow cells. The Empty Rectangle Intersection is the orange cell on D6. Our X and Y are the two green cells H6 and H9. It doesn't matter that there are a ton of 6s in column 6, what is important is there are two 6s in row H. H6 can see the ERI and H9 can see the intersection of H9 and the ERI - which is D9. That is the 6 we can remove.
X-Cycle equivalent
X-Cycle equivalent

Here is the equivalent Grouped X-Cycle based on candidate 6.
Double Empty Rectangle
Double Empty Rectangle : Load Example or : From the Start

Here is a fascinating double Empty Rectangle. The ERI cell on E3 can see two strong links on candidate 6. There is one in row G between G3 and G9 and in the other in column 8 between E8 and H8. The first one allows us to remove 6 from E9 and the second from H3. Double the fun!
X-Cycle equivalent
X-Cycle equivalent

The equivalent Grouped X-Cycle based on candidate 6 gets many more eliminations since it's an off-chain continuous Nice Loop.
Centered Empty Rectangle
Centered Empty Rectangle : Load Example or : From the Start

This awesome sudoku has great examples of certain strategies including a symetrical Hidden Quad. At this stage there is a perfectly centered Empty Rectangle based on candidate 3 with no clues in the rectangle cells. I have Klaus Brenner to thank for sending me this puzzle.
Triple Empty Rectangle
Triple Empty Rectangle : Load Example or : From the Start

Klaus Brenner has found some very nice looking triple Empty Rectangles. I didn't think we'd find any beyond the double one above, but no, here is one of five he has made.

The solver normally returns just the first instance of an advanced strategy but in the case of Empty Rectangles it will return all instances. Klaus also found a puzzle that returns three different Empty Rectangles in a single step. You can load the puzzle from this link.
Go back to Aligned Pair ExclusionContinue to Exocet



... by: Orsurbo Kreeg

Wednesday 19-Jul-2023
There is a typo in the "here is a real example" -- you are missing the 6 candidate in (D6) the ERI cell. When you show the grouped x-cycle you have the candidate 6 in the ERI clearly shown. Very confusing.
Andrew Stuart writes:
Quite correct. Now fixed - [Del]
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... by: Anonymous

Friday 10-Dec-2021
All instances of this strategy is completely overshadowed by Grouped X-Cycles, and it should probably be deprecated and removed from the solver as it isn't very useful.

... by: Mark Holt

Saturday 5-Oct-2019
If the X And Y are in the same box, then the Z will fall in the same box as the ERI. In this case, your assertion does not hold.
Andrew Stuart writes:
True - [Del]
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... by: Norris Barnes

Tuesday 11-Sep-2018
I encountered the same problem as mentioned in the comment below from 'halb'. When the candidate was eliminated that was in the same box as the empty rectangle, the solution was erroneous. In my case there were 7 'empty cells' in the box and in the remaining 2 boxes were b-vals "1/2", satisfying the 'at least 2 candidates condition'. Please comment.

... by: Thinkist

Tuesday 19-Apr-2016
It seems nearly every instance of an empty rectangle can be solved by a grouped X-cycle in your solver. Maybe this strategy should be moved higher up, perhaps as a "Diabolical" strategy instead of an "Extreme" one. It would increase its usefulness for sure.
Andrew Stuart writes:
Yeah true. Advances elsewhere have made this almost redundant. Personally I think it’s a horrible strategy and don’t want it earlier in the list. There are quite a few overlapping strategies now - [Del]
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... by: Bob

Thursday 11-Feb-2016
halb wesen halb ding asked on 20 Sep 2015, "Is there a problem when the other end of the strong link and the ERI intersect inside the box with the empty rectangle?" This is an excellent question because I find this happens quite a bit and it appears that I have had problems if I eliminated a candidate under these circumstances, but nothing definitive. You stated you have not tested that condition yet--are you planning to?
Andrew Stuart writes:
Not done so yet! - [Del]
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... by: Raphael

Sunday 11-Oct-2015
I am really sorry, but I may not understand the explanation well. I had a number of cases where this Empty Rectangle strategy does not apply.

I wish I had the ability to copy these cases.

Rgds, Raphael

... by: halb wesen halb ding

Sunday 20-Sep-2015
Is there a problem if the other end of the strong link and the ERI intersect inside the box with the empty rectangle?
Andrew Stuart writes:
There could well be. I certainly don't look for ER with that sort of pattern, but I've not tested it either. - [Del]
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... by: Roman M

Thursday 7-Feb-2013
The power of ERs can be illustrated by using your examples in your Franken Sword-Fish strategy pdf,namely ,RHS diagrams 3 and 5.
By looking at the conjugate pair E5 and H5 , and the consequences of one and other being true,
and applying "intersection removal",all the red Xs will result - the Franken fin is not needed.

... by: Roman

Sunday 1-Apr-2012
In your Single Chain PDF, Fig. 1 of the 5-bi-location links,
also has a good example of an ER - based on the G3-G7
link, eliminates 5's in E# AND E7.

... by: Werner Cejnek

Wednesday 10-Feb-2010
The explanations are great. I bought your book. Will the book also have a new edition ?
Thanks, Werner cejnek
Article created on 12-April-2008. Views: 173813
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