## Skyscrapers, Kites & Empty Rectangles

Everything about Sudoku that doesn't fit in one of the other sections

### Skyscrapers, Kites & Empty Rectangles

Puzz1.png (25.69 KiB) Viewed 629 times

.2.736.416.14..73.73481..6.5.76.42132431...7616.3274..3...716.4412.683.7.76.431..
I have long been interested in whether these 3 different elimination methods can be worked together in a single process, since they all depend on the same type of data, involving the interaction of boxes with one or two columns or rows that have conjugate pairs (just two possible instances of a given candidate) in separate boxes. Clearly they overlap anyway. Numerous times I have found that the same elimination I made with a Kite was found by a solver with an ER. In fact I notice that some solvers often find ERs but never mention Kites of Skyscrapers. Also, since they are relatively simple to work with, I think they are the logical first line of attack for Pen & Paper Solvers after the Basics.
In this puzzle the Kite Map shows:
KiteMap.png (5.05 KiB) Viewed 629 times

There were no Skyscrapers, but the 8Kite r6c378 / r15c7 => r1c3<>8. This expands the list of 8 Conjugate Pairs to 8C137, 8R145
but there are no more Kites or Skyscrapers, so we can move on to try for Empty Rectangles. The ER Map is the same except that Box3 has no 9ER and 5s are now included.
ERMap.png (5.06 KiB) Viewed 629 times

However, none of the 5CPs combine with the ERs in Boxes 129 to eliminate anything, so it’s back to looking for eliminations in 8 or 9.
8ER in Box3 + c1 => r9c9<>8, which further expands the list of 8CPs to 8C1379, 8R1456.
Now we have 8ER in Box4 + c9 => r2c2<>8 => 8r1c1 stte.
It is true that the puzzle as shown can be solved more simply with the Contradiction Chain 8r4c2 => 8r2c2 Not Possible! So 8r6c3, but that was not the purpose of this exercise.
BTW, what’s the standard hieroglyph for ‘Not Possible,’?
Here’s another example: .47.8..61.6..........6..7..62..1357......5..6.1..6....28..4.....9.1...4.....2.69.
2-String Kite (5)r1c14 / r238 => r8c1<>5 OR Empty Rectangle (5)Box8(r8->c4) - r1c4=r1c1 Remove candidate 5 from r8c1
Yogi
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### Re: Skyscrapers, Kites & Empty Rectangles

Yogi,

skyscrapers, kites, empty rectangles and turbot fish are all special cases of "two strong (grouped) links". But of course (at least for beginners) a skyscraper is easier to spot than a kite with grouped strong links.
Note that your ER elimination can be done with a simple kite too:
8r9c1=r1c1-r2c2=r2c9 => r9c9<>8
[edit: typo]
Last edited by eleven on Tue Sep 06, 2016 7:59 pm, edited 1 time in total.
eleven

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### Re: Skyscrapers, Kites & Empty Rectangles

Don't forget X-Wings. They also fit the pattern.

On the other hand, the way people typically search for these various things is often quite different. X-Wings and skyscrapers typical are quick to spot in searches by digit, by row/column. While turbot fish are typically found by examining the conjugate pairs you have and seeing if they combine in a useful way. While, empty rectangles are typically a secondary method for lengthening chains once you already have a chain in mind.

JasonLion
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### Skyscrapers, Kites & Empty Rectangles

Eleven wrote:
8r9c9=r1c1-r2c2=r2c9 => r9c9<>8
I don’t quite understand this but if it’s a typo and supposed to be something like
8r9c1=r1c1-r2c2=r2c9 then I recognise this as a kite described in chaining language as two conjugate pairs weakly linked in Box1.
And indeed it does do the same job as the ER I mentioned which eliminates 8 from r9c9.
Thanx!
Yogi
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Location: New Zealand

### Re: Skyscrapers, Kites & Empty Rectangles

yes, sorry for the typo.
eleven

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Joined: 10 February 2008