A record named "maximum" of Sudoku list

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

A record named "maximum" of Sudoku list

Postby Eioru » Thu Aug 17, 2006 4:11 am

There are many puzzles I found or quote other users' list.

Like the first one :
7..3..8...4..5..9...2..4..62.....1...7.....4...6.....21..8..6...9..3..5...5..7..8
This puzzle needs using APE 3 times when solving it fast, only if you choose more difficult methods to solve it.
Though there are many puzzles might also need 3 times, but I didn't find any puzzle needing 4 times yet.
Trying to find 4-times or more-times puzzle, if there is


I will renew it when there is a new puzzle discovered by everyone
renew on 2007/3/2 11:00am
You can analyse them on Sudoku Explainer (which I use) to check.

3 Aligned Pair Exclusion 7..3..8...4..5..9...2..4..62.....1...7.....4...6.....21..8..6...9..3..5...5..7..8
3 Aligned Triplet Exclusion 9..5..8......4......1..8..92..3..4...6.....5...8..7..13..9..2......6......4..1..7
6 Bidirectional Cycle 3...7...9.4...2.1....8.......7..4...2...3...8.1.6..2.......1....5.2...8.9...8...3
3 Bidirectional X-Cycles .......4.89.2..5...2..35..9..1.7.6...4...6..7..8.1.3...1..54..325.9..4.........7.
8 Bidirectional Y-Cycles 3..6..9......1......5..2..38..3..1...7.....6...2..4..59..8..4......7......6..1..2
2 Bivalue Universal Graves 6.79..2......5.34.3...82.911.....8...56...41...8.....656.31...4.81.4......4..91.5
13 Cell Forcing Chains 1..3.6....2..4..6...9...4..8..6....3.4.....8.7....4..9..8...5...6..8..2....9.7..6
7 Claiming .5..2..8.6..5....4..78..3.....9.453.3.......1.951.3.....4..92..9....5..3.2..1..9.
12 Direct Hidden Pair 4.2..96.1...3.8............3.1...9.8.........9.4...2.7..........9.8.1.5.7.62.34..
4 Direct Hidden Triplet 6.13.24.5...9.8............1.3...5.6.........8.5...3.7..........6.8.9.3.7.42.16.9
2 Direct Pointing 8.9.5..........57.4.......3..71..29.6..5........3..1.7.85....1.......8....1.6...5
14 Dynamic Cell Forcing Chains .....1..2.1..3..4.5..6.......4.....1.7..8..3.6.....8..2....6.......5..7...39..5..
3 Dynamic Cell Forcing Chains(+) 1.......2.3..4..5....6.7.....4...6...7..8..4...9...8.....9.8....2..7..1.5.......3
36 Dynamic Contration Forcing Chains 1....23...4..5......26....7..7.....6.8.....2.9.....5..5....31......9..4...47....8
19 Dynamic Contration Forcing Chains(+) 1.......2.3..4..5...6...7.....1.4....8..7..4....5.8.....7...6...5..3..8.2.......1
4 Dynamic Double Forcing Chains 1.......2.3..4..5...6...7.....1.3....4..6..8....4.5.....2...9...8..5..4.7.......1
8 Dynamic Double Forcing Chains(+) 1.......2.3..4..5....6.7.....4...6...7..8..4...9...8.....9.8....2..7..1.5.......3
27 Dynamic Region Forcing Chains .....1..2.3..2..4.5..6.......6.....7.7..1..3.8.....5..7....5.......3..2...98..6..
10 Dynamic Region Forcing Chains(+) 1.......2.3..4..5...6...7.....1.3....8..7..4....4.6.....2...6...5..3..8.9.......1
31 Forcing Chains ..........4..8..5...3..1..8...9..5...9..6..2...1..8..4...3..7...6..2..9...5..7..1 1..2..3.............4..1..53..6..2...7.....3...9..4..82..8..7......6......8..5..9
5 Forcing X-Chains .1....2.......3..45...2..1.2...6..5...7..8....9.3..........48....49....36......2.
7 Hidden Pair 1.......2.3..4..5...6...7.....1.4....5..2..4....5.3.....2...6...4..8..3.7.......1
1 Hidden Quad .......1.21...348..398..2...6.3.49.............16.7.4...8..217..267...98.9.......
3 Hidden Triplet 4..9..3......5......8..7..93..1..5...8.....6...7..6..82..3..1......2......6..4..7
2 Jellyfish ..8.125..3.146..2.5..8....18..5......32...65......6..84....3..5.8..512.4..598.1..
8 Naked Pair .9.2.8..62.36.9...4...75....1..8.3......5..8..34...1...79.....2....2.8.7.........
1 Naked Quad 2..9..8...5..7..6...4..1..73.....1...7.....5...9.....68..5..2...9..3..4...1..2..3
5 Naked Single ..2.8.1.....7.4...9...6...4.1.....5.2.5...6.3.7.....4.8...5...9...1.8.....6.9.2..
5 Naked Triplet 8..5..6......8......3..9..75..4..8...7..1......2..7..94..6..1......9......1..3..4
11 Nishio Forcing Chains ....6......62.13...278.456..61...45.4.......6.52...13..184.672...47.28......8....
15 Pointing .6.....4.7..3.8..9...2.7....29...13...........43...6.....8.2...8..6.9..5.1.....7.
23 Region Forcing Chains 1.......2.3..4..5...6...7.....1.4....8..2..4....5.8.....2...6...4..3..8.7.......1 1..2..3......1......4.....56..7..2...8.....3...7..5..92..6..9......3......9..8..4
5 Swordfish 1..2..3......4......5..6..27..3..8...8.....9...4..2..58..7..4......5......9..1..6
8 Turbot Fish 2..4..1...3..2..9...6..1..78.....7...5.....4...1.....96..7..4...2..8..3...8..6..5
2 Unique loops ..7..3.4..53...6.298.6...5...4..5..6....3....3..2..9...9...4.812.1...49..4.1..3..
3 Unique rectangles .....923.....5..91..1..65.7.....79.3.5..2..1.4.95.....9.76..8..83..7.....658.....
6 X-Wing 5..2..8......9......6..3..47..5..2...2.....3...3..6..14..6..9......7......1..4..8
6 XY-Wing 54..6..239.3...4.7......6......37...3..2.6..1...48......2......7.5...3.261..2..95
4 XYZ-Wing ..4.7.1....12.45..5.......46.5.2.7.1...7.6...4.7.5.6.31.......7..21.53....3.9.2..

I don't list two methods "Hidden Single" and "Direct Claiming",
because Hidden Single's record is 64 certainly, and Direct Claiming is used not unnatural told by http://diuf.unifr.ch/people/juillera/Sudoku/InterestingSudokus.html

Maximum solving methods : 100 ( except hidden single )
.....1..2.3..2..4.5..6.......6.....7.7..1..3.8.....5..7....5.......3..2...98..6..
Maximum solving kinds : 20
7..8..1......6......3..9..54..2..7...5.....1...6..8..38..4..2......1......9..7..6

These two upside are added to become my new record.
First is that you need use plenty of methods to solve.
Second is that you need use many kinds of methods to solve.

And, as long as this post need to renew or to alter ( like it is not found a method you have, you find a puzzle can replace the puzzle I list, or there are wrong things ) , please list to tell me, and I correct it soon.


Thanks for everyone help this work or read this post at the last.
Last edited by Eioru on Fri Mar 02, 2007 12:51 pm, edited 34 times in total.
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Postby ravel » Thu Aug 17, 2006 11:51 am

Hi Eioru,

welcome in the forum and thanks for the hard sudoku, i will check next week, if it qualifies for my hardest list.

I dont really understand your question. There are many ways to solve a hard sudoku, so we never can say exactly, how often a technique is needed.
E.g. your puzzle can be solved with a swordfish, 2 forcing chains, 1 dynamic cell and 3 dynamic contradiction forcing chains, where the chains give the following eliminations/placements:
r6c9<>9,r8c9<>9, r6c6=9, r2c9<>1, r2c6<>4, r1c7<>6.
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technics

Postby Papy » Thu Aug 17, 2006 12:23 pm

In fact you have two kinds of technics:
elimination and hypothese

elimination are nacked paires, X-wing where you eliminates candidat
It' the base of the sudoku

When you are stopped on a grid you use hypothese techniuc: coloroing, arianne, nishio...

Some time you do not need hypothese only elimination but you always use elimination (forced)
The challenge is to find new elimibation methode to never use hypothese. It's the goal!

You have an other kind of methode: the Brute Force. In fact when poeple speak of Brute Force they make a mistake. They use recursives techniques not brut force

In a recursive technique you use the state of the grid adding a number and again will you are not bocked. It's not a brut force!
In a brute force you dont use a partial result: on only test if the total solution is good!

A brute force for a sudoko solver must do that

put 1 in all the 81 than controlte solution
put 2 in 81 and control
Put 3....
It'is creasy But there are other ways to write Brut Force and the result is faster than a recursive method (not always but often)

With the Brut force we always finf the result There is no impossible grid. and the time change with the numbers of clues.

Papy
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Explain my thinking

Postby Eioru » Fri Aug 18, 2006 6:09 am

Though my example need other techniques,
(Cell/Region/--) Forcing Chain & Dynamic Forcing Chains are used more than others. ( 12 times and 20 times )
Are there sudokus have to use one kind of techniques 25 times even over 30 times except "single" analysed by Sudoku Explainer.
Techniques include Naked Single, Pointing, Claiming, X-Wing, XY-Wing, XYZ-Wing, Swordfish, Unique rectangles and loops, Jellyfish, Bivalue Universal Graves (BUG), Aligned Pair Exclusion (APE), (Cell/Region/--) Forcing Chains, Nishio, Dynamic (Cell/Region/Contradiction) Forcing Chains.
(I didn't see any sudoku solved by using over 30 times techniques before)
Last edited by Eioru on Fri Aug 18, 2006 5:57 am, edited 3 times in total.
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Most frequent

Postby Papy » Fri Aug 18, 2006 9:12 am

I don't know if that is the answer to your question burt here is a grid
that you can resolve ONLY WITH single technique (forced or hide simple)

It's a 29 clues grid so you have to use a single technique 52 times!!!

400319006
001000900
067400021
700050004
000142000
200070008
020000060
004000800
100508007
A wheel

Sure you can solve it with paires,X-wing and co
PAPY
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Postby ab » Fri Aug 18, 2006 1:59 pm

there are many 17s that solve with just singles, so they require 64 applications. Also singles is two techniques as you pointedd out, there are naked singles and hidden singles.
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Number of technique

Postby Papy » Fri Aug 18, 2006 2:21 pm

sorry,

I not understood:

If you need two techniques 63 times, every one of the to technicw is se more thant 30 times.
It was not the question?
Papy
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Postby JPF » Fri Aug 18, 2006 4:18 pm

ab wrote:there are many 17s that solve with just singles, so they require 64 applications. Also singles is two techniques as you pointedd out, there are naked singles and hidden singles.


Here is a puzzle which can be solved by nacked singles (59) only :

Code: Select all
 1 . . | . . 4 | . . .
 . 2 . | . . . | . . 8
 . 8 4 | . 5 . | 1 6 .
-------+-------+-------
 . . . | . . . | 8 . .
 7 . 5 | 9 . . | . . 3
 . . . | 1 6 . | . . .
-------+-------+-------
 . 9 . | . . 5 | 4 . .
 . 4 . | . . . | 6 . .
 . . . | . . 2 | . . .


r5c6=8 - r5c7=2 - r6c2=3 - r6c6=7 - r4c6=3 - r5c5=4 - r5c8=1 - r3c6=9 - r4c5=2 - r5c2=6 - r8c6=1 - r2c6=6 - r3c1=3 - r4c2=1 - r4c3=9 - r4c4=5 - r2c3=7 - r2c4=3 - r2c5=1 - r4c1=4 - r4c8=7 - r4c9=6 - r1c2=5 - r1c3=6 - r2c1=9 - r2c7=5 - r2c8=4 - r6c7=9 - r6c8=5 - r6c9=4 - r9c2=7 - r9c7=3 - r1c7=7 - r3c9=2 - r1c5=8 - r1c9=9 - r3c4=7 - r8c4=8 - r9c5=9 - r9c8=8 - r1c4=2 - r1c8=3 - r7c4=6 - r7c8=2 - r8c8=9 - r9c3=1 - r9c4=4 - r9c9=5 - r7c1=8 - r7c3=3 - r7c5=7 - r7c9=1 - r8c3=2 - r8c5=3 - r8c9=7 - r9c1=6 - r6c1=2 - r6c3=8 - r8c1=5

JPF
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Re: Explain my thinking

Postby Ocean » Fri Aug 18, 2006 9:41 pm

Eioru wrote:Are there sudokus have to use one kind of techniques 25 times even over 30 times except "single" analysed by Sudoku Explainer.

Here is one of the highest counts of forcing chains I could find. The 'dynamic' chains add to 49. Don't know how many of them that are really 'needed', but the numbers come from Explainer.
Code: Select all
 *-----------*
 |...|..1|..2|
 |.3.|.2.|.4.|
 |5..|6..|...|
 |---+---+---|
 |..6|...|..7|
 |.7.|.1.|.3.|
 |8..|...|5..|
 |---+---+---|
 |7..|..5|...|
 |...|.3.|.2.|
 |..9|8..|6..|
 *-----------*

#
Analysis results
Difficulty rating: 9,2 / 10
This Sudoku can be solved using the following logical methods:
57 x Hidden Single
2 x Naked Single
1 x Direct Hidden Triplet
8 x Pointing
5 x Claiming
4 x Naked Pair
1 x X-Wing
3 x Hidden Pair
1 x Swordfish
6 x Forcing Chain
2 x Bidirectional Cycle
5 x Nishio Forcing Chains
3 x Cell Forcing Chains
10 x Region Forcing Chains
27 x Dynamic Region Forcing Chains
17 x Dynamic Contradiction Forcing Chains
4 x Dynamic Cell Forcing Chains
1 x Dynamic Double Forcing Chains
#


The set of 32 puzzles recently published in the 'hardest sudokus' thread contains one puzzle with 33 x Dynamic Contradiction Forcing Chains. Another of those puzzles contains 54 x 'dynamic' forcing chains (when the various types are added). The counts given by Sudoku Explainer's analysis is only an indication of how many chains that are "needed", since the numbers may change when the order of application is altered (typically the case for permuted versions of the same puzzle).
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Postby Eioru » Sat Aug 19, 2006 5:25 am

Umm, that is a great puzzle. Thanks, Ocean.

27 x Dynamic Region Forcing Chains
17 x Dynamic Contradiction Forcing Chains
4 x Dynamic Cell Forcing Chains
1 x Dynamic Double Forcing Chains

Dynamic Forcing Chain is the hardest technique i saw before, total is 49 Dynamic Forcing Chains, and become the record to wait for breaking, other techniques, too.

It also has 5 Nishio, a distinct article.
Every technique might have their own puzzle which needs the highest time then others to be used.
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Postby Mike Barker » Sun Aug 27, 2006 2:15 am

Here's a puzzle with at least 9 possible X-wings:
Code: Select all
 8 . 5 | . 9 . | . . .
 4 . 7 | 5 . . | 1 2 .
 . 2 . | . . . | 3 . .
 ------+-------+------
 . . . | . . . | 6 3 .
 . . 1 | . . 4 | . . 7
 . 5 8 | . . . | . 4 .
 ------+-------+------
 . . . | 6 2 . | . . .
 . 1 4 | . . 7 | 8 . .
 . . . | . 8 . | . 1 .

And one with at least 8 possible Swordfish including Franken and a Big Fin Varieties:
Code: Select all
 . 6 7 | 8 . 2 | . . .
 9 . 8 | 1 . 7 | . . 4
 . . . | . 4 . | . . 5
 ------+-------+------
 2 3 . | . . 1 | . 7 .
 . . . | . . . | 3 . .
 . . . | . . 4 | . . .
 ------+-------+------
 8 . . | . . 3 | . . 7
 . 1 . | 5 . 8 | . 3 .
 . . . | . . . | 2 . .

This one contains 6 XY-wings
Code: Select all
 9 . . | . 6 . | . . .
 . . . | . . 1 | 8 4 .
 . . 8 | . 7 . | . 5 .
 ------+-------+------
 . . . | . . 7 | . . 5
 . 7 . | 4 . . | 2 . .
 . 1 . | 5 9 . | 4 . .
 ------+-------+------
 . . 1 | . 8 . | . . .
 6 . . | . . . | . . 3
 . 4 . | 1 5 . | . . 6

This puzzle contains 6 XYZ-wings:
Code: Select all
 . . . | . 6 . | . 2 .
 . 6 . | . 1 5 | . . .
 . . . | 8 . 7 | 3 . .
 ------+-------+------
 . 2 . | . 8 . | . . .
 7 . . | . . . | . 4 .
 . 5 . | 7 . . | . . 8
 ------+-------+------
 . . . | 6 . . | . . 3
 . 3 . | . 5 . | . . .
 4 . . | . . 2 | 1 7 5

And these contain 9 URs:
Code: Select all
 . . 7 | . . . | . 3 .
 4 . . | 5 . 9 | . . .
 . 3 . | 8 . . | . 1 .
 ------+-------+------
 . . . | . . . | 1 . 4
 . . 5 | 1 4 . | . . .
 . . 4 | . . 3 | . . 6
 ------+-------+------
 . 2 . | . . 5 | . 7 .
 9 . . | . . . | . . 1
 . . . | . . 2 | . 9 .

 . . 7 | . . . | . 3 .
 4 . . | 5 . 9 | . . .
 . 3 . | 8 . . | . 1 .
 ------+-------+------
 . . . | . . . | 1 . 4
 . . 5 | 1 4 . | . . .
 . . 4 | . . 3 | . . 6
 ------+-------+------
 . 2 . | . . 5 | . 7 .
 9 . . | . . . | . . 1
 . . . | . . 2 | . 9 .
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Postby Eioru » Fri Sep 22, 2006 11:59 am

9/22 8:00pm ...
I alter the first post, hoping you will like this.
I separate Forcing Chains to 2 ( cell, region ) Dynamic Forcing Chains to 4 ( contration, cell, region, double ) for old post.
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Postby Eioru » Fri Sep 22, 2006 12:24 pm

I use SE to check the puzzle
JPF wrote:Here is a puzzle which can be solved by nacked singles (59) only :
Code: Select all
 1 . . | . . 4 | . . .
 . 2 . | . . . | . . 8
 . 8 4 | . 5 . | 1 6 .
-------+-------+-------
 . . . | . . . | 8 . .
 7 . 5 | 9 . . | . . 3
 . . . | 1 6 . | . . .
-------+-------+-------
 . 9 . | . . 5 | 4 . .
 . 4 . | . . . | 6 . .
 . . . | . . 2 | . . .

JPF

58 x Hidden Single 1 x Direct Hidden Pair

And for Mike Barker
Code: Select all
 8 . 5 | . 9 . | . . .
 4 . 7 | 5 . . | 1 2 .
 . 2 . | . . . | 3 . .
 ------+-------+------
 . . . | . . . | 6 3 .
 . . 1 | . . 4 | . . 7
 . 5 8 | . . . | . 4 .
 ------+-------+------
 . . . | 6 2 . | . . .
 . 1 4 | . . 7 | 8 . .
 . . . | . 8 . | . 1 .

51 x Hidden Single 2 x Direct Hidden Pair 4 x Pointing 2 x Claiming 1 x Naked Pair 2 x X-Wing 2 x Hidden Pair 1 x Naked Triplet 1 x XY-Wing 3 x Turbot Fish 1 x Bidirectional Cycle 5 x Forcing Chain
Code: Select all
 . 6 7 | 8 . 2 | . . .
 9 . 8 | 1 . 7 | . . 4
 . . . | . 4 . | . . 5
 ------+-------+------
 2 3 . | . . 1 | . 7 .
 . . . | . . . | 3 . .
 . . . | . . 4 | . . .
 ------+-------+------
 8 . . | . . 3 | . . 7
 . 1 . | 5 . 8 | . 3 .
 . . . | . . . | 2 . .

53 x Hidden Single 2 x Direct Hidden Pair 1 x Naked Single 2 x Pointing 1 x Claiming 2 x Naked Pair 1 x Naked Triplet 1 x XY-Wing 1 x Unique Rectangle type 1 1 x Bidirectional X-Cycle 1 x Turbot Fish 4 x Forcing X-Chain 1 x Bidirectional Y-Cycle 1 x Forcing Chain
Code: Select all
 9 . . | . 6 . | . . .
 . . . | . . 1 | 8 4 .
 . . 8 | . 7 . | . 5 .
 ------+-------+------
 . . . | . . 7 | . . 5
 . 7 . | 4 . . | 2 . .
 . 1 . | 5 9 . | 4 . .
 ------+-------+------
 . . 1 | . 8 . | . . .
 6 . . | . . . | . . 3
 . 4 . | 1 5 . | . . 6

56 x Hidden Single 6 x Pointing 1 x X-Wing 1 x Swordfish 6 x XY-Wing
Code: Select all
 . . . | . 6 . | . 2 .
 . 6 . | . 1 5 | . . .
 . . . | 8 . 7 | 3 . .
 ------+-------+------
 . 2 . | . 8 . | . . .
 7 . . | . . . | . 4 .
 . 5 . | 7 . . | . . 8
 ------+-------+------
 . . . | 6 . . | . . 3
 . 3 . | . 5 . | . . .
 4 . . | . . 2 | 1 7 5

56 x Hidden Single 1 x Direct Hidden Triplet 2 x Pointing 2 x Claiming 3 x Naked Pair 1 x X-Wing 1 x XYZ-Wing 1 x XY-Wing 1 x Turbot Fish 3 x Forcing X-Chain 1 x Forcing Chain
Code: Select all
 . . 7 | . . . | . 3 .
 4 . . | 5 . 9 | . . .
 . 3 . | 8 . . | . 1 .
 ------+-------+------
 . . . | . . . | 1 . 4
 . . 5 | 1 4 . | . . .
 . . 4 | . . 3 | . . 6
 ------+-------+------
 . 2 . | . . 5 | . 7 .
 9 . . | . . . | . . 1
 . . . | . . 2 | . 9 .

54 x Hidden Single 2 x Direct Hidden Pair 1 x Naked Single 6 x Pointing 3 x Claiming 1 x Naked Pair 1 x XY-Wing 3 x Forcing Chain
Code: Select all
 . . 7 | . . . | . 3 .
 4 . . | 5 . 9 | . . .
 . 3 . | 8 . . | . 1 .
 ------+-------+------
 . . . | . . . | 1 . 4
 . . 5 | 1 4 . | . . .
 . . 4 | . . 3 | . . 6
 ------+-------+------
 . 2 . | . . 5 | . 7 .
 9 . . | . . . | . . 1
 . . . | . . 2 | . 9 .

54 x Hidden Single 2 x Direct Hidden Pair 1 x Naked Single 6 x Pointing 3 x Claiming 1 x Naked Pair 1 x XY-Wing 3 x Forcing Chain

Final, I bold 6 x XY-Wing and renew my list.
Last edited by Eioru on Sat Sep 30, 2006 1:02 pm, edited 1 time in total.
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Re: Explain my thinking

Postby Eioru » Fri Sep 22, 2006 12:32 pm

To Ocean:
Your puzzle has a record 27 x Dynamic Region Forcing Chains on my 9/22 renew-list.
Ocean wrote:
Code: Select all
 *-----------*
 |...|..1|..2|
 |.3.|.2.|.4.|
 |5..|6..|...|
 |---+---+---|
 |..6|...|..7|
 |.7.|.1.|.3.|
 |8..|...|5..|
 |---+---+---|
 |7..|..5|...|
 |...|.3.|.2.|
 |..9|8..|6..|
 *-----------*

#
Analysis results
Difficulty rating: 9.2 / 10
This Sudoku can be solved using the following logical methods:
57 x Hidden Single
2 x Naked Single
1 x Direct Hidden Triplet
8 x Pointing
5 x Claiming
4 x Naked Pair
1 x X-Wing
3 x Hidden Pair
1 x Swordfish
6 x Forcing Chain
2 x Bidirectional Cycle
5 x Nishio Forcing Chains
3 x Cell Forcing Chains
10 x Region Forcing Chains
27 x Dynamic Region Forcing Chains
17 x Dynamic Contradiction Forcing Chains
4 x Dynamic Cell Forcing Chains
1 x Dynamic Double Forcing Chains
#

Eioru
 
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Re: A record named "maximum" of Sudoku list

Postby udosuk » Fri Sep 22, 2006 3:10 pm

Eioru wrote:5 Naked Single ..2.8.1.....7.4...9...6...4.1.....5.2.5...6.3.7.....4.8...5...9...1.8.....6.9.2..

...

I don't list two methods "Hidden Single" and "Direct Claiming",
because Hidden Single's record is 64 certainly, and Direct Claiming is...

I'm pretty sure somebody posted a puzzle with way more than 5 naked single steps (I think over 60, perhaps even 64) but I couldn't find it...

Also, are you sure the Hidden Single's record is 64? I only know that at the end of a puzzle there's generally a bunch of naked singles... But I'm not too sure all 64 empty cells from a 17-clue puzzle could be solved from hidden singles...

Any example would be much appreciated...
udosuk
 
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