The New Sudoku Players' Forum

[vidarino]

There's another one of my home-made "monsters".

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Vidar's Monster #2
+-------+-------+-------+
| 8 . . | 4 . . | 6 . . |
| . 3 . | . 6 . | . 7 . |
| . . 7 | . . . | . . 2 |
+-------+-------+-------+
| 1 . . | 6 . . | 3 . . |
| . 7 . | . 4 . | . 8 . |
| . . 3 | . . 9 | . . 6 |
+-------+-------+-------+
| . . . | 8 . . | 4 . . |
| . 9 . | . 5 . | . 2 . |
| . . 1 | . . . | . . 5 |
+-------+-------+-------+


While my solver can solve it without resorting to blind guesses, it has to use a whole bunch of pretty deep forcing chains to get anywhere (so deep they almost equal blind trial and error in my book, but hey). And in case you wondered, the fun starts immediately!

So, I'm very interested in seeing a more elegant solution to this one, so get cracking, folks. (Are you up to the challenge again, Carcul? )

Vidar

[tarek]

Hi Vidar,

very good puzzle, one question though:

Look at the following puzzles from the top1465:

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 7 . . | . . . | 4 . . 
 . 2 . | . 7 . | . 8 . 
 . . 3 | . . 8 | . . 9 
-------+-------+------
 . . . | 5 . . | 3 . . 
 . 6 . | . 2 . | . 9 . 
 . . 1 | . . 7 | . . 6 
-------+-------+------
 . . . | 3 . . | 9 . . 
 . 3 . | . 4 . | . 6 . 
 . . 9 | . . 1 | . . 5 

 2 . . | 5 . . | 6 . . 
 . 9 . | . . . | . 3 . 
 . . 1 | . . 9 | . . 4 
-------+-------+------
 4 . . | 9 . . | 7 . . 
 . 5 . | . 4 . | . 8 . 
 . . 7 | . . 1 | . . 6 
-------+-------+------
 . . . | 1 . . | 3 . . 
 . 2 . | . 7 . | . 9 . 
 . . 8 | . . 3 | . . 5 

 4 . . | 6 . . | 3 . . 
 . 1 . | . 2 . | . 6 . 
 . . 8 | . . 7 | . . 1 
-------+-------+------
 9 . . | 8 . . | 5 . . 
 . 4 . | . 5 . | . 1 . 
 . . . | . . 2 | . . 7 
-------+-------+------
 5 . . | . . . | 6 . . 
 . 3 . | . 8 . | . 4 . 
 . . . | . . 9 | . . 5 


 5 . . | 4 . . | 8 . . 
 . . . | . 9 . | . 1 . 
 . . 2 | . . 1 | . . 5 
-------+-------+------
 6 . . | 3 . . | 4 . . 
 . 5 . | . 7 . | . . . 
 . . 4 | . . . | . . 8 
-------+-------+------
 3 . . | 6 . . | 7 . . 
 . 6 . | . . . | . 8 . 
 . . 8 | . . 2 | . . 1 

 . . . | . . . | 8 . . 
 . 7 . | . 9 . | . 1 . 
 . . 3 | . . 7 | . . 5 
-------+-------+------
 1 . . | 6 . . | . . . 
 . 5 . | . 2 . | . 4 . 
 . . 8 | . . 9 | . . 7 
-------+-------+------
 4 . . | . . . | 2 . . 
 . 6 . | . 1 . | . 5 . 
 . . . | . . 6 | . . 3 

 9 . . | 7 . . | 3 . . 
 . 5 . | . 2 . | . 1 . 
 . . 7 | . . . | . . 9 
-------+-------+------
 3 . . | 6 . . | 7 . . 
 . 6 . | . 5 . | . 3 . 
 . . 2 | . . 8 | . . 6 
-------+-------+------
 1 . . | 4 . . | 6 . . 
 . 4 . | . 6 . | . 8 . 
 . . 6 | . . 3 | . . 4 

 . . . | 3 . . | 5 . . 
 . 5 . | . 1 . | . 3 . 
 . . 7 | . . 4 | . . 1 
-------+-------+------
 2 . . | . . . | 4 . . 
 . 6 . | . 9 . | . . . 
 . . 1 | . . 6 | . . 2 
-------+-------+------
 8 . . | 7 . . | 2 . . 
 . 9 . | . 8 . | . 5 . 
 . . 5 | . . 9 | . . 7 

 6 . . | . . . | 3 . . 
 . 5 . | . 9 . | . 8 . 
 . . 2 | . . 6 | . . 9 
-------+-------+------
 8 . . | . . . | 7 . . 
 . 7 . | . 5 . | . 4 . 
 . . . | . . 1 | . . 5 
-------+-------+------
 1 . . | 3 . . | 5 . . 
 . 4 . | . 2 . | . 6 . 
 . . 8 | . . 7 | . . 2 

The configuration of these puzzle (which are among the top hardest puzzles) is quite close to yours, do you think that this configuration will generate the hardes known puzzle, & were you inspired by the configuration so that you programmed your generator to generate it following the same lines or was it just a happy accident???

Tarek

[vidarino]

Heh, wow, I didn't know about those.

I just accidentally noticed that when looking for puzzles of this configuration, my solver had to resort to forcing chains and guesses a lot more often than with the other patterns I tried. So I let the generator run for a while to see what came out of it. Afterwards I reran the solver on the generated puzzles to rate them properly, and found Monster #2 among the results.

It's pretty interesting to see that it's easy to generate puzzles from some patterns, and a lot harder from others, and that their difficulty is completely unrelated to that property. I.e. I have a few nice patterns that appear to be extremely rare, but the puzzles that come out of them are all dead easy.

Vidar

[Carcul]

Hi Vidarino.

That is a really hard puzzle. Congratulations one more time. I have an extremely complicated solution for you:

[r9c7](-8-[r9c2]=8=[r8c3]-8-[r4c3])(-8-[r8c4|r8c7|r8c9]-1,3,7-[r8c1|r8c6])(-8-[r2c7|r3c7])-8-[r8c9]=8=[r2c9]=4=[r4c9](-4-[r4c2|r4c8|r6c8])=7=[r6c7](-7-[r6c4]=7=[r8c4|r9c4]-7-[r7c6])-7-[r8c7](-1-[r8c4|r8c9])-1-[r2c7|r3c7]-5,9-[r1c8|r1c9](-1-[r1c5])-1-[r1c2]=1=[r3c2](-1-[r3c5])=6=[r3c1](-6-[r5c1]=6=[r5c3]-6-[r7c3])-6-[r8c1](-4-[r8c6]-6-[r7c6])-4-[r6c1]{=4=[r6c2](=8=[r4c2])=8=[r6c5](-8-[r3c5]=8=[r3c6])=1=[r7c5]-1-[r7c6]}{XY-Wing:[r6c1]-5-[r5c1]-9-[r5c9]-1-[r6c8]-5-[r6c1]}-5-[r6c1](-2-[r2c1|r2c7]-5-[r2c3|r2c4|r2c6])-2-[r4c3|r4c8]-5-[r4c6]{Nice Loop: [r5c1]-5-[r4c3]=5=[r4c8]-5-[r6c8]=5=[r6c4]-5-[r3c4]=5=[r3c7]-5-[r2c7]=5=[r2c1]-5-[r5c1]}-5-[r5c1]-9-[r2c1]=9=[r1c3]-9-[r4c3](-5-[r4c8])-5-[r7c3]-2-[r7c6]-3-[r8c4]-7-[r8c9]-3-[r1c9]-1-[r5c9]-9-[r4c8]

This "monster" chain means that, if r9c7=8 then r4c8 wouldn't have any possible candidate. So, r9c7<>8 and that solve the puzzle.

Are you up to the challenge again, Carcul?

Now the real challenge is to find a longer and simpler solution for this puzzle.

Regards, Carcul

[vidarino]

Holy sh*t, Batman!

That is ONE BIG chain you got there! I'm deeply impressed. Well done!

How on earth did you do it??

Did you hunt for the magic cell first, and then try to find a way to crack it? (I have yet to look for the magic cells, so perhaps there are others.)

Vidar

[Wolfgang]

Another one is r4c3 (but i dont know how to eliminate 8 from r8c3).
An alternative to fix r9c2=8 is to show that you can eliminate 2,4, and 6 there. This leads to 3 simple, but long chains, which i am too lazy to denote.

[vidarino]

OK, I just checked, and there are a total of 2 or 5 magic cells. 2 if you regard a cracked puzzle as requiring absolutely nothing but singles, and 5 if you allow the use of locked candidates in the "endgame". (Both of the magic cells mentioned above needed locked candidates.)

[Jeff]

I have an extremely complicated solution for you:

[r9c7](-8-[r9c2]=8=[r8c3]-8-[r4c3])(-8-[r8c4|r8c7|r8c9]-1,3,7-[r8c1|r8c6])(-8-[r2c7|r3c7])-8-[r8c9]=8=[r2c9]=4=[r4c9](-4-[r4c2|r4c8|r6c8])=7=[r6c7](-7-[r6c4]=7=[r8c4|r9c4]-7-[r7c6])-7-[r8c7](-1-[r8c4|r8c9])-1-[r2c7|r3c7]-5,9-[r1c8|r1c9](-1-[r1c5])-1-[r1c2]=1=[r3c2](-1-[r3c5])=6=[r3c1](-6-[r5c1]=6=[r5c3]-6-[r7c3])-6-[r8c1](-4-[r8c6]-6-[r7c6])-4-[r6c1]{=4=[r6c2](=8=[r4c2])=8=[r6c5](-8-[r3c5]=8=[r3c6])=1=[r7c5]-1-[r7c6]}{XY-Wing:[r6c1]-5-[r5c1]-9-[r5c9]-1-[r6c8]-5-[r6c1]}-5-[r6c1](-2-[r2c1|r2c7]-5-[r2c3|r2c4|r2c6])-2-[r4c3|r4c8]-5-[r4c6]{Nice Loop: [r5c1]-5-[r4c3]=5=[r4c8]-5-[r6c8]=5=[r6c4]-5-[r3c4]=5=[r3c7]-5-[r2c7]=5=[r2c1]-5-[r5c1]}-5-[r5c1]-9-[r2c1]=9=[r1c3]-9-[r4c3](-5-[r4c8])-5-[r7c3]-2-[r7c6]-3-[r8c4]-7-[r8c9]-3-[r1c9]-1-[r5c9]-9-[r4c8]

This "monster" chain means that, if r9c7=8 then r4c8 wouldn't have any possible candidate. So, r9c7<>8 and that solve the puzzle.

Well done Carcul, This is a monster indeed.

[Carcul]

I'm deeply impressed. Well done!

Well done Carcul, This is a monster indeed.

I sincerily appreciate those words, but I am sorry to say that I don't deserve them. Although most of that chain was constructed in my head, at a certain stage I needed to write a separate grid with the implications of r9c7=8, and then in this grid I spoted the other two "sub-loops" and the corresponding implications.

How on earth did you do it??

After some time analysing the puzzle (and finding three loops that did't advanced the puzzle) I noted the relation between r9c7=8, the ALS r8c479, the almost ALS's in box 3, and so on, and started to construct in the head all those implications. At a certain stage I was tired of keeping track of so many implications at the same time, and I have decided to write a separate grid.
So, I have cheating a bit and I do not found my solution very legitimate. Please accept my apologies for that.
However, I finded very interesting that, in such a hard puzzle, the only thing "needed" to solve the puzzle is to prove that a candidate must not be in a given cell, and so I decided to post it.
In the meantime, perhaps someone else can find a trully legitimate and elegant solution.

Regards, Carcul

[tso]

Sudoku Susser could not find a single candidate elimination (without tabling) from the starting grid of Monster #2. That's impressive. Even the so-called "toughest known" included with the application allows a few candidate eliminations.

[aeb]

Sudoku Susser could not find a single candidate elimination (without tabling) from the starting grid of Monster #2. That's impressive. Even the so-called "toughest known" included with the application allows a few candidate eliminations.

Still, there are a few simple forcing chains. For example
(4,3)!8 > (8,3)8 > (8,9)!8 > (2,9)8 > (4,9)4 > (4,3)!4
shows that r4c3 is not 4. And
(6,5)!8 > (6,2)8 > (9,2)!8 > (9,7)8 > (8,9)!8 > (2,9)8 > (4,9)4 > (6,7)7 > (6,5)!7
shows that r6c5 is not 7.
(After that I have nothing simple anymore. The next step shows (5,9)!9 with a very messy argument.)

[Carcul]

Still, there are a few simple forcing chains. For example...

Besides those two, I have also identified the following one:

[r5c1|r5c3](-2-[r4c2|r6c2])-2-[r5c7]=2=[r6c7]=7=[r4c9]=4=[r2c9]=8=[r8c9]-8-[r8c3]=8=[r4c3]-8-[r4c2|r6c2]-4,5-[r6c1]-2-[r5c1|r5c3], => r5c1,r5c3<>2.

However, this deduction is completely useless.

Carcul

[ronk]

Still, there are a few simple forcing chains. For example
(4,3)!8 > (8,3)8 > (8,9)!8 > (2,9)8 > (4,9)4 > (4,3)!4
shows that r4c3 is not 4.

Since (4,3) has five candidates, and (4,3)!8 and (4,3)!4 is not a contradiction, how does the above forcing chain show anything?

Sudoku Susser could not find a single candidate elimination (without tabling) from the starting grid of Monster #2.

Ditto for my solver, and I see the same result for tarek's 2nd and 3rd gleanings from the top1465. How does Susser do on these two?

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2..5..6...9.....3...1..9..44..9..7...5..4..8...7..1..6...1..3...2..7..9...8..3..5
4..6..3...1..2..6...8..7..19..8..5...4..5..1......2..75.....6...3..8..4......9..5

[edit: In order, the puzzles are #244 and #125 of the top1465.]

TIA, Ron

[aeb]

Also:
[r5c1|r5c3](-2-[r4c2|r6c2])-2-[r5c7]=2=[r6c7]=7=[r4c9]=4=[r2c9]=8=[r8c9]-8-[r8c3]=8=[r4c3]-8-[r4c2|r6c2]-4,5-[r6c1]-2-[r5c1|r5c3], => r5c1,r5c3<>2.

Yes. As an exercise in understanding this notation, let me see how I would write this.
a) (5,[13])2 > ([46],2)!2
b) (5,[13])2 > (5,7)!2 > (6,7)2 > (4,9)7 > (2,9)4 > (8,9)8 > (8,3)!8 > (4,3)8 > ([46],2)!8
Combining a) and b): (5,[13])2 > ... > ([46],2)!28 > ([46],2)45 > (6,1)2 > (5,[13])!2,
so that (5,[13])!2.

[aeb]

For example
(4,3)!8 > (8,3)8 > (8,9)!8 > (2,9)8 > (4,9)4 > (4,3)!4
shows that r4c3 is not 4.

Since (4,3) has five candidates, and (4,3)!8 and (4,3)!4 is not a contradiction, how does the above forcing chain show anything?

Think about it. If (4,3)!8 implies (4,3)!4, then (4,3)!4.