## The hardest sudokus

Everything about Sudoku that doesn't fit in one of the other sections
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`top1465/Vidars #89:...3..5...5..1..3...7..4..12.....4...6..9......1..6..28..7..2...9..8..5...5..9..7r8c1    =  7     Hidden Singler258    -  2     Swordfishr4c3    =  9     Forcing Chains on [r5c1] `

I'm attempting to manually create the above forcing chains from my solver. Are these chains even remotely accurate?

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`[r5c1]=5=[r6c1]                  =9=[r4c3]-9-[r12c3], => [r4c3]=9 and [r12c3]<>9[r5c1]-5-[r5c9]=5=[r4c9]=6=[r4c8]                          [r4c89]=9=[r4c3]-9-[r12c3], => [r4c3]=9 and [r12c3]<>9*-----------------------------------------------------------------------------*| 1469    1248    468-9   | 3       267     78      | 5       246789  4689    || 469     5       2468-9  | 2689    1       278     | 6789    3       4689    || 369     238     7       | 5689    256     4       | 689     2689    1       ||-------------------------+-------------------------+-------------------------|| 2       378     9-38    | 158     357     13578   | 4       16789   35689   || 345     6       348     | 12458   9       123578  | 1378    178     358     || 3459    3478    1       | 458     3457    6       | 3789    789     2       ||-------------------------+-------------------------+-------------------------|| 8       134     346     | 7       3456    135     | 2       1469    3469    || 7       9       2346    | 1246    8       123     | 136     5       346     || 1346    1234    5       | 146     2346    9       | 1368    1468    7       |*-----------------------------------------------------------------------------*`
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

Wow, you guys seem to know each puzzle of the top1465 by heart

So thanks, Ocean, for identifying the 9.5-rated puzzle and for the link to the nice solution.
When i reformulate it, it looks like this:

r5c236=7 => triple 145 in r5c789 => r5c4=9 => r4c5=5
r5c236=7 => r5c789<>7 => r46c7=7 => r2c7<>7 => quad 3469 in r2c1237 => r2c5=5

r4c7=7 => r2c7<>7 => quad 3469 in r2c1237 => r2c5=5 => pair 12 in r7c15 => r7c7=7

r4c5=5 => r2c5<>5 => quad 3469 in r2c1237 => r2c7=7 => r7c7<>7 => pair 12 in r7c17 => r7c5=5

Coloring solves then.

So the Explainer rating here (as for other puzzles, which can be solved using triples and quads in short chains) seems too high for me.

--
ravel

Posts: 998
Joined: 21 February 2006

daj95376 wrote:Are these chains even remotely accurate?

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`[r5c1]=5=[r6c1]                  =9=[r4c3]-9-[r12c3], => [r4c3]=9 and [r12c3]<>9[r5c1]-5-[r5c9]=5=[r4c9]=6=[r4c8]                          [r4c89]=9=[r4c3]-9-[r12c3], => [r4c3]=9 and [r12c3]<>9`

Add four cells to pick-up an almost-locked-set and you have this nice loop:

r4c3=9=r4c89-9-{ALS:[r5c789|r6c78]=9|5=r5c9}-5-r5c1=5=r6c1=9=r4c3, => r4c3=9
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

For me this is the simplest deduction to show that r4c3=9:
r6c1=9 => r5c1=5 => r5c9<>5 => r4c9=5 => r4c8=6 => r4c3=9
How do you write it as NL ? Does it really need a 5-cell-ALS ?
ravel

Posts: 998
Joined: 21 February 2006

ravel wrote:For me this is the simplest deduction to show that r4c3=9:
r6c1=9 => r5c1=5 => r5c9<>5 => r4c9=5 => r4c8=6 => r4c3=9
How do you write it as NL ?

Well, there is the simpler errornet expression ...

r4c3=9=r6c1=5=r5c1-5-r5c9=5=r4c9=6=r4c8=9=r4c3, => r4c3=9

... which is an asymmetrical expression, meaning it can only be read left-to-right (L->R). Nice loops must (I think) be readable both L->R and L<-R.

Hence the ALS. There's probably a simpler NL for the same exclusion, but I don't see it.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

daj95376 wrote:I'm attempting to manually create the above forcing chains from my solver. Are these chains even remotely accurate?

You can find information on the solution to #89 here
It may be better to discuss the solution of this puzzle there. I used #89 puzzle in this thread to evaluate the Sudoku Explainer, which has become a good candidate for rating hard puzzles.

/Viggo
Viggo

Posts: 60
Joined: 21 April 2006

ravel wrote:When i reformulate it, it looks like this:
...
r4c7=7 => r2c7<>7 => quad 3469 in r2c1237 => r2c5=5 => pair 12 in r7c15 => r7c7=7

r4c5=5 => r2c5<>5 => quad 3469 in r2c1237 => r2c7=7 => r7c7<>7 => pair 12 in r7c17 => r7c5=5

Did you remove the candidate 3 in r7c5 in some way? Otherwise I cannot see that the deduction is valid.

/Viggo
Viggo

Posts: 60
Joined: 21 April 2006

Viggo wrote:Did you remove the candidate 3 in r7c5 in some way? Otherwise I cannot see that the deduction is valid.

Yes, after the elimintion of the 3 7's you get a hidden pair 57 in r69c2, an 8 in r8c2, and you can remove the 3's in r8c2 and r479c2 by box/line interaction. Sorry, that i did not say that, my intention was only to show, that the solution is relatively short.
ravel

Posts: 998
Joined: 21 February 2006

ravel wrote:Yes, after the elimintion of the 3 7's you get a hidden pair 57 in r69c2, an 8 in r8c2, and you can remove the 3's in r8c2 and r479c2 by box/line interaction. Sorry, that i did not say that, my intention was only to show, that the solution is relatively short.

Thank you - I see that now. Yes, it is a relatively short solution and with some very productive ALS's. The Explainer solution has some very long chains and did not discover the ALS's.

/Viggo
Viggo

Posts: 60
Joined: 21 April 2006

It has been a pleasure to read Viggo's thorough evaluation of the ratings from Sudoku Explainer, and the analysis of top1465#89. Insightful comments as usual from daj95376, r.e.s., ronk and ravel, and interesting discussions. It seems that Explainer's rating and Ravel's stepcounts can be regarded as complementary measures of a puzzle's difficulty.

Here is an innocent-looking beast ... (this time not the diagonal pattern, but a "fully symetric 20"-pattern where ab introduced the first puzzle to my knowledge.)
Code: Select all
`# oceanM20 #10111 *-----------* |..1|...|2..| |.3.|.4.|.5.| |6..|...|..7| |---+---+---| |...|1.3|...| |.8.|...|.3.| |...|6.4|...| |---+---+---| |2..|...|..6| |.4.|.5.|.8.| |..7|...|1..| *-----------*`

Analysis results
Difficulty rating: 9,8 / 10
This Sudoku can be solved using the following logical methods:
60 x Hidden Single
1 x Naked Single
3 x Pointing
4 x Claiming
3 x Naked Pair
1 x Hidden Pair
2 x Swordfish
1 x Jellyfish
1 x Forcing X-Chain
7 x Forcing Chain
1 x Nishio Forcing Chains
1 x Cell Forcing Chains
10 x Region Forcing Chains
18 x Dynamic Contradiction Forcing Chains
12 x Dynamic Region Forcing Chains
1 x Dynamic Cell Forcing Chains
1 x Dynamic Contradiction Forcing Chains (+)

Nearly all elimination steps (63 of them) take place before the first placement, and 32 of those steps are rated 9 or higher. So unless these steps can be bypassed by simpler methods, it's definitely not an easy starter.
Ocean

Posts: 442
Joined: 29 August 2005

Ocean,

When I ran your puzzle through my fledgling Templates-based solver, it reported 3784 ways to arrange 9s without conflicting with the givens. Ironically, there are only two ways (each) to arrange 1s and 6s.

Puzzles with a missing given value are often more difficult because they increase the number of candidates in every unsolved cell. Your [Edit: remove non-symmetric] puzzle nearly gave my general solver a coronary. It'll be interesting to see what Carcul, et. al., devise for solutions!

[Addendum:] I'm adding this information in hopes that others may succeed where I failed. Since the 1/6s overlap in bivalue cells [r28c6], all that's needed to resolve all 1/6s is to resolve one of them. I started with the assumption [r2c6]=1=[r8c6] and searched for a contradiction. Had I succeeded, then four additional elimination chains would have cracked this puzzle.
Last edited by daj95376 on Thu Jul 13, 2006 12:25 pm, edited 3 times in total.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

Ocean,

its always astonishing, what you bring in here. What a Beautiful Beast! I love to put it on the top of my list.

- it is fully symmetric and does not have the diagonal pattern
- it has only 20 clues (all others above 10 steps have 23 or more)
- before applying a swordfish all (61) cells have at least 3 candidates
- Explainer cannot place a number before the 9.8 hint (but i think it missed a long singles chain elimination to solve it with 9.5)
- like only a few others, susser 2.5.4 does not solve it with standard settings
- it cannot be solved with a lucky guess and basic methods
- i needed 6 tries to get it down to 15 steps.
I looked a bit closer to a 16- and the 15-step solution.
The first one can be reduced to 11 steps by applying 4 swordfishes and starts with 3 probably very long chains (one using pair and box elimination). 5 of the chains are relatively short (less than about 10 cells).
The second one is reduced to 14 steps by a swordfish and starts with 4 very long chains (2 using a pair). Then there are 5 shorter and rather similar eliminations followed by 2 very short ones (about 5 cells). At the end again 3 long singles chains are needed.
I did not check, if steps were redundant. (If someone wants to analyse the 15-17 steps solutions, i can PM the eliminations.)
[Edit:]
gsf wrote:... has backdoor size 2 (up to xy cycles, no uniqueness) -- the first one I've seen ...
Last edited by ravel on Tue Jul 18, 2006 10:30 am, edited 1 time in total.
ravel

Posts: 998
Joined: 21 February 2006

Thank you for analyzing this puzzle! It has been around for a few months (available here), but it took a while to discover its unusual properties.
Ocean

Posts: 442
Joined: 29 August 2005

The two second highest Explainer Ratings found in the symmetrical pattern (20 clues) were 9.7 and 9.6, with twelve and ten chains of type 'dynamic+', respectively. These are the only puzzles I have seen with 9.7/9.6-ratings so far, so maybe they can add some variability to the puzzle set.

#
001000200030040050600000007000103000080000030000506000700000006050030080009000100 #ER=9.7
#

Analysis results
Difficulty rating: 9,7 / 10
This Sudoku can be solved using the following logical methods:
61 x Hidden Single
5 x Pointing
5 x Claiming
3 x Naked Pair
2 x Hidden Pair
1 x Naked Triplet
2 x Swordfish
1 x Aligned Pair Exclusion
1 x Bidirectional Cycle
10 x Forcing Chain
10 x Region Forcing Chains
1 x Cell Forcing Chains
3 x Dynamic Cell Forcing Chains
9 x Dynamic Contradiction Forcing Chains
1 x Dynamic Region Forcing Chains
3 x Dynamic Contradiction Forcing Chains (+)
9 x Dynamic Region Forcing Chains (+)

#
001000200030040050600000007000105000040000080000904000700000006050030040002000100 #ER=9.6
#

Analysis results
Difficulty rating: 9,6 / 10
This Sudoku can be solved using the following logical methods:
60 x Hidden Single
1 x Naked Single
14 x Pointing
3 x Claiming
1 x Naked Pair
3 x Hidden Pair
2 x Naked Triplet
1 x Swordfish
2 x Hidden Triplet
1 x XY-Wing
1 x Turbot Fish
1 x Forcing X-Chain
3 x Bidirectional Y-Cycle
17 x Forcing Chain
17 x Region Forcing Chains
1 x Cell Forcing Chains
3 x Dynamic Region Forcing Chains
5 x Dynamic Contradiction Forcing Chains
6 x Dynamic Contradiction Forcing Chains (+)
4 x Dynamic Region Forcing Chains (+)
Ocean

Posts: 442
Joined: 29 August 2005

The 2 puzzles needed 4 and 9 steps.
I will insert them, when i am through with the 50 hard ones from the Fully Symmetrical thread (excluding those 2 and the 3 ones already in the list).
ravel

Posts: 998
Joined: 21 February 2006

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