The hardest sudokus

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

Postby Allan Barker » Tue Nov 11, 2008 7:04 am

Mauricio wrote:How hard is really this sudoku?
Code: Select all
+-------+-------+-------+
| . . . | . . 1 | . . 2 |
| . . 3 | . . . | . 4 . |
| . 5 . | . 6 . | 7 . . |
+-------+-------+-------+
| . . . | 8 . . | . 7 . |
| . . 7 | . . 3 | 8 . . |
| 9 . . | . 5 . | . . 1 |
+-------+-------+-------+
| . . 6 | . 8 . | 2 . . |
| . 4 . | 6 . . | . . 7 |
| 2 . . | . . 9 | . 6 . |
+-------+-------+-------+ ED=11.3, ER=?

To solve it, I only need to suppose uniqueness and to use not more than hidden pair.


Champagne, thank you for pointing this one out.

Congratulations to whoever found this, this, this.... monster!, not to mention finding an easy solution. I have so far solved the first 12 candidates but that's enough to see something about the nature of the puzzle. Using standard Sudoku logic, it is a very difficult puzzle but not the worst it seems like what I now expect from an 11.3 rating.

What's different is its uniform difficulty, there don't seem to be any weak spots. Solution paths from any starting point are all about the same and elimination to elimination is much the same as well. The paths are unusually long averaging above 70 total sets in length (set +cover sets). Several of the first eliminations have large embedded broken wing like structures, long odd length loops of strong links.

Code: Select all
candidate    sets/
(RCN)        linksets
--------------------------
338          64     BW logic
262          79     BW logic
662          73     BW logic
228          70     BW logic
745          75     BW logic
211          83     BW logic
226          78     BW logic
416          49
275          79
473          71
559          71
971          72


Maybe later I can post a picture in the thread on set-logic monster solutions, but they are not very pretty. I am truly amazed that a uniqueness / symmetry argument can decimate such a puzzle so easily. Lots to learn.

Edit: What's the name??
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Postby Glyn » Tue Nov 11, 2008 8:09 am

Allan Barker For your further amusement some more on a slightly different theme of symmetry may be found here. http://forum.enjoysudoku.com/viewtopic.php?t=6355&postdays=0&postorder=asc&start=0 Once again Mauricio is one of the inspirations for these. Links to earlier discussions may be found there.
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Postby Mauricio » Tue Nov 11, 2008 8:39 am

Mauricio wrote:
Code: Select all
+-------+-------+-------+
| . . . | . . 1 | . . 2 |
| . . 3 | . . . | . 4 . |
| . 5 . | . 6 . | 7 . . |
+-------+-------+-------+
| . . . | 8 . . | . 7 . |
| . . 7 | . . 3 | 8 . . |
| 9 . . | . 5 . | . . 1 |
+-------+-------+-------+
| . . 6 | . 8 . | 2 . . |
| . 4 . | 6 . . | . . 7 |
| 2 . . | . . 9 | . 6 . |
+-------+-------+-------+ ED=11.3, ER=?


Let's call it "Shining Mirror".
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Postby susume » Tue Nov 11, 2008 5:02 pm

Only a monster until you apply the looking glass. Then it turns out to be a puppy.:D

Clued in by the mention of symmetry, I found it very easy. Who knows if I would ever have noticed that on my own, however. The first placements were two naked singles which followed immediately after this conjecture: Every cell on the axis of symmetry must contain one of the digits which is its own mirror. I'm sure one of the maths wizards here could prove this, or maybe it was already proved long ago among Gurth's emeralds. After that the hardest thing I used was an x-wing (no uniqueness methods AFAIK).
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Postby Mauricio » Tue Nov 11, 2008 5:06 pm

susume wrote:The first placements were two naked singles which followed immediately after this conjecture: Every cell on the axis of symmetry must contain one of the digits which is its own mirror.
... After that the hardest thing I used was an x-wing (no uniqueness methods AFAIK).

The thing is, to prove the conjecture you need to assume uniqueness.
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Postby m_b_metcalf » Tue Nov 11, 2008 11:36 pm

Mauricio wrote:ED=11.3, ER=?

Mauricio, it's 11.3/11.3/11.3. After 16 Dynamic Contradiction Forcing Chains, SE sets r7c2 to 9. After that, the hardest step is 10.6.

Regards,

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Postby tarek » Wed Nov 12, 2008 12:35 am

I've been attempting to rate all the my diamond puzzle collection (formally known as the Pearly6000), I'm currently at 3500.

Here are the the diamonds of ED>=11.0, the puzzles are mostly known ... The 1st 3045 are already incorporated under the q1 & q2 taxonomy lists under the labels starting with "tarek-" if they were not incorporated under an earlier label.

The 1st set are the puzzle with ratings, the second set contains the same puzzles but with a more comprehensive data.


........6..5..18...9...8.7....8.2.....3.1.2..4..5.3....6.....9...83..1..7.......4 11.8/11.8/11.5 tarx0075
2.......6.5..8..1...4...9...7.3.1......82.......7.5.3...9...4...8..1..5.6.......2 11.8/11.8/11.4 ultra0203
.......39.....1..5..3.5.8....8.9...6.7...2...1..4.......9.8..5..2....6..4..7..... 11.9/11.9/11.3 Golden Nugget
1.......9.4...3.8...2...6...7..58.......2.......7.4.5...6...2...3.8...7.9.......1 11.7/11.7/11.2 Pearly6000-2803
8.......5.6...2.3...7...9......31.2.....7.....1.6.4...5.....7...3.1...4...9.....8 11.4/11.4/11.2 Pearly6000-2153
8.......5.1...3.2...4...7.......1.3....28.....9.3.6...5.....4...2.9...6...7.....8 11.4/11.4/11.2 Pearly6000-3025
3.....9...7...1.5...2.....4....76.1....3.5....6.81....4.....2...5.6...8...9.....3 11.4/11.4/11.2 Pearly6000-2191
1.......7.3...9.6...5...2......43.9.....1.....4.9.8...7.....5...6.4...8...2.....1 11.4/11.4/11.2 Pearly6000-3038
6.......8.2...3.1...7...4.......5.3....37.....5.2.9...8.....7...1.5...9...4.....6 11.3/11.3/11.2 Pearly6000-2842
3.......5.7...8.6...2...9...1...6......82.......1.7.4...9...2...8.6...1.5.......3 11.3/11.3/11.2 Pearly6000-1803
1.......4.2.8...7...9...3.....6.5.8....1......5..82...4.....9...7.5...6...3.....1 11.2/11.2/11.2 Pearly6000-2284
3.......1.4...2.7...5...8......76.4.....5.....6.2.9...1.....5...7.6...9...8.....3 11.6/11.6/11.1 Pearly6000-1847
3.......8.7.5...1...6...4...9.2.1.......4.......97..2.4.....3...5...2.7...8.....6 11.6/11.6/11.1 ultra0313
9.......7.1...8.3...2...5.......6.1....32.....6.1.4...7.....2...3.6...4...5.....9 11.5/11.5/11.1 Pearly6000-2164
5.......8.3...2.4...9...1......27.3....5.4....7.63....8.....9...4.7...6...1.....5 11.5/11.5/11.1 Pearly6000-2228
6.......8.3...7.4...9...1.....3.2.7.....14....2.75....8.......9.4.2...5...1...6.. 11.3/11.3/11.1 Pearly6000-2224
1.......4.2...6.5...3...9.....2.8.6.....95....8.67....4.....3...5.8...7...9.....1 11.3/11.3/11.1 Pearly6000-2221
4.2.....1.7...9.2...8...6.....7.5.9......2....5.93....1.....8...2.5...3...6.....4 11.2/11.2/11.1 Pearly6000-3036
8.3.....7.6...1.3...9...5...2...6.1....3........124...7.....9...3.2...4...5.....8 11.2/11.2/11.1 Pearly6000-3012
1.4.....9.3...7.4...6...2......83.7....4......8.7.5...9.....6...4.8...5...2.....1 11.2/11.2/11.1 Pearly6000-3014
.7......1..29...5.8.....4....625.........7......3.6.9.4.....7...1......8..5.3..6. 11.4/11.4/11.0 Pearly6000-1839
.5......4..39...7.1.....6....837........95......2...9.6.....5...4......1..7.2..8. 11.2/11.2/11.0 Pearly6000-2804



Puzzle,ER/EP/ED,Labels,Sx9,Sxt,Q1,Q2
........6..5..18...9...8.7....8.2.....3.1.2..4..5.3....6.....9...83..1..7.......4,11.8/11.8/11.5,pearly6000-3091;tarx0075,2974,2042,95057,98321
2.......6.5..8..1...4...9...7.3.1......82.......7.5.3...9...4...8..1..5.6.......2,11.8/11.8/11.4,Pearly6000-1225; ultra0203,2198,1238,12977,95268
.......39.....1..5..3.5.8....8.9...6.7...2...1..4.......9.8..5..2....6..4..7..... ,11.9/11.9/11.3,Pearly6000-1812;Golden Nugget,3464,2161,95657,99220
1.......9.4...3.8...2...6...7..58.......2.......7.4.5...6...2...3.8...7.9.......1,11.7/11.7/11.2,Pearly6000-2803,1609,749,95425,95886
8.......5.6...2.3...7...9......31.2.....7.....1.6.4...5.....7...3.1...4...9.....8 ,11.4/11.4/11.2,Pearly6000-2153,1354,659,95002,95175
8.......5.1...3.2...4...7.......1.3....28.....9.3.6...5.....4...2.9...6...7.....8,11.4/11.4/11.2,Pearly6000-3025,1314,559,11030,89064
3.....9...7...1.5...2.....4....76.1....3.5....6.81....4.....2...5.6...8...9.....3 ,11.4/11.4/11.2,Pearly6000-2191,1817,875,86523,97500
1.......7.3...9.6...5...2......43.9.....1.....4.9.8...7.....5...6.4...8...2.....1,11.4/11.4/11.2,Pearly6000-3038,1178,678,9618,56993
6.......8.2...3.1...7...4.......5.3....37.....5.2.9...8.....7...1.5...9...4.....6,11.3/11.3/11.2,Pearly6000-2842,1192,703,79842,93757
3.......5.7...8.6...2...9...1...6......82.......1.7.4...9...2...8.6...1.5.......3 ,11.3/11.3/11.2,Pearly6000-1803,1703,771,98870,99161
1.......4.2.8...7...9...3.....6.5.8....1......5..82...4.....9...7.5...6...3.....1,11.2/11.2/11.2,Pearly6000-2284,1491,889,53841,95092
3.......1.4...2.7...5...8......76.4.....5.....6.2.9...1.....5...7.6...9...8.....3 ,11.6/11.6/11.1,Pearly6000-1847,1413,733,95295,98045
3.......8.7.5...1...6...4...9.2.1.......4.......97..2.4.....3...5...2.7...8.....6,11.6/11.6/11.1,Pearly6000-1202;ultra0313,1431,659,95262,98805
9.......7.1...8.3...2...5.......6.1....32.....6.1.4...7.....2...3.6...4...5.....9 ,11.5/11.5/11.1,Pearly6000-2164,1538,708,94074,97570
5.......8.3...2.4...9...1......27.3....5.4....7.63....8.....9...4.7...6...1.....5 ,11.5/11.5/11.1,Pearly6000-2228,1831,994,73873,97572
6.......8.3...7.4...9...1.....3.2.7.....14....2.75....8.......9.4.2...5...1...6.. ,11.3/11.3/11.1,Pearly6000-2224,1233,647,77438,95400
1.......4.2...6.5...3...9.....2.8.6.....95....8.67....4.....3...5.8...7...9.....1 ,11.3/11.3/11.1,Pearly6000-2221,1209,591,78275,95027
4.2.....1.7...9.2...8...6.....7.5.9......2....5.93....1.....8...2.5...3...6.....4,11.2/11.2/11.1,Pearly6000-3036,1465,678,9641,38948
8.3.....7.6...1.3...9...5...2...6.1....3........124...7.....9...3.2...4...5.....8,11.2/11.2/11.1,Pearly6000-3012,1336,635,11708,45920
1.4.....9.3...7.4...6...2......83.7....4......8.7.5...9.....6...4.8...5...2.....1,11.2/11.2/11.1,Pearly6000-3014,1299,606,11631,45543
.7......1..29...5.8.....4....625.........7......3.6.9.4.....7...1......8..5.3..6. ,11.4/11.4/1.01,Pearly6000-1839,1605,663,95351,98858
.5......4..39...7.1.....6....837........95......2...9.6.....5...4......1..7.2..8.,11.2/11.2/11.0,Pearly6000-2804,1400,635,95325,97974
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Postby Glyn » Wed Nov 12, 2008 2:55 am

If one is speculating with 'inspired' guesses then exploring rotation and one particular digit remapping might be very beneficial to solving 'Fata Morgana'.
There is something very interesting between two of the rows as well, but you'd need to really cheat to find that.
Knowing both 'Fata' would become a 'Size 0':)
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Postby tarek » Wed Nov 12, 2008 4:07 am

whatever Mauricio's using to scan for the 1st move ED rating is very powerful (probably also used for the patterns game) ....

the version which I have been using measures the whole puzzle difficulty as well ... which takes minutes if not hours ... That is why most of the recent postings are ER-less:D

How long does it take to give the result for your last puzzle Mauricio ? is it less than 5 min ?

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Postby JPF » Wed Nov 12, 2008 4:41 am

gsf wrote:
tarek wrote:g.r.emlin is the best to do all of this ....

g.r.emlin can verify q1,q2,Sx9 & Sxt ... SE121 rating has to be left for the poster's honesty.

If g.r.emlin is not up for it (for whatever devilish reasons:D ) Then I'm ready to do the same job.

My suggestion:

*Top 10 of each category in both line & graphic format
*categories are q1,q2,Sx9,Sxt,SE121
* verify q1,q2,Sx9,Sxt but not SE121 (verify SE131 if possible)
* SE121 & other q1,q2,Sx9,Sxty have to be provided by poster or that entry is disqualified


there has to be a database of posted puzzles (an updated q2 taxonomy list is the best) ... failing that .... I would use my own database (if organising this falls to me, that include famous puzzle lists to date )

sounds like a good start
might take 2 weeks to get the pieces together


Where do we stand with this project ?

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Postby m_b_metcalf » Wed Nov 12, 2008 10:10 am

tarek wrote:whatever Mauricio's using to scan for the 1st move ED rating is very powerful (probably also used for the patterns game) ....

Well, gsf doctored the original SE to provide ER/EP/ED. Anyone capable of doctoring it further could make a version that stops after evaluating ED. That, indeed, would be very useful in the Patterns Game:!: So, if anyone has that, maybe he could make it generally available.
tarek also wrote:
How long does it take to give the result for your last puzzle Mauricio ? is it less than 5 min ?

I obtained the 11.3/11.3/11.3 (see above) by stepping through, which took about 2 hours of real time. I would think well over an hour would be required if running in batch mode (there are several 11.3 steps each requiring many minutes).

Regards,

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Postby Mauricio » Wed Nov 12, 2008 11:43 am

m_b_metcalf wrote:Mauricio, it's 11.3/11.3/11.3. After 16 Dynamic Contradiction Forcing Chains, SE sets r7c2 to 9. After that, the hardest step is 10.6.
Thanks! I was expecting something higher:(
tarek wrote:whatever Mauricio's using to scan for the 1st move ED rating is very powerful (probably also used for the patterns game) ....

the version which I have been using measures the whole puzzle difficulty as well ... which takes minutes if not hours ... That is why most of the recent postings are ER-less:D

How long does it take to give the result for your last puzzle Mauricio ? is it less than 5 min ?
I use my own rating based on tabling. My method can't solve my previous puzzle, and it stops after 8 seconds.

One more automorphic puzzle
Code: Select all
+-------+-------+-------+
| . . . | . . 1 | . . 2 |
| . 1 . | . 3 . | . 4 . |
| . . 5 | 6 . . | . . . |
+-------+-------+-------+
| . . 2 | 7 . . | 8 . . |
| . 8 . | . . 4 | . 3 . |
| 1 . . | . 9 . | . . . |
+-------+-------+-------+
| . . . | 3 . . | 5 . . |
| . 9 . | . 8 . | . 1 . |
| 6 . . | . . . | . . 7 |
+-------+-------+-------+ ER=11.3/11.3/10.5
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Postby gsf » Wed Nov 12, 2008 5:47 pm

m_b_metcalf wrote:
tarek wrote:whatever Mauricio's using to scan for the 1st move ED rating is very powerful (probably also used for the patterns game) ....

Well, gsf doctored the original SE to provide ER/EP/ED. Anyone capable of doctoring it further could make a version that stops after evaluating ED. That, indeed, would be very useful in the Patterns Game:!: So, if anyone has that, maybe he could make it generally available.

that's a good idea
I'll add that to the todo list
also on that list is a general mechanism for collecting puzzles with specific properties
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Postby m_b_metcalf » Wed Nov 12, 2008 9:11 pm

gsf wrote:
m_b_metcalf wrote:
tarek wrote:whatever Mauricio's using to scan for the 1st move ED rating is very powerful (probably also used for the patterns game) ....

Well, gsf doctored the original SE to provide ER/EP/ED. Anyone capable of doctoring it further could make a version that stops after evaluating ED. That, indeed, would be very useful in the Patterns Game:!: So, if anyone has that, maybe he could make it generally available.

that's a good idea
I'll add that to the todo list

Well, it's an idea, but on reflection I'm not so sure it's a good one. If the game is to be a test of programming skills and strategey, using a third-party product seems to be underhand. Mauricio has gone to the trouble of developing his own algorithm, and now that he's revealed his approach it's open to any of us to make our own attempt (my own attempt to find a fast proxy for ED has so far failed).

Regards,

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Postby eleven » Thu Nov 13, 2008 5:59 am

Mauricio wrote:One more automorphic puzzle
Code: Select all
+-------+-------+-------+
| . . . | . . 1 | . . 2 |
| . 1 . | . 3 . | . 4 . |
| . . 5 | 6 . . | . . . |
+-------+-------+-------+
| . . 2 | 7 . . | 8 . . |
| . 8 . | . . 4 | . 3 . |
| 1 . . | . 9 . | . . . |
+-------+-------+-------+
| . . . | 3 . . | 5 . . |
| . 9 . | . 8 . | . 1 . |
| 6 . . | . . . | . . 7 |
+-------+-------+-------+ ER=11.3/11.3/10.5
Again very easy to solve. After clearing the diagonal there is a UR 1.1 (12 or 16), then only 2 pairs to the end.
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