## hardest 6x6?

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

### Re: hardest 6x6?

My last venture (possibly ) into one-trick-pony land:
Code: Select all
`.3..6...5..4.4....2....13......1.2..   ED=7.1/7.1/7.12....6....2....3...3....1...5..4.1..   ED=3.0/3.0/3.042..1......2......1....4..35...5....   ED=6.8/6.6/2.6`

tarek

tarek

Posts: 3762
Joined: 05 January 2006

### Re: hardest 6x6?

In the land of the Mostly Weak, the one-trick-pony is king!

Mathimagics
2017 Supporter

Posts: 1926
Joined: 27 May 2015
Location: Canberra

### Re: hardest 6x6?

Random generation can bring out some of the hardest but will not be as concise as Mathimagics enumeration.

The 1st TW is most likely in the AllTW list but I also found a few PW puzzles so I'm pretty sure Mathimagics bigger file will have some nice puzzle there.

Code: Select all
`....1..5.2...14...6.....24...3..3... ED=8.3/2.0/2.031...4..4....6..1..2.5...3...1...... ED=8.3/1.2/1.22....5..4..11......2..3.6..2........ ED=7.6/1.2/1.2`

tarek

tarek

Posts: 3762
Joined: 05 January 2006

### Re: hardest 6x6?

Hi tarek,

I haven't actually got a separate extraction of PW(1-5) puzzles that would include those 3 examples (I just counted them), but can produce one if you wish to search it?

Mathimagics
2017 Supporter

Posts: 1926
Joined: 27 May 2015
Location: Canberra

### Re: hardest 6x6?

Thanks Mathimagics,

No need. I had enough time on this one for the moment. I managed to tweak a few things in my solver/generator which was satisfying. I suspect the PW batch to be fairly large & therefore would take a considerable time to rate!!!

tarek

tarek

Posts: 3762
Joined: 05 January 2006

### Re: hardest 6x6?

hey guys, just caught up on this topic. was in the middle of a complicated house move and still don't have internet set up at my new place, i can browse these forums fine with mobile data but every time i try to post it logs me out whenever i click submit post and then everything i've written is lost. it's so frustrating

1to9only you are amazing. that was an incredibly quick turnaround time to modify SE to work on 6x6s and those are some great results. well done!

the next step would be to take a look at the pw puzzles with only a few singles at the start to see if there are any higher rated ones 8.4+ - i'm optimistic as there are so many more of them

I'm also surprised that the ratings list contains no 3.6 naked triplets or 3.8 swordfish in it, those would make some nice puzzles for manual solvers

mathimagics: well done on enumerating all possible 6x6 puzzles in your previous posts, that's impressive work too

Mathimagics wrote:.
Sudoku 6x6 is "Sudoku for kids"?

Apparently not - unless the kid is pretty smart …

if you allow pencilmark grids in sudokus like tarek is interested in, you can get even smaller ones like this

Code: Select all
`12  .   | .   ..   124 | .   34--------+---------.   .   | 14  .23  .   | .   .SE 8.3/8.3/7.1.   134 | .   3412  .   | .   .--------+---------234 134 | 234 ..   .   | .   23SE 7.3/7.3/6.6`
although at this puzzle size it is questionable if se is a good measure of rating stuff like this since a lot of ratings aren't possible e.g. 7.4-8.1 are almost certainly not possible for any move

edit: singles backdoor size 1 doesn't necessarily mean 1 trick pony because the backdoor could be a candidate that isn't part of a strong link so it needs multiple eliminations to make that placement. but with puzzle sizes this small, it's certainly going to be much easier to find nice short solution paths for them without long chains
999_Springs

Posts: 591
Joined: 27 January 2007
Location: In the toilet, flushing down springs, one by one.

### Re: hardest 6x6?

.
Well, if anybody wants to get down and dirty, I can always extract those PW 1-3 or 4 for you to play with …

Hmm, Sukaku 6x6 is interesting, but how do you rate them?

A further thought, we might find harder 6x6 puzzles for SudokuP, but then we'd have a new rating problem

Mathimagics
2017 Supporter

Posts: 1926
Joined: 27 May 2015
Location: Canberra

### Re: hardest 6x6?

sure, go for it although i'm sure that if you posted those then 1to9only would get there first with a full rating scan, leaving me with nothing to do, just like tarek! i'd be interested to know what comes up

pencilmark 6x6s would be way harder i'm sure, since if even 4x4s can get to se 8.3 i wonder what 6x6s will bring, but they will be tough to rate (maybe easier than the pencilmark 9x9s ? - i remember 1to9only was running into some kind of overflow issue with the pencilmark 9x9s which won't happen here with not many cells remaining) - and even harder to generate as there is no way you can do an exhaustive search i'll leave that one to someone else then

and finally, if you get the first 8.2/6.6/6.6 from the file:
Code: Select all
`..3..6...23...1.4..4.3...12...5..1..`
which remains SE 8.2 after the first placement r1c3=3 - and satisfies gurth's symmetry theorem which is nice

turns out you can embed (the complement of) this puzzle into a 9x9 grid by shoving 7,8,9's into random places just like i was doing earlier in the first post - but this time you actually get a valid puzzle:
Code: Select all
`..3..7689...89.237789236154..8.4.791.7.9..348941378526.97...8128..1..975512789463 SE 8.2, 57 clues`
and with 57 clues, this smashes the previous record for SE 8.2 in the "maximum clues per se rating" thread! i'm sure there are others out there like this - it seems a very strange way to go about generating hard 9x9s with lots of clues, but if it works it works

edit: here's another one using the same method
Code: Select all
`1.3.75689.869..715957186234..97.8543......897874359126.98..7461......978761894352 SE 7.8, 58 clues`
999_Springs

Posts: 591
Joined: 27 January 2007
Location: In the toilet, flushing down springs, one by one.

### Re: hardest 6x6?

999_Springs wrote: ... and finally, if you get the first 8.2/6.6/6.6 from the file:
Code: Select all
`..3..6...23...1.4..4.3...12...5..1..`
which remains SE 8.2 after the first placement r1c3=3 - and satisfies gurth's symmetry theorem which is nice

... which makes it easy:
if r35c4 is not 56, but 4, then also r2c3=4, and r1c5,r2c6=15 => -5r1c4,-6r6c3
eleven

Posts: 3067
Joined: 10 February 2008

### Sudoku6 PW Data

I have attached the grids for PW1 and PW2 types.

The files for PW3 and PW4 when zipped exceed 256Kb so they can't be posted here. If anybody would like these PM me giving an email address that I can send them to. I think any rating gems will most likely be found in these first 2 files.

[EDIT] These files (and the TW file) are inaccurate. See the update below.
Attachments
Sudoku6-PW2.zip
Sudoku6-PW1.zip
Last edited by Mathimagics on Sun Apr 21, 2019 7:41 pm, edited 1 time in total.

Mathimagics
2017 Supporter

Posts: 1926
Joined: 27 May 2015
Location: Canberra

### Re: hardest 6x6?

The stats for the last 2 posted files:

- the highest rated grids:
Code: Select all
`....5.4....3...5.1....4.3.2.1..6..3. ED=8.4/1.2/1.2....5.4....3...5.1...34.3.2.1..6.... ED=8.4/1.2/1.2.2......61..2.4....1.....4.6.5.6..3. ED=8.4/1.2/1.2.2......61..2.4...61.....4.6.5....3. ED=8.4/1.2/1.2.2.4....6.....4...61......2..5..123. ED=8.4/1.2/1.2.2.4....6...2.4...61.........5..123. ED=8.4/1.2/1.21...5......3.....1...34...2..556...4 ED=8.4/1.2/1.21...5......3...5.1...34...2...56...4 ED=8.4/1.2/1.2`

- the breakdown by difficulty ratings:
Code: Select all
`        Sudoku6-PW1.txt                                 Sudoku6-PW2.txt      8 ED=8.4          ED=8.4/1.2/1.2     50 ED=8.3          ED=8.3/2.0/2.0               70 ED=8.3          ED=8.3/1.5/1.5      1 ED=8.2          ED=8.2/1.2/1.2                                                      5 ED=7.8          ED=7.8/1.2/1.2     12 ED=7.7          ED=7.7/1.2/1.2                9 ED=7.7          ED=7.7/1.2/1.2     26 ED=7.6          ED=7.6/1.5/1.5               29 ED=7.6          ED=7.6/1.2/1.2      1 ED=7.5          ED=7.5/1.2/1.2     10 ED=7.3          ED=7.3/1.5/1.5               18 ED=7.3          ED=7.3/2.0/2.0   2749 ED=7.2          ED=7.2/2.0/2.0             2820 ED=7.2          ED=7.2/2.0/2.0   6824 ED=7.1          ED=7.1/2.0/2.0             8660 ED=7.1          ED=7.1/2.0/2.0     39 ED=7.0          ED=7.0/2.0/2.0               98 ED=7.0          ED=7.0/2.0/2.0     85 ED=6.9          ED=6.9/2.0/2.0              105 ED=6.9          ED=6.9/2.0/2.0    357 ED=6.8          ED=6.8/2.0/2.0              529 ED=6.8          ED=6.8/2.0/2.0    420 ED=6.7          ED=6.7/2.0/2.0              570 ED=6.7          ED=6.7/2.0/2.0   8509 ED=6.6          ED=6.6/2.0/2.0            14973 ED=6.6          ED=6.6/2.0/2.0     52 ED=6.5          ED=6.5/1.5/1.5              120 ED=6.5          ED=6.5/1.5/1.5     62 ED=6.2          ED=6.2/2.0/2.0               60 ED=6.2          ED=6.2/2.0/2.0                                                      1 ED=5.8          ED=5.8/1.5/1.5     81 ED=5.7          ED=5.7/2.0/2.0              161 ED=5.7          ED=5.7/2.0/2.0    535 ED=5.6          ED=5.6/2.0/2.0              923 ED=5.6          ED=5.6/2.0/2.0     48 ED=4.6          ED=4.6/2.0/2.0              133 ED=4.6          ED=4.6/2.0/2.0    622 ED=4.5          ED=4.5/2.0/2.0              931 ED=4.5          ED=4.5/2.0/2.0    269 ED=4.4          ED=4.4/2.0/2.0              651 ED=4.4          ED=4.4/2.0/2.0   6718 ED=4.2          ED=4.2/2.0/2.0            10120 ED=4.2          ED=4.2/2.0/2.0      4 ED=3.8          ED=3.8/1.2/1.2                3 ED=3.8          ED=3.8/1.2/1.2      8 ED=3.6          ED=3.6/1.2/1.2     40 ED=3.4          ED=3.4/2.0/2.0               30 ED=3.4          ED=3.4/2.0/2.0    151 ED=3.2          ED=3.2/2.0/2.0              208 ED=3.2          ED=3.2/2.0/2.0    257 ED=3.0          ED=3.0/2.0/2.0              218 ED=3.0          ED=3.0/2.0/2.0    122 ED=2.8          ED=2.8/2.0/2.0               79 ED=2.8          ED=2.8/2.0/2.0   1023 ED=2.6          ED=2.6/2.0/2.0             1622 ED=2.6          ED=2.6/2.0/2.0    130 ED=2.5          ED=2.5/2.0/2.0              205 ED=2.5          ED=2.5/2.0/2.0   1521 ED=2.0          ED=2.0/2.0/2.0             2170 ED=2.0          ED=2.0/2.0/2.0    193 ED=1.7          ED=1.7/1.7/1.7              405 ED=1.7          ED=1.7/1.5/1.5  -----                 --------------            -----                 --------------  30297                 highest rating            45926                 highest rating`

Some 3.6 naked triplets and 3.8 swordfishes in there, even 1x 5.8 BUG type 3 and 1x 7.5 ATE (Aligned Triplet Exclusion)!

1to9only

Posts: 4174
Joined: 04 April 2018

### Re: hardest 6x6?

Is an explusion an exclusive explosion (like a North Korean nuclear test) or an explosive exclusion (like Lance Armstrong getting kicked out of cycling) ?

[EDIT] Hey, you ruined my joke by fixing that typo

Mathimagics
2017 Supporter

Posts: 1926
Joined: 27 May 2015
Location: Canberra

### Re: hardest 6x6?

very nice work, mathimagics and 1to9only! I'll look into this and maybe the higher pw+n puzzles and stuff further when i get internet properly set up at my new place, but that 8.4 looks delicious (are they essentially different or the same puzzle after the singles?) correctly guessed there was a better one out there

Also yeah i spotted that typo too

edit:
Mathimagics wrote:.
Perhaps the 3 cases of NC=8 and FM = 1,3,4 are worth a look?

Code: Select all
`..34...5.....3......1..2.....56..........6.5.1..2..3.......1......6...4.1.........3........34.2......5.6..4.`
4.2
4.2
2.6
999_Springs

Posts: 591
Joined: 27 January 2007
Location: In the toilet, flushing down springs, one by one.

### Corrections: TW/PW Table

999_Springs reported a discrepancy in the table I posted above, and I found that there was a bug with the singles-counting process.

The corrected figures are:

Code: Select all
`MP table (total minimal puzzles = 72425047) +----+-----------+-----------+--------+---------+---------+---------+---------+---------+---------+---------+ | NC |   Minimal |    Strong | TW(0fm)| PW(1fm) | PW(2fm) | PW(3fm) | PW(4fm) | PW(5fm) | PW(6fm) | PW(7+fm)| +----+-----------+-----------+--------+---------+---------+---------+---------+---------+---------+---------+ |  8 |      2619 |      2447 |      0 |       6 |       0 |       2 |       6 |       2 |       4 |     152 | |  9 |    607072 |    576686 |     97 |     432 |     958 |    1263 |    1546 |    1621 |    1408 |   23061 | | 10 |  12884906 |  12256735 |   4893 |   14065 |   21115 |   28385 |   34713 |   39759 |   45134 |  440107 | | 11 |  35853981 |  34019016 |  24842 |   57964 |   78375 |  108043 |  133235 |  148572 |  172422 | 1111512 | | 12 |  20408957 |  19330960 |  18638 |   39077 |   63172 |   94941 |  104120 |  121106 |  121871 |  515072 | | 13 |   2601185 |   2472427 |   2377 |    5484 |   10649 |   15953 |   18530 |   17600 |   16770 |   41395 | | 14 |     65547 |     62403 |    159 |     265 |     393 |     304 |     291 |     304 |     928 |     500 | | 15 |       780 |       780 |      0 |       0 |       0 |       0 |       0 |       0 |       0 |       0 | +----+-----------+-----------+--------+---------+---------+---------+---------+---------+---------+---------+ |    |  72425047 |  68721454 |  51006 |  117293 |  174662 |  248891 |  292441 |  328964 |  358537 | 2131799 | +----+-----------+-----------+--------+---------+---------+---------+---------+---------+---------+---------+`

So there are 3 times as many TW puzzles as reported earlier, and so the file "AllTW.zip" is inaccurate, as are the "PW1", "PW2" etc files.

Apologies for any inconvenience!

I now have a file with all TW+PW grids (72,425,047 - 68,721,454 = 3,703,593 grids), tagged with the NC and FM counts, which I can mail to anybody on request (PM me with an email address). The zipped file size is 19Mb.

Mathimagics
2017 Supporter

Posts: 1926
Joined: 27 May 2015
Location: Canberra

### Re: hardest 6x6?

by filling in all the singles, i can independently confirm that the numbers in mathimagics's table above are right.

here are some more stats:
number of puzzles by clue count before doing singles:
Code: Select all
`clue count, puzzles8  1729  3038610 62817111 183496512 107799713 12875814 314415 0`

number of initial singles:
Code: Select all
`initial singles, puzzles0  510061  1172932  1746623  2488914  2924415  3289646  3585377  3757088  3791709  38046110 33488211 25692412 20615213 13404114 5084315 1286816 73817 12`

after filling in all the singles, and discarding duplicates, there were 542257 puzzles in total, out of the original 3703593. here is the count of number of puzzles by clue count, and number of unique puzzles:
Code: Select all
`clue count, puzzles, unique puzzles8  0      09  103    10310 5325   532311 39867  3877812 98986  8489513 149762 10122914 213196 9835615 280265 8212916 316389 5857617 365304 3590318 377135 1962219 381553 971520 429132 461321 379550 207522 255367 62523 261761 24224 95972  5925 53926  14`

i'm in the middle of converting the big pile of 542257 puzzles to 9x9's where possible, and rating them. i'll post when/if i find something good. so far i'm going through the batch in descending order of clue count. found a nice 16-clue SE 8.4...
999_Springs

Posts: 591
Joined: 27 January 2007
Location: In the toilet, flushing down springs, one by one.

PreviousNext