The New Sudoku Players' Forum

Website · by **denis_berthier** » Sun Jun 13, 2021 1:39 pm

creint wrote:
denis_berthier wrote:I also wonder how you arrive at the mentioned resolution states. To be more specific, the resolution state after Singles and whips[1] is
Hidden Text: Show
Code: Select all
14589ACD 125789AB 3 1456789ACD 159AC 5689D 15679ABC 1289ACD 14589ACD 12589AD 1256789BC 1689BC 5789CD 6 12579BC 2589ABCD 1789ACD 12359ACD 13589B 234789C 1579ACD 13589ABCD 12589ACD 123589 34789ABC 1589ACD 159BC 13589C 5689ABCD 15689ACD 129ACD 13489D 1245689ABC 2345689CD 13569ABCD 7 3589CD 13489AD 123589AB 123589ABC 4 589CD 156789ACD 12379ACD 16789BD 135689ABC 1235689AC 35689BC 359ABCD 135689CD 123789AD 12579 13589AD 12589AC 5689ACD B 124579ACD 135689D 126789AC 12345689C 34589AC 45689CD 13579D 169C 1235789ACD 12589CD 125789 2789C 3 1459CD 156789BD 125689BC 789CD 15689CD 124589B A 1679BCD 15689BCD 123589BCD 6 25789D 1589C 159CD A 35789BC 135789CD 189BD 1259BC 589BC 3789BCD 4 139ACD 1589ABC 589ACD 15789AD 1359C 2 1345789C 1345789AD 1569BC 14689ABCD 1356789BCD 1389AC 356789BCD 2589BCD 1359ABC 4 1589C 6 135789 1589AB 1579C 13589ABCD 13589ABCD 1289CD 123789ABCD 12389AC 7 389 69 89A B C D 1689A 2 13689A 4 5 1689A 3589ABC D 1 2 359A 459 34589A 35689AC 7 3589C 389C 689ABC 689ABC 1249C 3579AC 569AC 4579D 8 13569D 13579AC B 13569AC 19ACD 1579CD 12369ACD 1235679ACD 3589ABCD 1789AC 25789ABD 14569C 23579CD 135679BD 1235689AC 135789ACD 13489AD 1245689ABC 15689BCD 13789CD 135689BC
It seems you are missing some locked singles, whip[1].
For example 3r10 -> -3r5c2, -3r25c10

I don't see what you mean. These 3 candidates are already eliminated.

by **creint** » Sun Jun 13, 2021 2:22 pm

denis_berthier wrote:I don't see what you mean. These 3 candidates are already eliminated.

I forgot to add the given pencilmarks, ignore that part.

Your solver incorrectly removes 2 from r7c11.

Website · by **denis_berthier** » Sun Jun 13, 2021 3:27 pm

creint wrote:Your solver incorrectly removes 2 from r7c11.

no: r7c11 still contains 1259BC

by **creint** » Sun Jun 13, 2021 5:17 pm

denis_berthier wrote:
creint wrote:Your solver incorrectly removes 2 from r7c11.

no: r7c11 still contains 1259BC

That is cell r7c10

123589BCD 6 25789D 1589C 159CD A 35789BC 135789CD 189BD 1259BC 589BC 3789BCD 4

Website · by **denis_berthier** » Sun Jun 13, 2021 5:25 pm

creint wrote:
denis_berthier wrote:
creint wrote:Your solver incorrectly removes 2 from r7c11.

no: r7c11 still contains 1259BC
That is cell r7c10

123589BCD 6 25789D 1589C 159CD A 35789BC 135789CD 189BD 1259BC 589BC 3789BCD 4

OK, I had copied the resolution state for the wrong puzzle. It's now corrected. Line 7 is:
123589BCD 6 25789D 1589C 159CD A 35789BC 135789CD 189BD 1259BC 123589BC 3789BCD 4

Website · by **denis_berthier** » Mon Jun 14, 2021 6:27 am

.
It seems to me that dealing with 13x13 puzzles may be useful to stress test our solvers, but it is globally putting the cart before the horse. 7x7 and 11x11 puzzles, even if the solution grids are in small numbers (modulo isomorphisms), can still teach us a lot in terms of resolution paths (if we consciously ignore the iso results). In terms of complexity, 7x7 are the closest to Sudoku(9).

I've searched the web (mainly GitHub and Sourceforge) for LatinSquares generators. Surprisingly, while there are hundreds of Sudoku generators, there are only few LS generators and those I've found run only on Windows. I've found only one that can generate Pandiags (on Windows). As I'm focused on pattern-based resolution, I'm not interested in developing a generator myself.

Could anyone generate a substantial collection (> 100 or 1000) of (non isomorphic) of 7x7 puzzles in the 5 - 9 SER range? From a solver's POV, whether they are minimal or not doesn't matter.

by **1to9only** » Mon Jun 14, 2021 8:14 am

From a run (took about 20 mins!) of 10000 randomly generated PD7 grids, these are the only ones ED>5.0.
I dont think the SE generator I'm using will produce ratings higher than ED=7.7!!

104 PD7 grids: Show

Code: Select all: ...4........................2.............7.3.15. ED=7.7/7.7/2.9 6..3....................1...7..2..........4...... ED=7.7/7.7/2.9 ............7........3.......5............24.6... ED=7.7/7.7/2.9 ...............2..7...............41...6...3..... ED=7.7/7.7/2.9 .................56.3...471...................... ED=7.7/7.7/2.9 ....5..............167...4...........3........... ED=7.7/7.7/2.9 ................4.7...5.126...................... ED=7.7/7.7/2.9 ................3754....2...........6............ ED=7.7/7.7/2.9 .....................34.51..7.............6...... ED=7.7/7.7/2.9 3......4.1...72...........................5...... ED=7.7/7.7/2.9 2....6........3.7.....1.......5.................. ED=7.7/7.7/2.9 .....2..........34......................7......16 ED=7.5/7.5/2.9 ....5...........................6.1...7.32....... ED=7.5/7.5/2.9 ..4.1.......2.....6..............3...7........... ED=7.5/7.5/2.9 .........2.7.........6.5.....3...4............... ED=7.5/7.5/2.9 ............1..2...5........34.6................. ED=7.5/7.5/2.9 ..6.....257....4...........................3..... ED=7.5/7.5/2.9 .5...432....7.............................1...... ED=7.5/7.5/2.9 ................1.73..2.............56........... ED=7.5/7.5/2.9 ..35.7...........................14.............6 ED=7.5/7.5/2.9 ........6....3.....7.......2............5..1..... ED=7.5/7.5/2.9 .......6.......4..2.5...........3.7.............. ED=7.5/7.5/2.9 .......4...1...........63....5..7................ ED=7.5/7.5/2.9 ...5...2..76................4.........1.......... ED=7.5/7.5/2.9 ...4.........5..17.23............................ ED=7.1/7.1/2.9 ................................72.....15....3.6. ED=7.1/7.1/2.9 ...........1.................2..6.5........7....4 ED=7.1/7.1/2.9 ..4.................2.....53......6........1..... ED=7.1/7.1/2.9 .................3...........6...2.5......41..... ED=7.1/7.1/2.9 ........5....................4.67......3...1..... ED=7.1/7.1/2.9 ....374......6.......................5...1....... ED=7.1/7.1/2.9 ...6.....471......................3......5....... ED=7.1/7.1/2.9 ...............7...5......6.............2.....14. ED=7.1/7.1/2.9 ............4........1.......36..2...........7... ED=7.1/7.1/2.9 ...............243....16......5.................. ED=7.1/7.1/2.9 ...12......7...........................4....5.6.. ED=7.1/7.1/2.9 ......2.............5......4........1..6....7.... ED=6.5/6.5/2.9 ..................6................13.5...4...7.. ED=6.5/6.5/2.9 .....6..........2..3....................5.....74. ED=6.5/6.5/2.9 6.....47....3......1...........................5. ED=6.5/6.5/2.9 ..1...5.....7.................6..2.....4......... ED=6.5/6.5/2.9 2.....1.........7......4.......6.5............... ED=6.5/6.5/2.9 .2.....3..............46..5..7................... ED=6.5/6.5/2.9 ..........2.....34.6........5......7............. ED=6.5/6.5/2.9 ....6.............1...2..37......5............... ED=6.5/6.5/2.9 2...5.1...........3...........4.7................ ED=6.5/6.5/2.9 .....5....6.2......4.1....3...................... ED=6.5/6.5/2.9 ..6....5..........2....................7......14. ED=6.5/6.5/2.9 ................2......34.....7.............6..5. ED=6.5/6.5/2.9 .....2.......7.....3..64.......................5. ED=6.5/6.5/2.9 ..3......7....5..................61....4......... ED=6.5/6.5/2.9 .3....................2...1.....54....6.......... ED=6.5/6.5/2.9 ......................5.....1.42....3..........6. ED=6.5/6.5/2.9 ....5...2..64...7...............3................ ED=6.5/6.5/2.9 ..................74..6..512..................... ED=6.5/6.5/2.9 1.76......4.............3..........5............. ED=6.5/6.5/2.9 3.....2.....5...........4.6...7.................. ED=6.5/6.5/2.9 ..................5......1...........2.43....6... ED=6.5/6.5/2.9 ............3.2..7...561......................... ED=6.5/6.5/2.9 ..............63.....72.5.............4.......... ED=6.5/6.5/2.9 ....5.14....3........6......7.................... ED=6.5/6.5/2.9 ..................6.........1..47........5..3.... ED=6.5/6.5/2.9 ...2......................13..........54.....7... ED=6.5/6.5/2.9 .................3.42...7....1...............5... ED=6.5/6.5/2.9 .........4.............6.................32....51 ED=6.5/6.5/2.9 .............2........1.35...............67...... ED=6.5/6.5/2.9 .7.....24......5....................3...1........ ED=6.5/6.5/2.9 ................4......23.65...1................. ED=6.5/6.5/2.9 ....2.......................5......74....6..3.... ED=6.5/6.5/2.9 .2....13.4...............7......5................ ED=6.5/6.5/2.9 ....1...............24.....3......5.....6........ ED=6.5/6.5/2.9 ...6.5.................4..12....3................ ED=6.5/6.5/2.9 ...2.....6.................1........3.4.7........ ED=6.5/6.5/2.9 ......................2......75...6...........1.4 ED=6.5/6.5/2.9 .......7...4...........56......2...............3. ED=6.5/6.5/2.9 ................2..........63............15.....4 ED=6.5/6.5/2.9 ...5...............3...........46.1.....2........ ED=6.5/6.5/2.9 ................................7.1..........4625 ED=6.5/6.5/2.9 ..5...........1...37......................42..... ED=6.5/6.5/2.9 ......5......7......1.........3.....4.2.......... ED=6.5/6.5/2.9 ..............26.3.....1...............5......4.. ED=6.5/6.5/2.9 ......12....................3.75......4.......... ED=6.5/6.5/2.9 ..........4.5...1..7.........................6.2. ED=6.5/6.5/2.9 .........67.5....................3..1....2....... ED=6.5/6.5/2.9 ..17.......................................564..3 ED=6.5/6.5/2.9 ...6....3..4..................5.....2.......1.... ED=6.5/6.5/2.9 4.6..1.......7..................2..............5. ED=6.5/6.5/2.9 ..............13.6............72......4.......... ED=6.5/6.5/2.9 ..3.5................6......1................7.2. ED=6.5/6.5/2.9 ...........7....36.5...........4......2.......... ED=6.5/6.5/2.9 .....6........7..32............5......4.......... ED=6.5/6.5/2.9 ..1..................5.............73..2.......4. ED=6.5/6.5/2.9 ................4.......571.............6.3...... ED=6.5/6.5/2.9 .....5...7..........2................436......... ED=6.5/6.5/2.9 .........3..1...............7.2...6.........5.... ED=6.5/6.5/2.9 .........3.......56...........1...........42..... ED=6.5/6.5/2.9 .1..4.......3..............7.................56.. ED=6.5/6.5/2.9 ...3....................1...2.4.7...5............ ED=6.5/6.5/2.9 ...............3.46..........7.............5....2 ED=6.5/6.5/2.9 .............71...2......4.............3..6...... ED=6.2/2.3/2.3 ........45...3...6............................1.7 ED=6.2/2.3/2.3 4............................6.2.............7.53 ED=6.2/2.3/2.3 ..5...1.......3...2.......................7.4.... ED=6.2/2.3/2.3 .........................4.......2.5...3..1....7. ED=6.2/2.3/2.3

Website · by **denis_berthier** » Mon Jun 14, 2021 8:35 am

Hi 1to9only

Thanks; I'll try this list.

One way to produce harder (and less biased towards easy) puzzles is the controlled-bias variant of the top-down algorithm. Unfortunately, it's terribly slow.

Website · by **denis_berthier** » Mon Jun 14, 2021 10:53 am

.
All the puzzles in the above list are in Z2, except two in Z3:

Code: Select all: 6..3....................1...7..2..........4...... ED=7.7/7.7/2.9 #2 .................56.3...471...................... ED=7.7/7.7/2.9 #5

What's more interesting is when I activate only (Naked, Hidden and Super-Hidden (i.e. Fish)) Subsets. Apart from the above two, all the puzzles can be solved with Subsets[3].

Here is my preferred example (#4 in the list)

Code: Select all: . . . . . . . . . . . . . . . 2 . . 7 . . . . . . . . . . . . . . . 4 1 . . . 6 . . . 3 . . . . . ...............2..7...............41...6...3..... ED=7.7/7.7/2.9

Code: Select all: Resolution state after Singles: 24567 14567 25 13567 12345 234567 135 3457 1457 13567 256 2345 123456 12567 345 2 13456 3456 7 1456 1356 3567 14567 3457 123456 1245 256 12357 23567 567 12356 157 1235 12357 4 1 45 2457 23457 6 357 2357 2456 3 124567 12457 15 1457 2567 181 candidates, 1563 csp-links and 1563 links. Density = 9.59%

Notice the very high density, typical of Pandiags.

The first steps are a mix of all the types of whips[1] (i.e. rn, cn, dn, an):

Code: Select all: whip[1]: a7n4{r7c6 .} ==> r7c3 ≠ 4 whip[1]: r7n4{c6 .} ==> r2c6 ≠ 4 whip[1]: d4n1{r7c5 .} ==> r7c3 ≠ 1 whip[1]: r7n1{c6 .} ==> r1c5 ≠ 1 whip[1]: d3n3{r5c6 .} ==> r5c1 ≠ 3 whip[1]: d5n3{r6c7 .} ==> r3c7 ≠ 3 whip[1]: a3n6{r4c6 .} ==> r4c4 ≠ 6, r2c6 ≠ 6, r4c2 ≠ 6 whip[1]: r4n6{c6 .} ==> r3c7 ≠ 6 whip[1]: c7n6{r7 .} ==> r1c1 ≠ 6, r1c6 ≠ 6 whip[1]: c6n6{r4 .} ==> r7c3 ≠ 6 whip[1]: r7n6{c7 .} ==> r3c4 ≠ 6 whip[1]: r3n6{c6 .} ==> r5c1 ≠ 6 whip[1]: c1n6{r7 .} ==> r2c3 ≠ 6 whip[1]: d6n2{r2c5 .} ==> r2c7 ≠ 2 whip[1]: d1n2{r6c3 .} ==> r1c5 ≠ 2

Code: Select all: Resolution state after Singles and whips[1]: 2457 14567 25 13567 345 23457 135 3457 1457 1357 256 2345 1235 1567 345 2 13456 345 7 1456 15 3567 1457 3457 12345 1245 256 12357 257 567 12356 157 1235 12357 4 1 45 2457 23457 6 357 2357 2456 3 257 12457 15 1457 2567

Then comes a nice mix of Swordfishes in columns, diagonals or anti-diagonals with transversal cover sets.
Notice that both base sets and cover sets are homogeneous in all the cases (i.e. of a single type: rn, cn, dn or an). (Fully mixed Fishes are not coded in LatinRules - and will probably never be, as there are too many possible cases. They are the exotic Fishes of Pandiagonal Latin Squares.)
Transversal cover sets merely means cover sets are of a different kind than base sets. Given any type x of homogeneous base sets (x = rn, cn, dn or an), a homogeneous transversal cover set can be of any homogeneous type (rn, cn, dn or an) except x.

Code: Select all: swordfish-in-columns-w-transv-anti-diags: n2{c1 c5 c7}{a2 a4 a1} ==> a4c3 ≠ 2, a2c6 ≠ 2, a1c4 ≠ 2 swordfish-in-diags-w-transv-columns: n4{d2 d3 d7}{c4 c1 c2} ==> d5c2 ≠ 4, d6c4 ≠ 4, d1c1 ≠ 4 swordfish-in-anti-diags-w-transv-rows: n3{a4 a5 a7}{r4 r2 r1} ==> r4a1 ≠ 3, r2a2 ≠ 3, r1a6 ≠ 3 swordfish-in-anti-diags-w-transv-columns: n1{a4 a5 a7}{c7 c6 c4} ==> a2c6 ≠ 1, a1c4 ≠ 1, a6c7 ≠ 1

Notice that for Fish patterns, I use the coordinate system that is most natural to each of them.

The end is less remarkable: Show

Considering the potentially very large number of Fishes to try, I feared long computation times, but the whole list (104 puzzles) was solved in 30s.

What I'm still missing is Jellyfishes (which means harder puzzles).
Anyway, I hope these puzzles by 1to9only show that the 7x7 case is interesting.

by **coloin** » Mon Jun 14, 2021 2:36 pm

Thanks Dennis for that update on the branching factrs and the relative incidence of puzzles which require more than 1 level of T&E

It seems we are in a very hard space with those 24 C puzzles and there will be numerous ways to remove 12 clues !!!!

Mathimagics wrote: OMG, not another $*#@*! 16-clues existence problem, please!

I think that you will find one - there are quite few puzzles per grid !!!!

but maybe not as many as i initially thought .... what proportion of the 13 " 12^12 ways to have 12 clues in an individual grid are valid ?

by **Mathimagics** » Mon Jun 14, 2021 9:01 pm

coloin wrote: what proportion of the 13 " 12^12 ways to have 12 clues in an individual grid are valid ?

I have no idea what that formula means! Thirteen square feet? :lol:

But I think I understand the gist of the question! Let's look at the general case of an (N-1) clue puzzle generator that selects a cell at random, puts a 1 there, then selects a vacant cell at random, puts a 2 there (if allowed), and so on ...

So we get a puzzle that has N-1 clues, conforms to the PD rules, and has N-1 different clue values, all of which are required for a valid puzzle.

Now, what are the chances that we get a puzzle that has 1 solution? 0 solutions? 2+ solutions?

Sampling many thousands of puzzles generated with this method gave the following results:

Code: Select all: N ns=0 ns=1 ns>1 ------------------------- 7 78% 20% 2% 11 99% 1% 13 4% 10% 86%

Hmm, kind of weird, why so many invalid (no solutions) for N=7 and N=11? The reason is that the only solution grids are cyclic. If we introduce a "pro-cyclic" feature to our generator, which forces clue patterns that are compatible with a cyclic solution, for those N, then the "no solution" puzzles disappear altogether:

Code: Select all: N ns=0 ns=1 ns>1 ------------------------- 7 88% 12% 11 99% 1% 13 4% 10% 86%

N=13 has mostly non-cyclic solution grids, and that's why it's possible to get no solutions sometimes.

So we have about a 10% chance, with these 12-clue puzzles, of a unique solution. That's why they are so easy to find, but hard to count ...

by **coloin** » Mon Jun 14, 2021 10:27 pm

Thats quick work ... well done and 10% ... that is quite an impressive number of puzzles !!!
if you start from a valid grid , i meant that you pick 12 clues from 13, and one of each.
so thats 13 * 13^12 ways to have 12 clues [hope thats right now] [13^13]

by **Mathimagics** » Tue Jun 15, 2021 4:35 am

My apologies! That actually makes perfect sense ....

So, to predict the likely number of ED 12-clue puzzles, I re-sampled using the fixed-grid approach, and found that for 8 of the ED grids, roughly 2% of puzzles were valid, and only 0.2% for the 2 cyclic grids.

So the estimated number of 12-clue puzzles for all the ED grids is roughly 16.4% of 13^13, which clocks in at a shade under 50 trillion !! ( 49,671,517,481,130 )

Well, I did say that there were quite a lot ...

by **Mathimagics** » Tue Jun 15, 2021 6:59 am

For N = 5/7/11, there is only one ED grid in each case. The number of possible (N-1) clue puzzles is N^N, and so we can easily enumerate these for N=5 and N=7, obtaining exact valid puzzle counts for those cases:

N=5: 3,125 possible 4-clue puzzles; 2,525 valid (80.8%)

N=7: 823,543 possible 6-clue puzzles; 720,300 valid (87.46%)

N=11: 285,311,670,611 possible 10-clue puzzles; estimated valid = 98.92%

Thanks again to coloin for pointing out the simple (N-1) clue, all distinct values, puzzle count formula!

[ EDIT ] corrected some minor errors in valid counts

by **1to9only** » Tue Jun 15, 2021 12:26 pm

Jellyfish found in 2 the PD13s I posted a few days ago - SudokuExplainer solving order.

Code: Select all: ..3..........6..........................4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D12....7........8..B....3............. 25 ED=9.7/1.5/1.5 ..3..........6..........................4..............B............3......A...6...A......4.....2.........4.6........789ABCD.2345..D12....7........8..B....3............. 5.2, Jellyfish: Cells R1C6,R1C7,R5C3,R5C7,R5C12,R6C6,R6C7,R6C12,R12C3,R12C12: 6 in 4 columns and 4 rows: r6c9<>6

Code: Select all: ..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...BC..2..5..D12....7........8..B....3............. 25 ED=9.1/1.5/1.5 ..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........789.BC..23.5..D12....7....4.6.8..B...23............. 5.2, Jellyfish: Cells R1C4,R1C7,R1C11,R1C13,R5C7,R5C11,R5C13,R8C4,R8C7,R8C11,R8C13,R12C4,R12C7,R12C11: 7 in 4 columns and 4 rows: r1c2<>7

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