Pandiagonal Latin Squares

Programs which generate, solve, and analyze Sudoku puzzles

Re: Pandiagonal Latin Squares

Postby 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.
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Re: Pandiagonal Latin Squares

Postby 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.
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Re: Pandiagonal Latin Squares

Postby denis_berthier » Sun Jun 13, 2021 3:27 pm

creint wrote:Your solver incorrectly removes 2 from r7c11.

no: r7c11 still contains 1259BC
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Re: Pandiagonal Latin Squares

Postby 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
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Re: Pandiagonal Latin Squares

Postby 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
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Re: Pandiagonal Latin Squares

Postby 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.
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Re: Pandiagonal Latin Squares

Postby 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
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Re: Pandiagonal Latin Squares

Postby 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.
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Re: Pandiagonal Latin Squares

Postby 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
naked-pairs-in-a-diagonal: d7{a1 a4}{n4 n5} ==> d7a2 ≠ 5, d7a2 ≠ 4, d7a6 ≠ 5, d7a5 ≠ 5, d7a7 ≠ 5
whip[1]: d7n5{r6c2 .} ==> r2c2 ≠ 5, r4c2 ≠ 5, r1c4 ≠ 5, r4c7 ≠ 5, r6c4 ≠ 5, r6c6 ≠ 5
whip[1]: d4n5{r7c5 .} ==> r4c5 ≠ 5
whip[1]: r7n4{c6 .} ==> r1c5 ≠ 4
whip[1]: r1n4{c6 .} ==> r6c4 ≠ 4
whip[1]: c4n4{r7 .} ==> r2c2 ≠ 4
whip[1]: r2n4{c5 .} ==> r4c3 ≠ 4
whip[1]: c3n4{r6 .} ==> r3c6 ≠ 4
naked-pairs-in-a-column: c2{r2 r4}{n1 n7} ==> r5c2 ≠ 7, r1c2 ≠ 7, r1c2 ≠ 1
whip[1]: r1n1{c7 .} ==> r4c7 ≠ 1, r5c4 ≠ 1
whip[1]: r5n1{c5 .} ==> r7c5 ≠ 1, r3c3 ≠ 1
stte


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.
Last edited by denis_berthier on Mon Jun 14, 2021 3:05 pm, edited 1 time in total.
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Re: Pandiagonal Latin Squares

Postby 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! :cry:

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 ?
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Re: Pandiagonal Latin Squares

Postby 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 ...
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Re: Pandiagonal Latin Squares

Postby 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]
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Re: Pandiagonal Latin Squares

Postby 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 ... :?
Last edited by Mathimagics on Tue Jun 15, 2021 9:46 am, edited 1 time in total.
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Re: Pandiagonal Latin Squares

Postby 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
Last edited by Mathimagics on Tue Jun 15, 2021 3:08 pm, edited 1 time in total.
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Jellies

Postby 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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