The New Sudoku Players' Forum

[RW]

The term "2-digit unavoidable" is used frequently in this thread. But what exactly does it mean?

It means a subset of the unavoidable sets, the unavoidables that consist only of two digits. In this thread we have only examined these smaller sets. The maximum amount found in any grid is so far 78 in the Pt-grid. As you noticed in the Megaclue-thread, there is actually a huge amount more unavoidables in any grid, most however use more than two digits.

Since some of the 2-digit unavoidable sets are quite large, it can't mean only two digits appear in the set. So does "2-digit" mean each cell in the set can only be one of two values?

The maximum size of a two digit unavoidable set is 18 (2x9). The 127 FE-grids are special as all two digit unavoidables are of size 18. Essentially it means that if you pick any two digits in any of the grids, all 18 instances will only form one minimal unavoidable set.

For example, consider this unavoidable... Is it a "2-digit unavoidable?"

No, it seems to be a 6-digit unavoidable.

How does one know which of the unavoidable sets output by checker are 2-digit unavoidables ... or are they all 2-digiters?

I suppose the only way of knowing is to compare them to the grid. They are not all 2-digiters, and there is 2-digiters that don't get listed by checker (the largest ones). I don't know of any publically available program that counts the 2-d unavoidables, but it's not hard to program one your self. Mine works on the simple principle:

Code: Select all

for i=1 to 8 {
  for j=i+1 to 9 {
    load the grid
    for all cells in the grid {
      if cell A = i {
        set A=0
        find all cells with value (j) seen by A and set their value to 0
        find all cells with value (i) seen by those cells, set value to 0
        find all cells with value (j) seen by the new cells, set value to 0
        ...until no more cells (i,j) can be seen by any of the cells
        increase unavoidable counter
      }
    }
  }
}

I figured that would be the easiest way to do it without having to code any solver or pattern recognizer. I've been trying to come up with a way to find 3-digit and larger unavoidables in a similar way, but it's just not that simple...

RW

[ronk]

RW, thanks for a very helpful reply, Ron

[RW]

I must say that the estimate I got in the beginning of my search for 21s in the Pt-grid turned out to be quite wrong... At 128/1024: 66 uniques found, at 1024/1024: 13151 uniques found! Over half of these turned up in the last quarter of the puzzle. After removing the duplicates under isomorphism I'm left with 8956 puzzles, so searching the whole grid should reveal 53736 unique 21s.

Note that I had some problems in the search and had to restart it a few times from the middle of the grid. There's a small chance that I have made some mistake there, which could have caused me to miss some puzzle, but I doubt it.

So what do we learn from this..? I don't know, maybe that there can be 9000 times more 21s than 20s in a grid... I'll have a look at the puzzles to see if I can find anything else interesting. I noticed that, no suprisingly, most of the puzzles were in regions around the 20, using the first 6 clues of the 20 gives 559 unique 21s (498 minimal).

[Edit: Ah, it's so late, forgot to remove the non minimal. With one 20 there should be 61 non minimal 21s, right? That means 8895/53370 puzzles.]

RW

[coloin]

Well done ....I did wonder that I got so many within 3 clues of one 20 puzzle....

But I know why there are so many.............

Its because there are so many ruddy puzzles per single grid.......[maybe 10^16 ?]

This means a real puzzle distribution might look like this....
Say the average is 25 !

Code: Select all

PT grid

20    6
21    53,370
22  ? 1,000,000,000
23  ? 500,000,000,000,000
24  ? 3,000,000,000,000,000
25  ? 4,000,000,000,000,000
26  ? 2,000,000,000,000,000
27  ? 500,000,000,000,000
28  ? 50,000,000,000
29  ? 1,000,000,000
30  ? 100,000,000
31  ? 10,000,000
32  ? 1,000,000
33  ? 10,000
34  ? 330 *                              * 55 34s x 6

Who really knows !

By the way I have sorted the clue fixing for the MC grid.........

See Canonical grid has no 19

C

[RW]

I noticed that, no suprisingly, most of the puzzles were in regions around the 20

On closer inspection, the average amount of common clues with the 20 (or any of its isomorphs) is 10.2. One puzzle has only four clues in common with the 20:

Code: Select all

100050000000009300080000004000000800740001000006030500000004070500010260002003000

All the other have at least 5 common clues. Guess this just shows once more how hard it is to define the grid with few clues.

This means a real puzzle distribution might look like this....
Say the average is 25 !

Your random grid generating that is proved to make more low clue puzzles made an average of 25.56. I guess the real average of the grid should be somewhere between 26 and 27. If there only was a way of counting realistic estimates on these numbers...

RW

[coloin]

Yes you are right....

I was assuming the average grid had a puzzles average size of 24 going on 25 . But the PT grid must be 25.5 going on 26.5 as you say.

I shall make up a new set of figures !!

Code: Select all

PT grid

20    6
21    53,370
22  ? 1,000,000,000
23  ? 10,000,000,000,000
24  ? 500,000,000,000,000
25  ? 1,500,000,000,000,000
26  ? 3,000,000,000,000,000
27  ? 3,000,000,000,000,000
28  ? 1,500,000,000,000,000
29  ? 500,000,000,000,000
30  ? 10,000,000,000
31  ? 1,000,000,000
32  ? 10,000,000
33  ? 10,000
34  ? 330 *                              * 55 34s x 6

I think the mc grid gives us the best chance to count the puzzles....
C

[algogocom]

Oh, My God, it is so abundant clues for us, thank you again for you guys.
Rgds,
http://www.algogo.com
Camera,Cellphone,MP3/MP4,Electronics store

[Mauricio]

Sorry if this has been answered before.

I searched with my generator a grid without 4 cell unavoidables AND 6 cell unavoidables, but I could not find one. Does someone know a grid that fulfills that characteristic?

My program can generate about 1200 filled grids per second(P4 2.8Ghz), and I pruned the search in a partially filled grid when it found a 4 or 6 cell unavoidable, and continued with the next option. My program searched for ~6 hours.

[RW]

I searched with my generator a grid without 4 cell unavoidables AND 6 cell unavoidables, but I could not find one. Does someone know a grid that fulfills that characteristic?

You also asked if box interactions helped in finding the "snakes". No, they didn't. But some other ideas from the search can be adapted to help answer the following long-standing conjecture ...

Conjecture: every grid has either a size 4 unavoidable set or a size 6 unavoidable set. (Most grids have both sizes)

... which I can now confirm is true. (More's the pity.)

Red Ed never mentioned what these other ideas were, I guess eveybody assumed him to be right (as he usually is), without further questioning. Showing that the FE-grids all have 6-digit unavoidables isn't enough to confirm the conjecture, so he should have some more accurate proof.

RW

[Red Ed]

Moschopulus asked me the same question at the time I posted that claim. Here, verbatim, is what I told him.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Oh, it was nothing fancy, just another computer search.

The first part of the search finds grids with no 4-cell unavoidables or 2-digit 6-cell unavoidables:

• Fix the position of all the 1s (the "1s template"). This fixed template happens to include a cell in row 1, column 1.
• Enumerate all 2s templates subject to the constraints that each (a) has a cell in row 1, column 2; (b) makes no 4-cell or 6-cell unavoidables with the 1s template. Call this latter property P(1,2); for the Ds and Es templates we'll call it P(D,E).
• Enumerate all 3s ... 9s templates similarly.
• Now run a 8-deep loop (perhaps I could convince myself that 7-deep is enough, but never mind) which looks something like this:

  Code: Select all

  for each 2s template {
    filter the 4s ... 9s templates w.r.t. the 2s
    for each filtered 3s template s.t. P(2,3) {
        filter the 5s ... 9s templates w.r.t. the 3s
        for each filtered 4s template s.t. P(3,4) {
            filter the 6s ... 9s templates w.r.t. the 3s
            for each filtered 5s template s.t. P(4,5) {
                filter the 7s ... 9s templates w.r.t. the 3s
                for each filtered 6s template s.t. P(5,6) {
                    filter the 8s,9s templates w.r.t. the 3s
                    for each filtered 7s template s.t. P(6,7) {
                        filter the 9s templates w.r.t. the 3s
                        for each filtered 8s template s.t. P(7,8) {
                            for each filtered 9s template s.t. P(8,9) {
                                reconstruct grid and save to file
}   }   }   }   }   }   }   }

• (Here, filtering the Es templates w.r.t. the Ds means temporarily discarding those for which P(D,E) is false.)That search gives I think 157000+ grids which, it turns out, are isomorphic to just 333 representatives.
  
  Now all we need to do is search each grid for 3-digit 6-cell unavoidables. It turns out that there are none without 6-cell unavoidables and essentially only two with <4 of them:

  Code: Select all

  123456789456789132789213645275941368638527914941638257394175826517862493862394571
Type 6a: 1
Type 6b: 2

123456789456789231789231564234968175867125493915374826348617952592843617671592348
Type 6a: 0
Type 6b: 3

  
  
  Ed.

[RW]

Thank you Red Ed for the explanation, that should settle the issue.

Now all we need to do is search each grid for 3-digit 6-cell unavoidables. It turns out that there are none without 6-cell unavoidables and essentially only two with <4 of them:

Isn't there still a small theoretical possibility that there exists some grid that has 4-cell unavoidables, but no 6-cell unavoidables?

RW

[coloin]

Isn't there still a small theoretical possibility that there exists some grid that has 4-cell unavoidables, but no 6-cell unavoidables?

Coincententally the dukuso15 grid has come up on another thread.

You looked at this grid before

Code: Select all

123568479
864791352
957243681
218657934
536489127
749312865
391825746
472136598
685974213

           2-perm.  4-perm. 8-perm. 16-perm.  U-4s    U-6s  2-digitunavoidables   
dukuso15 - 18       0       18      0         36      12           72

However it has 36 4-setunavoidables [U-4] - it would appear to be streets ahead of the rest in this respect ? I knew it had many but I didnt realize it was more than the PT grid.

These are the 12 6-set unavoidables [U-6] - None with 2-digits

Code: Select all

{12,15,19,22,25,29}
{13,16,18,33,36,38}
{21,23,41,43,81,83}
{24,26,44,46,84,86}
{27,28,47,48,87,88}
{31,32,51,52,71,72}
{34,35,54,55,74,75}
{37,39,57,59,77,79}
{42,45,49,62,65,69}
{53,56,58,63,66,68}
{73,76,78,93,96,98}
{82,85,89,92,95,99}

I was prompted to look at the dukuso15 as JPF found these three grids- each have 29 4-set unavoidables

Code: Select all

243658791587129346169473528921364857678592134354781962412937685896215473735846219
586324179149678523273159864418732956967815432352946718695287341834561297721493685
246193857951487236387652491763918542592346718814725369625839174478561923139274685


The rational being that if you have a U-4 in a band it excludes a U-6.

C

[coloin]

Isn't there still a small theoretical possibility that there exists some grid that has 4-cell unavoidables, but no 6-cell unavoidables?

Yes

This one does not have U6s !!

It does have 36 U4s - found by Red Ed here
It is one of the 8 other grids found to have 36 U4s.

Code: Select all

123456789
457189236
968372514
291738465
374265198
685941327
546813972
732694851
819527643

Found 150 unavoidable sets (36 of size 4 or 6).

The maximum # of disjoint unavoidable sets (max clique number -- MCN) is 12.
One such maximal collection is:
   {14,16,84,86}
   {18,19,58,59}
   {21,22,71,72}
   {23,25,33,35}
   {24,26,64,66}
   {43,47,53,57}
   {51,52,81,82}
   {68,69,78,79}
   {83,85,93,95}
   {11,12,31,32,41,42,61,62,91,92}
   {28,29,38,39,48,49,88,89,98,99}
   {34,36,44,46,54,56,74,76,94,96}

C

[RW]

Very nice find, coloin (or should I say Red Ed)! So the conjecture "every grid has either a size 4 unavoidable set or a size 6 unavoidable set" turned out to be true. I was thinking there for a second that it could be rewritten to "every grid has a size 6 unavoidable set", but you just proved this wrong!

RW

[coloin]

Reopening this thread [after a year!]

We now can look at essesntially different clue combinations in grids.....

there were 181 essentially different 2-rookeries, with widely differing incidence in the sample.

Code: Select all

01 .......12....12....12.............21....21....21.........1..2..1..2.....2.....1..      16 sol.
01 .......12....12....12............12....12....12.............2.1...2.1...2.1......      8 sol.
01 .....1..2..2....1..1..2.........21...2.1.....1......2.....1.2....12.....2.......1      2 sol.
01 .....1..2..2....1..1..2........1.2....12.....2.......1...1...2..2....1..1....2...      2 sol.
02 .......12....12....12............12....12....12............12....12.....2.......1      2 sol.
02 .......12..1..2.....2.1........21....1....2..2.....1.....1...2....2....112.......      8 sol.
02 .......12..1..2....2..1..........12...2..1...1..2...........2.1.1..2....2..1.....      4 sol.
02 .......12..1..2....2..1..........12...21.....1...2..........2.1.1.2.....2....1...      4 sol.
02 .......12..1..2....2..1..........12..1.2.....2....1.........2.1..21.....1...2....      4 sol.
03 .......12....12....12...........1.2...1.2.....2......1...1..2..1..2.....2.....1..      4 sol.
03 .......12....12....12...........12....12.....2.......1....2.1...2.1.....1......2.      2 sol.
03 .......12..1..2....2..1.........1.2..1.2.....2.....1......2...1..21.....1.....2..      2 sol.
04 .......12..1..2.....2..1.........12..1..2....2..1...........2.1.2..1....1..2.....      8 sol.
04 .......12..1..2.....2.1..........12..1.2.....2....1.........2.1.2.1.....1...2....      4 sol.
04 .......12..1..2....2..1.........12.....2....121..........12......2...1..1......2.      2 sol.
05 .......12....12....12............12....12....2.1............2.1...2.1...12.......      8 sol.
05 .......12....12....12...........1.2...12.....2.....1.....1..2...2......11...2....      2 sol.
05 .......12..1..2.....2.1.......1..2...1..2....2.......1...2..1...2...1...1......2.      2 sol.
05 .......12..1..2....2..1.........1.2...2...1..1..2.........2...1.1....2..2..1.....      2 sol.
05 .......12..1..2....2..1.........12...1.2.....2.......1....2.1....21.....1......2.      2 sol.
05 .......12..1..2....2..1.......1...2..1..2....2.....1.....2....1..2..1...1.....2..      2 sol.
05 .....1..2..2....1..1..2.........2..1..1...2..2..1.........1..2..2....1..1..2.....      2 sol.
06 .......12....12....12.............21....21...12..........1..2.....2..1..2.1......      16 sol.
06 .......12....12....12............12...1.2.....2...1.........2.11..2.....2..1.....      8 sol.
06 .......12....12....12...........1.2...1.2.....2......1...2..1..1.....2..2..1.....      4 sol.
06 .......12....12....12...........12....12.....2.......1...1...2..2....1..1...2....      2 sol.
06 .......12..1..2.....2..1.......1..2..1.2.....2.....1.....1..2...2......11...2....      4 sol.
06 .......12..1..2.....2.1..........12..1..2....2....1.........2.1.2.1.....1..2.....      4 sol.
06 .......12..1..2....2..1.........12.....2....121...........2.1....21.....1......2.      2 sol.
06 .......12..1..2....2..1.........12...1..2....2.......1...1...2...2...1..1..2.....      2 sol.
07 .......12....12....12.............21..12......2.1.........21...1.....2..2.....1..      16 sol.
07 .......12....12....12............12...1.2....2..1........2.1....2......11.....2..      2 sol.
07 .......12....12....12............12...12.....2....1.........2.1.2.1.....1...2....      4 sol.
07 .......12..1..2.....2..1.......1..2..1.2.....2.......1....2.1...2.1.....1.....2..      4 sol.
07 .......12..1..2.....2.1..........12..1.2.....2..1.........2...1.2...1...1.....2..      2 sol.
07 .......12..1..2.....2.1.........12......2.1..12..........1...2..1.2.....2.......1      4 sol.
07 .......12..1..2....2..1..........12...2..1....1..2.......1..2..1..2.....2.......1      4 sol.
07 .......12..1..2....2..1.........1.2...2...1...1.2.........2...11.....2..2..1.....      2 sol.
07 .......12..1..2....2..1.........12...1.2.....2.......1...1...2...2...1..1...2....      2 sol.
08 .......12....12....12.............211..2.....2..1..........12....1.2.....2....1..      8 sol.
08 .......12....12....12............12....12....12.............2.1..12.....2....1...      4 sol.
08 .......12....12....12............12..2.1.....1...2..........2.1..12.....2....1...      4 sol.
08 .......12..1..2.....2..1.......1..2..1.2.....2.......1...1..2...2....1..1...2....      4 sol.
08 .......12..1..2.....2.1.......12.....1.....2.2.....1.....2.1....2......11.....2..      4 sol.
08 .......12..1..2.....2.1.......12.....1....2..2.......1...2.1....2....1..1......2.      4 sol.
08 .......12..1..2....2..1..........12...21.....1...2.........12...1.2.....2.......1      2 sol.
08 .......12..1..2....2..1.........1.2....2..1..1.2.........12.....1....2..2.......1      2 sol.
09 .......12....12....12.............21...12....2.1...........12...2....1..1..2.....      4 sol.
09 .......12....12....12...........1.2...12.....2.....1......2...1.2.1.....1.....2..      2 sol.
09 .......12..1..2.....2..1..........21.1..2....2..1.........1.2.....2..1..12.......      8 sol.
09 .......12..1..2.....2..1.......1..2....2....112..........12.....1....2..2.....1..      8 sol.
09 .......12..1..2.....2.1........21....1.....2.2.....1.....1..2...2......11..2.....      4 sol.
09 .......12..1..2.....2.1........21....1.....2.2.....1.....2....1.2.1.....1.....2..      4 sol.
09 .......12..1..2.....2.1.......1...2..1..2....2.....1.....2....1.2...1...1.....2..      2 sol.
09 .......12..1..2....2..1.........1.2..1..2....2.....1.....1..2....2.....11..2.....      2 sol.
09 .......12..1..2....2..1.........1.2..1.2.....2.....1.....1..2....2.....11...2....      2 sol.
09 .......12..1..2....2..1.........12.....2....121..........1...2...2...1..1...2....      2 sol.
10 .......12....12....12.............21..1.2....2..1........2..1...2...1...1.....2..      4 sol.
10 .......12....12....12...........1.2...1.2....2.....1.....1..2...2......11..2.....      2 sol.
10 .......12....12....12...........1.2.1...2....2.......1...1..2....12......2....1..      4 sol.
10 .......12....12....12...........12....12......2....1.....1...2.1...2....2.......1      2 sol.
10 .......12..1..2.....2..1..........21.1..2....2..1.........1.2...2....1..1..2.....      8 sol.
10 .......12..1..2.....2..1.......1..2..1.2.....2.....1......2...1.2.1.....1.....2..      4 sol.
10 .......12..1..2.....2.1..........12..1..2....2..1........2....1.2...1...1.....2..      2 sol.
10 .......12..1..2.....2.1.........12...1.2.....2.......1....2.1...2.1.....1......2.      2 sol.
10 .......12..1..2....2..1..........12...2..1...1..2.........2...1.1....2..2..1.....      2 sol.
10 .......12..1..2....2..1..........12...2..1...1..2........12.....1....2..2.......1      2 sol.
10 .......12..1..2....2..1..........12..1.2.....2....1.......2...1..21.....1.....2..      2 sol.
10 .......12..1..2....2..1.........1.2.....2.1..1.2.........1..2...1.2.....2.......1      2 sol.
10 .......12..1..2....2..1.........1.2...2...1..1...2.......1..2...1.2.....2.......1      2 sol.
10 .......12..1..2....2..1.........1.2..1..2....2.....1.....2....1..21.....1.....2..      2 sol.
11 .......12....12....12.............21..1.2....2..1..........12...2....1..1..2.....      4 sol.
11 .......12..1..2.....2..1.......1..2....2..1..12..........12.....1....2..2.......1      4 sol.
11 .......12..1..2.....2.1.........1.2..1..2....2.....1.....2....1.2.1.....1.....2..      2 sol.
11 .......12..1..2....2..1..........12...21.....1...2.........12.....2....121.......      2 sol.
11 .......12..1..2....2..1.........1.2.....2.1..1.2.........2....1.1....2..2..1.....      2 sol.
11 .......12..1..2....2..1.........1.2.....2.1..21..........1..2....2.....11..2.....      2 sol.
11 .......12..1..2....2..1.........1.2....2..1..1.2..........2...1.1....2..2..1.....      2 sol.
11 .......12..1..2....2..1.........1.2....2..1..1.2.........1..2...1..2....2.......1      2 sol.
11 .......12..1..2....2..1.........1.2...2...1...1.2........1..2..1...2....2.......1      2 sol.
12 .......12....12....12.............21..1.2.....2...1......1..2..1..2.....2.....1..      8 sol.
12 .......12....12....12............12...1.2....2..1...........2.1.2...1...1..2.....      4 sol.
12 .......12..1..2.....2.1.........1.2..1.2.....2.......1....2.1.....1..2..12.......      2 sol.
12 .......12..1..2....2..1..........12...2..1...1..2.........2...1...1..2..21.......      2 sol.
12 .......12..1..2....2..1.........1.2...2...1..1..2........1..2...1..2....2.......1      2 sol.
13 .......12....12....12.............21...12....12............12.....2..1..2.1......      8 sol.
13 .......12....12....12.............21...12....2.1...........12.....2..1..12.......      8 sol.
13 .......12....12....12.............21..12.....2..1.........21....2....1..1.....2..      8 sol.
13 .......12....12....12............12....12....2.1...........12.....2....112.......      4 sol.
13 .......12....12....12...........1.2...1.2....2.....1.....2....1.2.1.....1.....2..      2 sol.
13 .......12....12....12...........1.2...12......2....1.....1..2..1...2....2.......1      2 sol.
13 .......12..1..2.....2.1.........1.2..1.2.....2.....1.....1..2...2......11...2....      2 sol.
14 .......12....12....12............12...12......2.1...........2.11...2....2....1...      8 sol.
14 .......12....12....12............12..2.1.....1...2.........12.....2....12.1......      2 sol.
14 .......12..1..2.....2.1..........12..1.2.....2....1.......2...1.2.1.....1.....2..      2 sol.
14 .......12..1..2.....2.1.........12...1.2.....2.......1...12.....2....1..1......2.      2 sol.
14 .......12..1..2....2..1..........12..1.2.....2....1.......2...1...1..2..1.2......      2 sol.
15 .......12....12....12.............21....21...12..........1..2....12.....2.....1..      8 sol.
15 .......12....12....12............12....12....2.1...........12...2......11..2.....      2 sol.
15 .......12..1..2.....2.1..........12..1..2....2..1..........12...2......11..2.....      2 sol.
15 .......12..1..2.....2.1..........12..1.2.....2..1..........12...2......11...2....      2 sol.
15 .......12..1..2.....2.1.........1.2..1.2.....2.....1......2...1.2.1.....1.....2..      2 sol.
15 .......12..1..2.....2.1.......1...2..1..2....2.....1.....2.1....2......11.....2..      2 sol.
15 .......12..1..2....2..1..........12...2..1...1...2.......1..2...1.2.....2.......1      2 sol.
15 .......12..1..2....2..1.........1.2....2..1..21...........2...1..21.....1.....2..      2 sol.
16 .......12....12....12...........1.2...12......2....1......2...11.....2..2..1.....      2 sol.
16 .......12..1..2.....2..1.......1..2....2..1..12...........2...1.1....2..2..1.....      4 sol.
16 .......12..1..2....2..1.........1.2....2..1..21..........1..2....2.....11...2....      2 sol.
17 .......12..1..2.....2..1.........12..1..2....2..1.........1.2...2......11..2.....      4 sol.
17 .......12..1..2.....2.1..........12..1..2....2..1...........2.1.2...1...1..2.....      4 sol.
17 .......12..1..2.....2.1.........1.2....2....112..........1..2...1..2....2.....1..      2 sol.
17 .......12..1..2.....2.1.........12...1.2.....2.......1...1...2..2....1..1...2....      2 sol.
18 .......12....12....12...........1.2.1...2....2.......1...2..1....1...2...2.1.....      4 sol.
18 .......12..1..2.....2..1.........12..1..2....2..1.........1.2.....2....112.......      4 sol.
18 .......12..1..2.....2..1.......1..2....2....112...........2.1...1....2..2..1.....      4 sol.
18 .......12..1..2.....2.1..........12..1.2.....2....1......1..2...2......11...2....      2 sol.
18 .......12..1..2.....2.1.........1.2.....2.1..12..........2....1.1....2..2..1.....      2 sol.
18 .......12..1..2.....2.1.........1.2....2....112...........2.1...1....2..2..1.....      2 sol.
18 .......12..1..2.....2.1.........1.2....2..1..12...........2...1.1....2..2..1.....      2 sol.
18 .......12..1..2.....2.1.........1.2..1..2....2.....1.....1..2...2......11..2.....      2 sol.
18 .......12..1..2.....2.1.........1.2..1.2.....2.....1.....12.....2......11.....2..      2 sol.
19 .......12....12....12............12....12....2.1............2.1.2...1...1..2.....      4 sol.
19 .......12....12....12............12...12.....2..1..........12...2......11...2....      2 sol.
19 .......12..1..2.....2.1..........12..1..2....2..1..........12.....2....112.......      2 sol.
19 .......12..1..2.....2.1.........12...1..2....2.......1...1...2..2....1..1..2.....      2 sol.
19 .......12..1..2.....2.1.......1..2...1..2....2.......1...2.1....2....1..1......2.      2 sol.
19 .......12..1..2....2..1..........12...2..1....1.2.........2...11.....2..2..1.....      2 sol.
20 .......12....12....12.............21..1.2.....2.1........2.1...1.....2..2.....1..      8 sol.
20 .......12....12....12.............21..12.....2....1.......2.1...2.1.....1.....2..      4 sol.
20 .......12....12....12.............21..12.....2..1..........12...2....1..1...2....      4 sol.
20 .......12....12....12...........1.2...12.....2.......1...1..2...2....1..1...2....      2 sol.
20 .......12..1..2.....2.1..........12..1..2....2..1........2.1....2......11.....2..      2 sol.
21 .......12....12....12............12...12.....2....1.......2...1.2.1.....1.....2..      2 sol.
21 .......12..1..2.....2..1.......1..2....2..1..12..........1..2...1..2....2.......1      4 sol.
21 .......12..1..2.....2.1..........12..1.2.....2....1.......2...1...1..2..12.......      2 sol.
21 .......12..1..2.....2.1.........1.2....2..1..12..........1..2...1..2....2.......1      2 sol.
21 .......12..1..2.....2.1.........1.2....2..1..12..........12.....1....2..2.......1      2 sol.
22 .......12....12....12............12...1.2....2....1......1..2...2......11..2.....      2 sol.
22 .......12....12....12............12...1.2....2..1........2....1.2...1...1.....2..      2 sol.
22 .......12..1..2.....2..1.......1..2..1.2.....2.......1....2.1.....1..2..12.......      4 sol.
22 .......12..1..2.....2.1..........12..1..2....2....1......1..2.....2....112.......      4 sol.
22 .......12..1..2.....2.1.........12.....2....112...........2.1...1.....2.2..1.....      2 sol.
22 .......12..1..2.....2.1.........12.....2....112..........12.....1.....2.2.....1..      2 sol.
22 .......12..1..2.....2.1.........12...1..2....2.......1...2..1...2.1.....1......2.      2 sol.
23 .......12....12....12.............21..1.2....2..1..........12.....2..1..12.......      4 sol.
23 .......12....12....12............12..2.1.....1..2..........12....1.2....2.......1      2 sol.
23 .......12....12....12...........1.2.1...2....2.....1.....1..2....12......2......1      2 sol.
23 .......12..1..2.....2.1.........1.2.....2.1..12..........1..2...1.2.....2.......1      2 sol.
23 .......12..1..2.....2.1.........1.2..1.2.....2.......1...1..2...2....1..1...2....      2 sol.
24 .......12....12....12.............21..12.....2..1..........12......2.1..12.......      8 sol.
24 .......12....12....12............12...1.2....2..1..........12...2......11..2.....      2 sol.
24 .......12....12....12............12..2.1.....1..2.........2...1..1...2..2....1...      2 sol.
24 .......12..1..2.....2.1..........12..1.2.....2..1..........12......2...112.......      4 sol.
26 .......12....12....12.............21..1.2.....2.1..........12..1..2.....2.....1..      4 sol.
26 .......12....12....12.............21..12......2.1..........12..1...2....2.....1..      8 sol.
26 .......12....12....12............12...12.....2..1.........2...1.2...1...1.....2..      2 sol.
26 .......12....12....12...........1.2...12.....2.......1....2.1...2.1.....1.....2..      2 sol.
26 .......12..1..2.....2.1.........1.2....2....112..........12.....1....2..2.....1..      4 sol.
27 .......12....12....12.............21...12....12............12....12.....2.....1..      4 sol.
27 .......12..1..2.....2.1..........12..1.2.....2....1......12.....2......11.....2..      2 sol.
28 .......12....12....12.............21..12.....2....1.......2.1.....1..2..12.......      4 sol.
28 .......12....12....12............12...1.2.....2.1...........2.11..2.....2....1...      4 sol.
28 .......12....12....12............12...1.2....2....1......1..2.....2....112.......      4 sol.
28 .......12....12....12............12...1.2....2....1......2....1.2.1.....1.....2..      2 sol.
28 .......12....12....12............12...1.2....2..1..........12.....2....112.......      2 sol.
28 .......12....12....12............12.1..2.....2..1..........12....1.2.....2......1      4 sol.
28 .......12....12....12...........1.2...1.2.....2....1.....2....11.....2..2..1.....      2 sol.
28 .......12..1..2.....2.1..........12..1..2....2....1......1..2...2......11..2.....      2 sol.
29 .......12....12....12.............21..1.2.....2.1........2..1..1.....2..2....1...      4 sol.
29 .......12....12....12.............21..12.....2..1.........2.1...2...1...1.....2..      4 sol.
29 .......12....12....12...........1.2.1...2....2.....1.....2....1..1...2...2.1.....      2 sol.
30 .......12....12....12............12...1.2....2....1.........2.1.2.1.....1..2.....      4 sol.
31 .......12....12....12...........1.2...1.2.....2....1.....1..2..1..2.....2.......1      2 sol.
31 .......12..1..2.....2.1..........12..1..2....2....1......2....1.2.1.....1.....2..      2 sol.
33 .......12....12....12............12...12......2.1.........2...11.....2..2....1...      4 sol.
34 .......12....12....12............12...1.2.....2.1..........12..1..2.....2.......1      2 sol.
35 .......12....12....12............12...1.2.....2.1........2....11.....2..2....1...      2 sol.
35 .......12....12....12............12...1.2.....2.1........2.1...1.....2..2.......1      4 sol.
35 .......12....12....12............12..2.1.....1...2.........12....12.....2.......1      2 sol.
36 .......12....12....12............12...1.2.....2...1......1..2..1..2.....2.......1      4 sol.
42 .......12....12....12............12...12......2.1..........12..1...2....2.......1      4 sol.


EDIT I managed to calculate the number of sols = perm.
the frequency is unexpected

Code: Select all

2 sol.   105
4 sol.    55
8 sol.    18
16 sol.    3

Red Ed elucidated the the 3-rookery number

Code: Select all

there are 259272 essentially-different 3-rookeries.

C