## Structures of the solution grid

Everything about Sudoku that doesn't fit in one of the other sections
Thanks for this! According to the numbers, random picks of 36 rookeries should give:
Code: Select all
`2 sol.     20,88 4 sol.     10,948 sol.      3,5816 sol.     0,60`

However, in 10M random grids I found:
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`2 sol.     20,724 sol.     11,728 sol.      3,2516 sol.     0,31`

Seems that the average 16-permutable rookery is used less than half as frequently as the average 4-perm. Wonder why this is...

RW
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

Well my sample size was small, but a bigger one didnt provide any more essentially different 2-rookeries.

I think the error with the 16-perm rookeries is reasonable ! I wouldnt think my numbers would have the power to show that difference.

Each 2-rookery has 6^8*2 isomorphs [3359232]
*2 for the two clue values = 6718464
*181 = 1216041984

Given that there are only 838501632 "different" 2-rookeries.

What is the degree of automorphism with each of our 181 "essentially different" 2-rookeries ?

[1216041984] / [838501632] = 1.4 average

Some of the 2-rookeries will have automorphic isomorphs

Is this the reason for the observed difference [x30-40] in incidence of some of the 181 2-rookeries ?

Might grids with 17s have a greater proportion of "rarer" 2-rookeries ?

C
coloin

Posts: 1666
Joined: 05 May 2005

Hmm... actually the fewer appearances of 16-perms seems to imply that there are fewer possible ways to fill in the remaining 7 digits to a 16-perm than a 2- or 4-perm. This must have something to do with the way the clues are placed, I'm guessing it's because of the digit distribution within the boxes. Two of the 16-perms don't have any diagonal boxes (boxes where the two digits aren't in either the same row or the same column). The third 16-perm has only one diagonal box. I haven't looked at them all, but I believe there's only one 8-perm without diagonal boxes, all 4-perms must have at least two diagonal boxes and all 2-perms at least three. The lack of diagonal boxes in the 16 perms probably reduces the possible ways to fill in the rest of the grid. This could of course be verified by counting the grid solutions for all 2-rookeries and comparing this to the amount of diagonal boxes in the rookery, but I don't (neither do I expect anyone else to) have the computing power to pull off a stunt like that...

Perhaps we also need to consider the option that the program I used to create the grids (gsf's program back in the summer of 2006) was biased not to create 2-rookeries without diagonal boxes, though I doubt this is the case.

RW
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

Indeed , the so called random grids made from random puzzles are biased to a degree in the number of U4 unavoidables.

A lot of work was done here

I did a few calculations some time ago [in the min clues thread]!!!

coloin wrote:have run a solver on a few grids of the following pattern [with different B2-B9]:
Code: Select all
`123 --- --- 456 1-2 --- 789 --- -12- --- -2- --1 -1- --- 2-- --2 --1 --- --1 --- -2- 2-- -1- --- --- 2-- 1-- `
coloin wrote: The analysis take 9 hours each! [Hopefully the B1 filling doesnt introduce bias - I cant see any]
Of the five I have done I got solution rates of approx.
2100,000,000
1220,000,000
1200,000,000
1080,000,000
989,000,000 {this is canonical - 1/2 in different row and column}
I dont have the actual grids pertaining here ! - But it is not a difficult "stunt" - very possible with suexk2.exe

I will run it with a few selected 2-rookeries [now we know what they are !]

Is there is a significant difference between the sol counts....?

I stumbled on this some time ago but didnt have the insight that we do now here

on page 4 of this thread Viggo wrote
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`Grid name:                             top1  top2  top3  ran1  ran2  ran3    SF   SFB Number of 2-permutable:                  20    19    26    18    23    25    28    36 Number of 4-permutable:                  13    13     9    16     9     8     8     0 Number of 8-permutable:                   2     4     1     2     3     3     0     0 Number of 16-permutable:                  1     0     0     0     1     0     0     0 Sum of solutions all 2-rookeries:       124   122    96   116   122   106    88    72 Sum of solutions all 3-rookeries:      3414  3384  2322  3252  3222  2574  1818  1194 Max numb. solutions of 3-rookeries:     360   258   252   264   264   120   102    54 Number of minimal 3-rookeries:            7     5    10     2     5    11    15    39        MCN:                                     12    10    10    10    12    10     9     8 `

I think he means
Sum of solutions all 2-rookeries: ~ Sum of solutions all 7-rookeries:
Sum of solutions all 3-rookeries: ~ Sum of solutions all 6-rookeries:

Code: Select all
`...749568456.8.9.77896.54....8567.4956.498.7.947...68587495...66.5.7489..9.8.6754   48 sol....849567456..78.9789.56..464.5987..97.46..855.8.7.946.9768.45.8657.4.9...49.5678   60 sol..46..8975785.96.4...945786.49.56.7.8..897.654657.84..95648.9..78..7.549697.64.58.   6 sol. .4798..5658.476.9..695..874896.45..7754..896....7694859.5..764847869.5..6..8547.9   72 sol...568.97464..79.587894.56....49687.596.5.784.857.4..964.6.9.587.7.854.695987.64..   12 sol.4.6..5987.756894..98..476.5.9.7.85468475.6.9.56..947.8.5987..646.495.87.7.846..59   30 sol.8569..7.47..456.9849.8.7.56.48.9.6759.5764.8..67.854.9.7.54896.684.795..5.96..847   12 sol.86.95..74.95.746.84.768.59..498.7.56..64.5789578.964..6..7498.575..6894.9845...67   18 sol...89.4576596.7.8.4.74586..97856..94..4..5968796.847.5.4..768.956594..7.88.7.9546.   12 sol..4.58976.675.4.98.8.9.675.44678...5959.6.4.78.8.975.467.8..64959.645.8.7.547986..   96 sol.`

these are 10 random 6-rookeries, with the sol. count which is the perm. count of the missing 3-rookery.

To note each completion has 6 isomorphic clue combinations.
And the 6 sol. must be a minimal 3-rookery, equivalent to an unentwined 3-rookery.

RW wrote:According to the numbers, random picks of 36 rookeries should give:
which numbers ?.......[not mine hopefully !]
my random 2-rookeries were taken from 2700 random 17-puzzles [one from each puzzle]
C
coloin

Posts: 1666
Joined: 05 May 2005

coloin wrote:
RW wrote:According to the numbers, random picks of 36 rookeries should give:
which numbers ?.......[not mine hopefully !]
my random 2-rookeries were taken from 2700 random 17-puzzles [one from each puzzle]

Yes, yours... I was under the impression that you gave a list of all essentially different 2-rookeries. Perhaps I was mistaken. Even now when I reread the post I get the impression that it contains them all....

RW
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

It is the list of ALL essentially different 2-rookeries.
coloin wrote:Well my sample size was small, but a bigger one didnt provide any more essentially different 2-rookeries.

I would have expected Red Ed to correct it if it was wrong !

I cant understand why some are way more common.....it might be automorphism [or lack of automorphism], but I cant visualise it.

The solution counts for a 6-rookery~ [6,12,18..30..60..72..102..120.......?360] average is around 36, there are 3! [3*2*1] equivalent solution grids therefore 36/6 = 6 ways to add a 3-rookrry on average.

What happens when you have a grid with 2 pairs of identical 2-rookerys.....it couldnt be an automorphic grid perhaps ??
C
coloin

Posts: 1666
Joined: 05 May 2005

The automorphic grid - duk15
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`123568479864791352957243681218657934536489127749312865391825746472136598685974213`
Its 36 2-rookeries
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`.......79...79....9.7...........79.......9..77.9.......9....7...7.....9....97..........4.9..4.9....9...4..........9.4...4.9....49.......9.....4.4......9....9.4.........47...47.......7.4.........7..4...4....774.............74.47...........74........8..98...9....9......8...8...9......89.....9...8...9.8............98.8.9..........8.7.8..7.......7....8...8..7.......8...77.....8.....8..7...7......8.8..7.........84..8.4..........4..8...8.....4...48.....4....8.....8...4.4.......8.8...4.......6...9.6..9....9.....6.....6..9....6..9.....9....6..9......6.....6.9.6..9.........6..7..6.7.......7...6.....6.7.....6.....77......6.......7.6.7...6...6...7........6.4...64..........4.6.....6....4..64......4.....6........464....6...6....4.......68...86.............68...86.......6.8..........86....8....6.....6..868..........5....9....9..5.95...........5.9..5....9.....9.....5.9...5.........59...59........5...7....7...5..57..........57...5.......77.......5.....57...7....5....5.7.......5..4....4....5..5..4........5...45..4......4......5.....5.4.4.....5....5..4......5.8...8......5..5.....8...8.5....5...8..........8.5...8.5.........5.8.85.........56.....6.....5..5....6.....65....5.6.............65.....5..6.....65..6.5........3.....9....9.3..9....3.........93..3...9.....93.....39...........3..9....9....3..3....7....7..3....7..3........7.3..3......77..3.....3.....7...7..3........7...3..3...4....4...3......43..........34.3.4......4.3.....3......4.4...3.........4..3..3..8...8.....3.......3.8...8....3..3..8.......3..8..3..8.........3...8.8......3..3.6.....6....3.......36.....6...3..36.........3...6.3.......6....36...6.......3..35...........35..5...3.......5..3.53..........3....53....5.......3.5....5.....3.2......9....9...29..2.....2.....9.......9.2...9..2....9..2......2....9....9..2...2.....7....7....2..72.....2....7..........277....2.......2.7...72..........7.2...2....4....4.....2...24....2.......4...4...2..4...2.......2..4.4.2...........42...2...8...8.......2...2...8.2.8..........8..2......28.....82......2.....8.8....2...2..6.....6......2...2..6..2..6.......6....2......2.6.....2...6..2..6...6.....2...2.5............52.5.2.....2...5....5......2......2..5....25.....2...5....5...2...23............3.2...2.3...2......3..3.....2....3.2...3...2......2.3..........2.31.......9....91...9.......1.1....9.......91....9.1.....91.........1...9....9...1.1......7....7.1.....7.....1.1...7.........1.77...1......1...7...7.1.........7..1.1.....4....4..1.......4...1.1......4...4..1...4..1......1....4.4..1..........4.1.1....8...8....1..........81.18..........8.1......1.8....18........1....8.8.....1.1...6.....6...1.........6.1.1.6.......6...1......1..6...1.....6...1.6...6......1.1..5..........1.5..5......1.1..5....5.....1......1...5..1..5......1..5....5....1.1.3...........13.......3..1.1.....3..3....1.....31....3.1.........13...........1312............1..2...2....121.............12.....12.....1.2......21...........21.`

the canonicalized 2-rookeries are:
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`000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012001002000020010000000001200000200001210000000000100020002000100100020000000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012000012000012000000000000021001200000200100000000021000020000100100000200`

Two types
Code: Select all
`000000012000012000012000000000000021001200000200100000000021000020000100100000200000000012001002000020010000000001200000200001210000000000100020002000100100020000`
From here
Code: Select all
`grid # automorphisms, band, and minlex index 123456789457189326869372514214965873635847192978213465381624957546798231792531648 # 36 240 5449015463`

Most grids [ 99.99%] are non-automorphic.
From gsfs work, here is the distribution.
Code: Select all
`   1 5472170387     [non-automorphic - 1 grid]   2     548449    3       7336    4       2826    6       1257    8         29    9         42   12         92   18         85   27          2   36         15     [including duk15]  54         11   72          2  108          3  162          1  648          1     [MC]`

C
coloin

Posts: 1666
Joined: 05 May 2005

Condor wrote:T(0) = 1
T(1) = 46656 = 6^6
T(2) = 838501632 = 6^6 * 2^2 * 4493
T(3) = 5196557037312 = 6^6 * 2^2 * 3433 * 8111
T(4) = 9631742544322560 = 6^6 * 2^7 * 3 * 5 * 19 * 5659037

T(9) = 6670903752021072936960 = 6^6 * 2^14 * 3^2 * 5 * 7 * 27704267971

Condor wrote:The only thing I can add is that for every one of the 46656 templates there is 17972 ways of placing the second template.

I generated all the placing of the "2"s for a fixed "1" template.
Code: Select all
`Here is the minlex ordered 2-rookeriesrookery i.d.  # puzz #   perm ,  number of isomorphs in 4493 rookeries#001#  .......12....12....12.............21....21....21.........1..2..1..2.....2.....1..  #   16 sol.     3 #002#  .......12....12....12.............21....21...12..........1..2.....2..1..2.1......  #   16 sol.     9 #003#  .......12....12....12.............21....21...12..........1..2....12.....2.....1..  #    8 sol.    18#004#  .......12....12....12.............21...12....12............12.....2..1..2.1......  #    8 sol.    18#005#  .......12....12....12.............21...12....12............12....12.....2.....1..  #    4 sol.    18#006#  .......12....12....12.............21...12....2.1...........12.....2..1..12.......  #    8 sol.    18#007#  .......12....12....12.............21...12....2.1...........12...2....1..1..2.....  #    4 sol.    18#008#  .......12....12....12.............21..1.2.....2...1......1..2..1..2.....2.....1..  #    8 sol.    18#009#  .......12....12....12.............21..1.2.....2.1..........12..1..2.....2.....1..  #    4 sol.    36#010#  .......12....12....12.............21..1.2.....2.1........2..1..1.....2..2....1...  #    4 sol.    36#011#  .......12....12....12.............21..1.2.....2.1........2.1...1.....2..2.....1..  #    8 sol.    36#012#  .......12....12....12.............21..1.2....2..1..........12.....2..1..12.......  #    4 sol.    36#013#  .......12....12....12.............21..1.2....2..1..........12...2....1..1..2.....  #    4 sol.    18#014#  .......12....12....12.............21..1.2....2..1........2..1...2...1...1.....2..  #    4 sol.    18#015#  .......12....12....12.............21..12......2.1..........12..1...2....2.....1..  #    8 sol.    36#016#  .......12....12....12.............21..12......2.1.........21...1.....2..2.....1..  #   16 sol.    18#017#  .......12....12....12.............21..12.....2....1.......2.1.....1..2..12.......  #    4 sol.    36#018#  .......12....12....12.............21..12.....2....1.......2.1...2.1.....1.....2..  #    4 sol.    36#019#  .......12....12....12.............21..12.....2..1..........12......2.1..12.......  #    8 sol.    36#020#  .......12....12....12.............21..12.....2..1..........12...2....1..1...2....  #    4 sol.    36#021#  .......12....12....12.............21..12.....2..1.........2.1...2...1...1.....2..  #    4 sol.    36#022#  .......12....12....12.............21..12.....2..1.........21....2....1..1.....2..  #    8 sol.    18#023#  .......12....12....12.............211..2.....2..1..........12....1.2.....2....1..  #    8 sol.    18#024#  .......12....12....12............12....12....12.............2.1...2.1...2.1......  #    8 sol.     1 #025#  .......12....12....12............12....12....12.............2.1..12.....2....1...  #    4 sol.     9 #026#  .......12....12....12............12....12....12............12....12.....2.......1  #    2 sol.     6 #027#  .......12....12....12............12....12....2.1............2.1...2.1...12.......  #    8 sol.     3 #028#  .......12....12....12............12....12....2.1............2.1.2...1...1..2.....  #    4 sol.    18#029#  .......12....12....12............12....12....2.1...........12.....2....112.......  #    4 sol.     9 #030#  .......12....12....12............12....12....2.1...........12...2......11..2.....  #    2 sol.    18#031#  .......12....12....12............12...1.2.....2...1.........2.11..2.....2..1.....  #    8 sol.    18#032#  .......12....12....12............12...1.2.....2...1......1..2..1..2.....2.......1  #    4 sol.    36#033#  .......12....12....12............12...1.2.....2.1...........2.11..2.....2....1...  #    4 sol.    36#034#  .......12....12....12............12...1.2.....2.1..........12..1..2.....2.......1  #    2 sol.    36#035#  .......12....12....12............12...1.2.....2.1........2....11.....2..2....1...  #    2 sol.    36#036#  .......12....12....12............12...1.2.....2.1........2.1...1.....2..2.......1  #    4 sol.    36#037#  .......12....12....12............12...1.2....2....1.........2.1.2.1.....1..2.....  #    4 sol.    36#038#  .......12....12....12............12...1.2....2....1......1..2.....2....112.......  #    4 sol.    36#039#  .......12....12....12............12...1.2....2....1......1..2...2......11..2.....  #    2 sol.    36#040#  .......12....12....12............12...1.2....2....1......2....1.2.1.....1.....2..  #    2 sol.    36#041#  .......12....12....12............12...1.2....2..1...........2.1.2...1...1..2.....  #    4 sol.    18#042#  .......12....12....12............12...1.2....2..1..........12.....2....112.......  #    2 sol.    36#043#  .......12....12....12............12...1.2....2..1..........12...2......11..2.....  #    2 sol.    36#044#  .......12....12....12............12...1.2....2..1........2....1.2...1...1.....2..  #    2 sol.    36#045#  .......12....12....12............12...1.2....2..1........2.1....2......11.....2..  #    2 sol.    18#046#  .......12....12....12............12...12......2.1...........2.11...2....2....1...  #    8 sol.    18#047#  .......12....12....12............12...12......2.1..........12..1...2....2.......1  #    4 sol.    36#048#  .......12....12....12............12...12......2.1.........2...11.....2..2....1...  #    4 sol.    36#049#  .......12....12....12............12...12.....2....1.........2.1.2.1.....1...2....  #    4 sol.     9 #050#  .......12....12....12............12...12.....2....1.......2...1.2.1.....1.....2..  #    2 sol.    36#051#  .......12....12....12............12...12.....2..1..........12...2......11...2....  #    2 sol.    36#052#  .......12....12....12............12...12.....2..1.........2...1.2...1...1.....2..  #    2 sol.    36#053#  .......12....12....12............12..2.1.....1...2..........2.1..12.....2....1...  #    4 sol.     9 #054#  .......12....12....12............12..2.1.....1...2.........12.....2....12.1......  #    2 sol.    18#055#  .......12....12....12............12..2.1.....1...2.........12....12.....2.......1  #    2 sol.    36#056#  .......12....12....12............12..2.1.....1..2..........12....1.2....2.......1  #    2 sol.    36#057#  .......12....12....12............12..2.1.....1..2.........2...1..1...2..2....1...  #    2 sol.    36#058#  .......12....12....12............12.1..2.....2..1..........12....1.2.....2......1  #    4 sol.    36#059#  .......12....12....12...........1.2...1.2.....2......1...1..2..1..2.....2.....1..  #    4 sol.     6 #060#  .......12....12....12...........1.2...1.2.....2......1...2..1..1.....2..2..1.....  #    4 sol.     6 #061#  .......12....12....12...........1.2...1.2.....2....1.....1..2..1..2.....2.......1  #    2 sol.    36#062#  .......12....12....12...........1.2...1.2.....2....1.....2....11.....2..2..1.....  #    2 sol.    36#063#  .......12....12....12...........1.2...1.2....2.....1.....1..2...2......11..2.....  #    2 sol.    18#064#  .......12....12....12...........1.2...1.2....2.....1.....2....1.2.1.....1.....2..  #    2 sol.    18#065#  .......12....12....12...........1.2...12......2....1......2...11.....2..2..1.....  #    2 sol.    18#066#  .......12....12....12...........1.2...12......2....1.....1..2..1...2....2.......1  #    2 sol.    18#067#  .......12....12....12...........1.2...12.....2.......1....2.1...2.1.....1.....2..  #    2 sol.    36#068#  .......12....12....12...........1.2...12.....2.......1...1..2...2....1..1...2....  #    2 sol.    36#069#  .......12....12....12...........1.2...12.....2.....1......2...1.2.1.....1.....2..  #    2 sol.    18#070#  .......12....12....12...........1.2...12.....2.....1.....1..2...2......11...2....  #    2 sol.    18#071#  .......12....12....12...........1.2.1...2....2.......1...1..2....12......2....1..  #    4 sol.    18#072#  .......12....12....12...........1.2.1...2....2.......1...2..1....1...2...2.1.....  #    4 sol.    18#073#  .......12....12....12...........1.2.1...2....2.....1.....1..2....12......2......1  #    2 sol.    18#074#  .......12....12....12...........1.2.1...2....2.....1.....2....1..1...2...2.1.....  #    2 sol.    36#075#  .......12....12....12...........12....12......2....1.....1...2.1...2....2.......1  #    2 sol.    18#076#  .......12....12....12...........12....12.....2.......1....2.1...2.1.....1......2.  #    2 sol.     6 #077#  .......12....12....12...........12....12.....2.......1...1...2..2....1..1...2....  #    2 sol.     6 #078#  .......12..1..2.....2..1..........21.1..2....2..1.........1.2.....2..1..12.......  #    8 sol.    18#079#  .......12..1..2.....2..1..........21.1..2....2..1.........1.2...2....1..1..2.....  #    8 sol.    18#080#  .......12..1..2.....2..1.........12..1..2....2..1...........2.1.2..1....1..2.....  #    8 sol.     9 #081#  .......12..1..2.....2..1.........12..1..2....2..1.........1.2.....2....112.......  #    4 sol.    36#082#  .......12..1..2.....2..1.........12..1..2....2..1.........1.2...2......11..2.....  #    4 sol.    36#083#  .......12..1..2.....2..1.......1..2....2....112...........2.1...1....2..2..1.....  #    4 sol.    36#084#  .......12..1..2.....2..1.......1..2....2....112..........12.....1....2..2.....1..  #    8 sol.    18#085#  .......12..1..2.....2..1.......1..2....2..1..12...........2...1.1....2..2..1.....  #    4 sol.    36#086#  .......12..1..2.....2..1.......1..2....2..1..12..........1..2...1..2....2.......1  #    4 sol.    36#087#  .......12..1..2.....2..1.......1..2....2..1..12..........12.....1....2..2.......1  #    4 sol.    18#088#  .......12..1..2.....2..1.......1..2..1.2.....2.......1....2.1.....1..2..12.......  #    4 sol.    36#089#  .......12..1..2.....2..1.......1..2..1.2.....2.......1....2.1...2.1.....1.....2..  #    4 sol.    18#090#  .......12..1..2.....2..1.......1..2..1.2.....2.......1...1..2...2....1..1...2....  #    4 sol.    18#091#  .......12..1..2.....2..1.......1..2..1.2.....2.....1......2...1.2.1.....1.....2..  #    4 sol.    18#092#  .......12..1..2.....2..1.......1..2..1.2.....2.....1.....1..2...2......11...2....  #    4 sol.    18#093#  .......12..1..2.....2.1..........12..1..2....2....1.........2.1.2.1.....1..2.....  #    4 sol.    18#094#  .......12..1..2.....2.1..........12..1..2....2....1......1..2.....2....112.......  #    4 sol.    36#095#  .......12..1..2.....2.1..........12..1..2....2....1......1..2...2......11..2.....  #    2 sol.    36#096#  .......12..1..2.....2.1..........12..1..2....2....1......2....1.2.1.....1.....2..  #    2 sol.    36#097#  .......12..1..2.....2.1..........12..1..2....2..1...........2.1.2...1...1..2.....  #    4 sol.    18#098#  .......12..1..2.....2.1..........12..1..2....2..1..........12.....2....112.......  #    2 sol.    36#099#  .......12..1..2.....2.1..........12..1..2....2..1..........12...2......11..2.....  #    2 sol.    36#100#  .......12..1..2.....2.1..........12..1..2....2..1........2....1.2...1...1.....2..  #    2 sol.    36#101#  .......12..1..2.....2.1..........12..1..2....2..1........2.1....2......11.....2..  #    2 sol.    36#102#  .......12..1..2.....2.1..........12..1.2.....2....1.........2.1.2.1.....1...2....  #    4 sol.    18#103#  .......12..1..2.....2.1..........12..1.2.....2....1.......2...1...1..2..12.......  #    2 sol.    36#104#  .......12..1..2.....2.1..........12..1.2.....2....1.......2...1.2.1.....1.....2..  #    2 sol.    36#105#  .......12..1..2.....2.1..........12..1.2.....2....1......1..2...2......11...2....  #    2 sol.    36#106#  .......12..1..2.....2.1..........12..1.2.....2....1......12.....2......11.....2..  #    2 sol.    36#107#  .......12..1..2.....2.1..........12..1.2.....2..1..........12......2...112.......  #    4 sol.    36#108#  .......12..1..2.....2.1..........12..1.2.....2..1..........12...2......11...2....  #    2 sol.    18#109#  .......12..1..2.....2.1..........12..1.2.....2..1.........2...1.2...1...1.....2..  #    2 sol.    18#110#  .......12..1..2.....2.1.........1.2.....2.1..12..........1..2...1.2.....2.......1  #    2 sol.    36#111#  .......12..1..2.....2.1.........1.2.....2.1..12..........2....1.1....2..2..1.....  #    2 sol.    36#112#  .......12..1..2.....2.1.........1.2....2....112...........2.1...1....2..2..1.....  #    2 sol.    36#113#  .......12..1..2.....2.1.........1.2....2....112..........1..2...1..2....2.....1..  #    2 sol.    36#114#  .......12..1..2.....2.1.........1.2....2....112..........12.....1....2..2.....1..  #    4 sol.    36#115#  .......12..1..2.....2.1.........1.2....2..1..12...........2...1.1....2..2..1.....  #    2 sol.    36#116#  .......12..1..2.....2.1.........1.2....2..1..12..........1..2...1..2....2.......1  #    2 sol.    36#117#  .......12..1..2.....2.1.........1.2....2..1..12..........12.....1....2..2.......1  #    2 sol.    36#118#  .......12..1..2.....2.1.........1.2..1..2....2.....1.....1..2...2......11..2.....  #    2 sol.    36#119#  .......12..1..2.....2.1.........1.2..1..2....2.....1.....2....1.2.1.....1.....2..  #    2 sol.    36#120#  .......12..1..2.....2.1.........1.2..1.2.....2.......1....2.1.....1..2..12.......  #    2 sol.    18#121#  .......12..1..2.....2.1.........1.2..1.2.....2.......1...1..2...2....1..1...2....  #    2 sol.    36#122#  .......12..1..2.....2.1.........1.2..1.2.....2.....1......2...1.2.1.....1.....2..  #    2 sol.    18#123#  .......12..1..2.....2.1.........1.2..1.2.....2.....1.....1..2...2......11...2....  #    2 sol.    36#124#  .......12..1..2.....2.1.........1.2..1.2.....2.....1.....12.....2......11.....2..  #    2 sol.    36#125#  .......12..1..2.....2.1.........12......2.1..12..........1...2..1.2.....2.......1  #    4 sol.    18#126#  .......12..1..2.....2.1.........12.....2....112...........2.1...1.....2.2..1.....  #    2 sol.    36#127#  .......12..1..2.....2.1.........12.....2....112..........12.....1.....2.2.....1..  #    2 sol.    36#128#  .......12..1..2.....2.1.........12...1..2....2.......1...1...2..2....1..1..2.....  #    2 sol.    36#129#  .......12..1..2.....2.1.........12...1..2....2.......1...2..1...2.1.....1......2.  #    2 sol.    36#130#  .......12..1..2.....2.1.........12...1.2.....2.......1....2.1...2.1.....1......2.  #    2 sol.    18#131#  .......12..1..2.....2.1.........12...1.2.....2.......1...1...2..2....1..1...2....  #    2 sol.    36#132#  .......12..1..2.....2.1.........12...1.2.....2.......1...12.....2....1..1......2.  #    2 sol.    36#133#  .......12..1..2.....2.1........21....1.....2.2.....1.....1..2...2......11..2.....  #    4 sol.    18#134#  .......12..1..2.....2.1........21....1.....2.2.....1.....2....1.2.1.....1.....2..  #    4 sol.    18#135#  .......12..1..2.....2.1........21....1....2..2.....1.....1...2....2....112.......  #    8 sol.     3 #136#  .......12..1..2.....2.1.......1...2..1..2....2.....1.....2....1.2...1...1.....2..  #    2 sol.    18#137#  .......12..1..2.....2.1.......1...2..1..2....2.....1.....2.1....2......11.....2..  #    2 sol.    36#138#  .......12..1..2.....2.1.......1..2...1..2....2.......1...2..1...2...1...1......2.  #    2 sol.    18#139#  .......12..1..2.....2.1.......1..2...1..2....2.......1...2.1....2....1..1......2.  #    2 sol.    36#140#  .......12..1..2.....2.1.......12.....1.....2.2.....1.....2.1....2......11.....2..  #    4 sol.    18#141#  .......12..1..2.....2.1.......12.....1....2..2.......1...2.1....2....1..1......2.  #    4 sol.    18#142#  .......12..1..2....2..1..........12...2..1....1..2.......1..2..1..2.....2.......1  #    4 sol.     9 #143#  .......12..1..2....2..1..........12...2..1....1.2.........2...11.....2..2..1.....  #    2 sol.    36#144#  .......12..1..2....2..1..........12...2..1...1...2.......1..2...1.2.....2.......1  #    2 sol.    36#145#  .......12..1..2....2..1..........12...2..1...1..2...........2.1.1..2....2..1.....  #    4 sol.     6 #146#  .......12..1..2....2..1..........12...2..1...1..2.........2...1...1..2..21.......  #    2 sol.    18#147#  .......12..1..2....2..1..........12...2..1...1..2.........2...1.1....2..2..1.....  #    2 sol.    36#148#  .......12..1..2....2..1..........12...2..1...1..2........12.....1....2..2.......1  #    2 sol.    18#149#  .......12..1..2....2..1..........12...21.....1...2..........2.1.1.2.....2....1...  #    4 sol.     3 #150#  .......12..1..2....2..1..........12...21.....1...2.........12.....2....121.......  #    2 sol.    18#151#  .......12..1..2....2..1..........12...21.....1...2.........12...1.2.....2.......1  #    2 sol.    18#152#  .......12..1..2....2..1..........12..1.2.....2....1.........2.1..21.....1...2....  #    4 sol.     3 #153#  .......12..1..2....2..1..........12..1.2.....2....1.......2...1...1..2..1.2......  #    2 sol.    18#154#  .......12..1..2....2..1..........12..1.2.....2....1.......2...1..21.....1.....2..  #    2 sol.    18#155#  .......12..1..2....2..1.........1.2.....2.1..1.2.........1..2...1.2.....2.......1  #    2 sol.    18#156#  .......12..1..2....2..1.........1.2.....2.1..1.2.........2....1.1....2..2..1.....  #    2 sol.    18#157#  .......12..1..2....2..1.........1.2.....2.1..21..........1..2....2.....11..2.....  #    2 sol.    36#158#  .......12..1..2....2..1.........1.2....2..1..1.2..........2...1.1....2..2..1.....  #    2 sol.    18#159#  .......12..1..2....2..1.........1.2....2..1..1.2.........1..2...1..2....2.......1  #    2 sol.    18#160#  .......12..1..2....2..1.........1.2....2..1..1.2.........12.....1....2..2.......1  #    2 sol.    18#161#  .......12..1..2....2..1.........1.2....2..1..21...........2...1..21.....1.....2..  #    2 sol.    36#162#  .......12..1..2....2..1.........1.2....2..1..21..........1..2....2.....11...2....  #    2 sol.    36#163#  .......12..1..2....2..1.........1.2...2...1...1.2.........2...11.....2..2..1.....  #    2 sol.    18#164#  .......12..1..2....2..1.........1.2...2...1...1.2........1..2..1...2....2.......1  #    2 sol.    36#165#  .......12..1..2....2..1.........1.2...2...1..1...2.......1..2...1.2.....2.......1  #    2 sol.    18#166#  .......12..1..2....2..1.........1.2...2...1..1..2.........2...1.1....2..2..1.....  #    2 sol.    18#167#  .......12..1..2....2..1.........1.2...2...1..1..2........1..2...1..2....2.......1  #    2 sol.    36#168#  .......12..1..2....2..1.........1.2..1..2....2.....1.....1..2....2.....11..2.....  #    2 sol.    36#169#  .......12..1..2....2..1.........1.2..1..2....2.....1.....2....1..21.....1.....2..  #    2 sol.    18#170#  .......12..1..2....2..1.........1.2..1.2.....2.....1......2...1..21.....1.....2..  #    2 sol.    18#171#  .......12..1..2....2..1.........1.2..1.2.....2.....1.....1..2....2.....11...2....  #    2 sol.    18#172#  .......12..1..2....2..1.........12.....2....121...........2.1....21.....1......2.  #    2 sol.    18#173#  .......12..1..2....2..1.........12.....2....121..........1...2...2...1..1...2....  #    2 sol.    18#174#  .......12..1..2....2..1.........12.....2....121..........12......2...1..1......2.  #    2 sol.     6 #175#  .......12..1..2....2..1.........12...1..2....2.......1...1...2...2...1..1..2.....  #    2 sol.    18#176#  .......12..1..2....2..1.........12...1.2.....2.......1....2.1....21.....1......2.  #    2 sol.    18#177#  .......12..1..2....2..1.........12...1.2.....2.......1...1...2...2...1..1...2....  #    2 sol.    18#178#  .......12..1..2....2..1.......1...2..1..2....2.....1.....2....1..2..1...1.....2..  #    2 sol.    18#179#  .....1..2..2....1..1..2.........2..1..1...2..2..1.........1..2..2....1..1..2.....  #    2 sol.     9 #180#  .....1..2..2....1..1..2.........21...2.1.....1......2.....1.2....12.....2.......1  #    2 sol.     6 #181#  .....1..2..2....1..1..2........1.2....12.....2.......1...1...2..2....1..1....2...  #    2 sol.     1                                                                                                        ------                                                                                                       4493     `

The number of isomorphs explains the frequency chart.

Given a full box with 9 clues these are the average solution rates for the 2-rookeries.
Code: Select all
`16 perm  - 2400000000 - 2600000000 sol.                                                                   8 perm  - 2100000000 - 2600000000 sol.                                                                   4 perm  - 1370000000 - 1900000000 sol.                                                                   2 perm  - 0940000000 - 1300000000 sol.`

Here are the different 2-rookeries [all 2-perm] in the SFB grid
Code: Select all
`1    # .......12....12....12............12...1.2....2..1........2.1....2......11.....2..#  045    181    # .......12....12....12...........12....12.....2.......1....2.1...2.1.....1......2.#  076     61    # .......12..1..2....2..1..........12...21.....1...2.........12...1.2.....2.......1#  151    181    # .......12..1..2....2..1..........12..1.2.....2....1.......2...1..21.....1.....2..#  154    181    # .......12..1..2....2..1.........1.2....2..1..21...........2...1..21.....1.....2..#  161    362    # .......12....12....12...........1.2...12.....2.....1......2...1.2.1.....1.....2..#  069    183    # .......12....12....12............12....12....2.1...........12...2......11..2.....#  030    183    # .......12....12....12...........1.2...12.....2.....1.....1..2...2......11...2....#  070    183    # .......12....12....12...........1.2.1...2....2.....1.....1..2....12......2......1#  073    183    # .......12....12....12...........12....12......2....1.....1...2.1...2....2.......1#  075    184    # .......12....12....12............12...1.2....2..1..........12.....2....112.......#  042    364    # .......12....12....12...........1.2...1.2.....2....1.....1..2..1..2.....2.......1#  061    364    # .......12....12....12...........1.2...1.2.....2....1.....2....11.....2..2..1.....#  062    365    # .......12....12....12...........1.2.1...2....2.....1.....2....1..1...2...2.1.....#  074    36-- 36 `

C
coloin

Posts: 1666
Joined: 05 May 2005

### Re: Structures of the solution grid

181 essentially different 2-rookeries ? I only have 170
and 92053 3-rookeries
how many 4-rookeries ? 5-rookeries ?

1 equivalence-class of 1-rookeries
170 equivalence classes of 2-rookeries
92053 equivalence-classes of 3-rookeries
...
1 equivalence-class of 8-rookeries
1 equivalence-class of 9-rookeries

they are equivalent when they can be mapped into each other by our symmetrygroup
of 6^8*2 elements (no permutation of symbols)

a k-rookery just consists of k*9 cells of an 9*9 square such that there are k cells from the
k-rookery in each row,column,block

so, in the above list it doesn't matter where there are 1s or 2s, we just have "filled" and "empty"
dukuso

Posts: 479
Joined: 25 June 2005

### Re: Structures of the solution grid

1 5472170387 [non-automorphic - 1 grid]
2 548449
3 7336
4 2826
6 1257
8 29
9 42
12 92
18 85
27 2
36 15 [including duk15]
54 11
72 2
108 3
162 1
648 1 [MC]

Condor wrote:T(0) = 1
T(1) = 46656 = 6^6
T(2) = 838501632 = 6^6 * 2^2 * 4493
T(3) = 5196557037312 = 6^6 * 2^2 * 3433 * 8111
T(4) = 9631742544322560 = 6^6 * 2^7 * 3 * 5 * 19 * 5659037

T(9) = 6670903752021072936960 = 6^6 * 2^14 * 3^2 * 5 * 7 * 27704267971

Red Ed elucidated the the 3-rookery number
there are 259272 essentially-different 3-rookeries.

fruitless-sudoku-discoveries-t1846-30.html

A k-rookery is defined by the cells given by k numbers. In general in each sudoku grid there is k!/9!(9-k)! k-rookeries. The contents of the rookery don't matter.

sets of templates
dukuso

Posts: 479
Joined: 25 June 2005

### Re: Structures of the solution grid

dukuso wrote:how many 4-rookeries ? 5-rookeries ?

Shouldn't there be exactly as many 5-rookeries as there is 4-rookeries?

RW
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

### Re: Structures of the solution grid

yes
dukuso

Posts: 479
Joined: 25 June 2005

### Re: Structures of the solution grid

So..... 181 different 2-rookeries.
dukuso says there are 170 2-rookery templates.

this means 11 different 2-rookery templates have two ways to insert a 2-rookery.

Solution Grids in U-Space (Unavoidable Sets)

has this breakdown of the 2-rookeries - and many worthy diagams relevant to this thread
Code: Select all
`1814+412+610+810+4+48+6+46+6+66+4+4+4 `

C
coloin

Posts: 1666
Joined: 05 May 2005

### Re: Structures of the solution grid

2-rookeries containing
this sub-2-rookery (rookery-band)

Code: Select all
`+---+---+---+|1..|2..|...||2..|...|1..||...|1..|2..|+---+---+---+`

will have 2 "solutions", you can exchange
1 and 2 to get a different 2-sudoku

so

Code: Select all
`+---+---+---+|x..|x..|...||x..|...|x..||...|x..|x..|+---+---+---+`

is an unavoidable set

you need 2 clues to solve such a 2 rookery uniquely.

making rookery-puzzles ...
how many clues are needed (minimum) to solve a 3-rookery,4-rookery,... ?

has it been done here ?

-----edit----------
checking gordon's 17s, the minimum numbers of clues required to uniquely
solve a 1,2,..,9 rookery are 0,1,2,3,5,7,10,13,17

so we can solve a 6-rookery with 7 clues !
Check all ~90000 6 rookeries , put 7 clues from {1,..,6} inside it so to solve it and 10 clues
from {7,8,9} outside it and check whether you get a 17 ?
dukuso

Posts: 479
Joined: 25 June 2005

### Re: Structures of the solution grid

dukuso wrote:checking gordon's 17s, the minimum numbers of clues required to uniquely
solve a 1,2,..,9 rookery are 0,1,2,3,5,7,10,13,17

so we can solve a 6-rookery with 7 clues !
Check all ~90000 6 rookeries , put 7 clues from {1,..,6} inside it so to solve it and 10 clues
from {7,8,9} outside it and check whether you get a 17 ?

this sounds like a plan for re-energizing the search for all 17s
random search seems to have taken all the low hanging fruit
gsf
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Location: NJ USA

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