## Sudoku space

Everything about Sudoku that doesn't fit in one of the other sections
ravel - can we look at the easy one first and see how many 17puzzles [if any] have the 9/9 template ?

Wont they all have at least these 10 equivalent clues ?
Code: Select all
`+---+---+---+|1..|...|...||...|2..|...||...|...|3..|+---+---+---+|.4.|.1.|...||...|.5.|...||...|...|.6.|+---+---+---+|..7|...|...||...|..8|...||...|...|..9|+---+---+---+i.e. are there always 2 similar clues in diagonal boxes - probably.?`

I suppose untill I find any of these 17s I cant proceed, and im kind of thinking that because of the symmetry there wont be a 17 clue puzzle.

Perhaps the 8/9 template can be subdivided easily into 4 groups.

Code: Select all
`no 9th clue, 1 empty box9th clue present, 1 empty box.no 9th clue no empty boxes9th clue present no empty boxes`

3 clues in a box options, or perhaps single clue boxes would also classify furthur.

Not sure how to go with this. Havard is confident he can do a 4-on search on a 13 template to get 17puzzles !

C
coloin

Posts: 1864
Joined: 05 May 2005

coloin wrote:ravel - can we look at the easy one first and see how many 17puzzles [if any] have the 9/9 template ?
Sorry for the late answer, i was off for some days.
I checked it for almost 16000 puzzles in Gordon's old list.

Code: Select all
`...5...1.7.6......4......8.....9.2......6...3.1..........8.4...2.....6.....1..7..`

The rest:
Code: Select all
`8/9:  3267/9: 78516/9: 77785/9:   42 `
I.e. 42 are not equivalent to any such 6-template, the first being this one:
Code: Select all
`.......421..7............8.6..3..5...4.....2....1.........6.1.5.9..4..........3..`

To get a feeling, how much common clues the puzzles have on average, these are the results of comparing the first 7 (and partly the 8th) with 16000 others:
Code: Select all
`16:     815:     614:    1713:   15212:  221411: 2000010: 60485 9: 16993 8:     6`
I am too busy now to work on your other questions.
ravel

Posts: 998
Joined: 21 February 2006

Thanks ravel

Red Ed has just given me this other one
Code: Select all
`...5...1.7.6......4......8.....9.2......6...3.1..........8.4...2.....6.....1..7..7....1......4.8...6.....2...3..6....2...9...........1...8...4....1..5.........7.6`

There isnt much concordance between the two puzzles on the 9/9 side.

Swapped and relabelled
Code: Select all
`1....6......2.7...9.....3...4..9....3...5...........6...7...2....6..8.........1.91...7.......2......8....3...4...3.....9.5...7.......6...7...........84..5.....8.91...........2...........3...4...........5...........6...7...........8...........9`

Given the 9/9 template the number of different ways to add 8 more clues
[approx [[70*6]^8] /8!] is reduced by the 6^8 isomorphic ways to portray the puzzle. Which might just make a difference.

This also goes on to an extent in the 8/9 template.

Thus if you want to search for 17 puzzles, the clues to remove in a 3off/3on search should include one of the clues which contribute to the 8/9 or two clues from the 9/9 template.

Thanks for the data on the spread, the 5/9 template puzzles do exist !

C
coloin

Posts: 1864
Joined: 05 May 2005

coloin wrote:Red Ed has just given me this other one
Code: Select all
`...5...1.7.6......4......8.....9.2......6...3.1..........8.4...2.....6.....1..7..7....1......4.8...6.....2...3..6....2...9...........1...8...4....1..5.........7.6`

There isnt much concordance between the two puzzles on the 9/9 side.

Swapped and relabelled
Code: Select all
`1....6......2.7...9.....3...4..9....3...5...........6...7...2....6..8.........1.91...7.......2......8....3...4...3.....9.5...7.......6...7...........84..5.....8.91...........2...........3...4...........5...........6...7...........8...........9`

I'm confused as to which puzzle is the one and which is the other
and how the first two puzzles relate to the second three puzzles
gsf
2014 Supporter

Posts: 7306
Joined: 21 September 2005
Location: NJ USA

The two puzzles have been scrambled to show the [1-9] 9/9-template. I also tried to line up the "three clues in a box" and "single box clues".

They could not be much more different , one has in-box-clues which are in line and the other has clues which tend to be diagonal in-box.

I dont really know where we can go with this, I cant see how it would be possible to standardize other puzzles by fixing a template and then finding a common representation of a proportion of the remaining clues.....

C.
coloin

Posts: 1864
Joined: 05 May 2005

i have an intresting idea on how to define sudoku space.

instead of looking at variations between puzzles.

we know the exact number of solutions to all puzzles.

u can catogrize the solution puzzles variances if u want.

but my idea is:

if a given solution from a puzzle can remove n many clues forming x many diffrent puzzles with the same solution.

the only variation between the puzzles produced from the oringinal solutions is the n clues removed.

the greater the distance from the starting position of solved grid translates to a grid with many wholes (more missing clues = further away)
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1088
Joined: 05 September 2006

I think I know what you are saying........except that it only applies to puzzles of the same grid.

We are dealing with different girds, and 40,000 all different puzzles each with 9! *6^8*2 isomorphs. We are trying to make some attempt at grouping puzzles which are "close" but because of the variations its impossible to filter them easily.

So if we squash the clues into the first part of the "text string" or ? max lex we will match up the holes to an extent and concordance will be improved.

It might be possible to pick out the most common partial grid template of 30 clues [or less] [this would mean 51 common holes] and restrict the clue insertions to these template clues.

Might work !

C
coloin

Posts: 1864
Joined: 05 May 2005

well if we know its solution we can group puzzles in relation to the solution.

yes i know each puzzle could have an issomorph to the some other puzzle.
if they are then they all came from the same orbit. what im going to term any given puzzle from the same soloar system where a system is diverised by it unique solution.

why not group similar puzzles (simial via solution) that would be your first distance relation ship.

how far each puzzle from the same issomproh solution is from the completed grid = orbital distance.

the next would be then the variation between completed solutions ( soloar system)

. say only a specific set of clues in any given solution is can modified to create the unique grid. what are those variations?

u could possible devise a sytem to find them easily by canalizing the solution.

marking each subseeding line and utilze a cordinal system to indicate variations in postions for each line.

so then u would have the total number of unique solutions count = # of soloar systems.

you could then classify the unique variations of the solutions compared to each other easier then u can an uncompleted puzzles.
becasuse there is no holes to catach.

(use a brute force cracking programs to obtain the solution for comparision)

thus i would form a space line.
as

solutions A compared to Solution B would = D = distrubution variation

puzzle a - 1 clue its orbital path around soloar sytem A distance = 1 - orbit( 1 clue missing from completeing)

puzzle a compared to solution b = Distance in orbit + D (variation of both solutions).

puzzle a compared to a puzzle b = distance in orbits (orbitts the same = 0) +D ( variation of distrution of solution.)

open question.

is it possible to propgate the number of possible "17" clues for each individual solution ?

if u can. could you say do that to a few hundred solutions creating a statistical bell curve and extraopolate the total number of 17 grids? based on said statistics".
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1088
Joined: 05 September 2006

Well interesting theory, StrmCkr.....but no answers !.

After a week off - a new slant might be easier ....

Lets continue to stick to the 17 clues sudoku space....and we have a reborn expert in representation theory of finite groups in our midst .

Havard is finding more and more 17 clue puzzles, with extensive search methods...

But we are dealing with finite groups here, in particular the finite group which includes all the 17 clue puzzles which share at least one 15 clue subpuzzle with another 15 clue subpuzzle from another 17 clue puzzle.[common-15 group]

The other as yet not-finite group is the number of [new] 17s which are in the remote-15 group

Im not sure even of the ratio in these two groups

When we start thinking about 14 clue subpuzzles the numbers are much bigger, and i wonder if all 17 clue puzzles will have a 3off/3on "cousin"

Indeed just how many essentially different [minimal] subpuzzles are there ? Well there are many and at least this many.
Code: Select all
`number of clues / number of different subpuzzles1                                12                                73                                704                                8675                                11269                              6                                1499357                                161124358                                -91011121314                              20 million from 1 27-clue puzzle alone !`

Here are the 7 with 2 clues
Code: Select all
`+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+   |12.|...|...| |1..|...|...| |1..|2..|...| |1..|...|...|   |...|...|...| |.2.|...|...| |...|...|...| |...|1..|...|   |...|...|...| |...|...|...| |...|...|...| |...|...|...|   +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+   |...|...|...| |...|...|...| |...|...|...| |...|...|...|   |...|...|...| |...|...|...| |...|...|...| |...|...|...|   |...|...|...| |...|...|...| |...|...|...| |...|...|...|   +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+   |...|...|...| |...|...|...| |...|...|...| |...|...|...|   |...|...|...| |...|...|...| |...|...|...| |...|...|...|   |...|...|...| |...|...|...| |...|...|...| |...|...|...|   +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+                                                             +---+---+---+ +---+---+---+ +---+---+---+                 |1..|...|...| |1..|...|...| |1..|...|...|                 |...|.2.|...| |...|...|...| |...|...|...|                 |...|...|...| |...|...|...| |...|...|...|                 +---+---+---+ +---+---+---+ +---+---+---+                 |...|...|...| |...|1..|...| |...|2..|...|                 |...|...|...| |...|...|...| |...|...|...|                 |...|...|...| |...|...|...| |...|...|...|                 +---+---+---+ +---+---+---+ +---+---+---+                 |...|...|...| |...|...|...| |...|...|...|                 |...|...|...| |...|...|...| |...|...|...|                 |...|...|...| |...|...|...| |...|...|...|                 +---+---+---+ +---+---+---+ +---+---+---+`

C
coloin

Posts: 1864
Joined: 05 May 2005

mathmatically each individual placement of any given singles create an obtainable comparisin. using hueristics.

check to see if any of the 9 completed patterns match the other 9.

there is only 508 possible hueristic grids for each completed set of numbers 1-9
blah still alot...

ignoring spiatial morphism ie possible positions in the same quad.
placement of n single - possibilties with limits concerned.

0 single: 1 way:
1 single: 9 possiblites:
2 singles: 36 possiblities
3 singles : 84 possible
4 singles : 126 possible.
5 singles: 126 possible
6 singles : 84 possible
7 singles : 36 possible
8 : 9 possiblities
9 : 1 way.

add all the pattern placements and u get : 512 hueristics for each completed "grid" for each individual number.

i noticed that when comparing all numbers the hueristics for each individual number creates
subsets for each non-completed hurristics limiting each other expressing singles, box lines etc.

thus i stumbled onto the conflict of placement for my massive mathmatical equation that ive been working on to solve cells mathmatically.

blah long daunting task to prove how and y 17 is the lowest number.

x^(81-N-P-C)
as (x approches 81)

where x = (1:9*9)
where N = # of placed canadiates of that type(1:9)
P = (# of boxes removed + single cells in line of sight, compared to the value of x selected)
C = constraints that generate 0 placemetents of x in any sub set of equations as x must be valid in all 9 subsets. (when looking specifically at x)

my attempt is to prove that all arangments of hueristics can't limit the pattern chain of 2 numbers without the 17th number or beyond using conflict of placements. based purely on math of limits.

finding all the conflict of placemetns accross all huresitics in incremental number of placed numbers and possible arangements. is taking a while.
Last edited by StrmCkr on Tue Dec 18, 2007 7:54 pm, edited 1 time in total.
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1088
Joined: 05 September 2006

Code: Select all
`1 single:  9 possiblites: 2 singles: 36 possiblities 3 singles: 84 possible 4 singles: 125 possiblities. 5 singles: 125 possiblities 6 singles: 84 possibilties 7 singles: 36 singles 8 : 9 possiblities 9 : self solves. `

For 1 clue - essentially the 9 possiblities are equivalent ~ 1
for 4 and 5 clues - perhaps it is 126 possibilities
not sure about 8 clues - you might well be right with 9 possibilities

from the past
Red Ed wrote:T(0) = 1 way of laying down no digits at all
T(1) = 46656 ways of laying down all the 1s
T(2) = 838501632 ways of laying down all the 1s,2s
T(3) = 5196557037312 ways of laying down all the 1s,2s,3s
T(4) = 9631742544322560 ways of laying down all the 1s,2s,3s,4s

T(8) = 6670903752021072936960 ways of laying down all the 1s-8s
T(9) = 6670903752021072936960 ways of laying down all the 1s-9s

Number of 1- rookeries = 46656 [all isomorphically equivalent]
Number of essentially different 2 -rookeries, dukuso mentioned "170"

although i randomly looked at a few using gsf 's -go0.0 function on a collection of 2-rookeries

total number in initial = 2708
here is the most common 2-rookery - occurred 42 times out of 2708
Code: Select all
`+---+---+---+|...|...|.12||...|.12|...||.12|...|...|+---+---+---+|...|...|12.||..1|2..|...||.2.|1..|...|+---+---+---+|...|..1|2..||1..|.2.|...||2..|...|..1|+---+---+---+  6 clue [B145] and 12 clue[B236789] unavoidable sets`

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

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

Code: Select all
`T(0) = 1 way of laying down no digits at all                     > 1 empty grid T(1) = 46656 ways of laying down all the 1s                      > 1 essentially different 1-rookeryT(2) = 838501632 ways of laying down all the 1s,2s               > 181 essentially different 2-rookeriesT(3) = 5196557037312 ways of laying down all the 1s,2s,3s        > ?T(4) = 9631742544322560 ways of laying down all the 1s,2s,3s,4s  >T(8) = 6670903752021072936960 ways of laying down all the 1s-8s  > 5472730538 essentially different 8-rookeriesT(9) = 6670903752021072936960 ways of laying down all the 1s-9s  > 5472730538 essentially different grids`

Ive looked at a few other subpuzzle patterns, and will post a few more, interestingly fewer than you would expect !

Code: Select all
`blah long daunting task to prove how and y 17 is the lowest number`
indeed, good luck.
Last edited by coloin on Fri Dec 14, 2007 8:36 pm, edited 1 time in total.
coloin

Posts: 1864
Joined: 05 May 2005

This template...
Code: Select all
`+---+---+---+    +---+---+---+                    |1..|...|...|    |1..|...|...|                     |...|1..|...|    |...|2..|...|       |...|...|1..|    |...|...|3..|              +---+---+---+    +---+---+---+                       |.1.|...|...|    |.4.|...|...|               |...|.1.|...| to |...|.5.|...|      |...|...|.1.|    |...|...|.6.|                   +---+---+---+    +---+---+---+                        |..1|...|...|    |..7|...|...|                     |...|..1|...|    |...|..8|...|      |...|...|..1|    |...|...|..9|      +---+---+---+    +---+---+---+       9^9 = 387420489 `

The number of ways to fill in these this template or rookery [with any clue] is 387420489.

Looking at my random 2708 templates again
only 331 essentially distinct lines with 64 unique lines [so almost certainly there are slightly more than 331]
the most common duplicate @66 out of 2708
Code: Select all
`+---+---+---+|...|...|..1||...|..1|...||..2|...|...|+---+---+---+|...|...|.2.||...|.3.|...||.4.|...|...|+---+---+---+|...|...|5..||...|6..|...||7..|...|...|+---+---+---+most common [2.5%]template`

EDIT, searching a bit harder I found 439 essentially different 9-clue templates.

the 8-clue template, still 9^9 possibilities
Code: Select all
`+---+---+---+|1..|...|...||...|1..|...||...|...|1..|+---+---+---+|.1.|...|...||...|.1.|...||...|...|.1.|+---+---+---+|..1|...|...||...|..1|...||...|...|...|+---+---+---+  going on for 624 essentially different 8-templates `

the 7-clue template, 9^7*18*2 possibilities
Code: Select all
`+---+---+---+|...|...|...||...|1..|...||...|...|1..|+---+---+---+|.1.|...|...||...|.1.|...||...|...|.1.|+---+---+---+|..1|...|...||...|..1|...||...|...|...|+---+---+---+  273 of this type [type A]+---+---+---+|...|...|...||...|...|...||...|...|1..|+---+---+---+|.1.|...|...||...|.1.|...||...|...|.1.|+---+---+---+|..1|...|...||...|..1|...||...|...|..1|+---+---+---+   273 of this type [type B]`

Final tally
Code: Select all
`9-clue templates - 4398-clue templates - 6247-clue templates - 5466-clue templates - 3875-clue templates - 1504-clue templates -  553-clue templates -  152-clue templates -   41-clue templates -   1`

The lesser templates are there for completeness.

Looking at a few 17-puzzles its pretty hard to find any 9-clue templates within them and the occaisional 8-clue is found. I have said before that I havnt found a puzzle which hasnt got a 5-clue template [with different clues] in B15689.

EDIT
Code: Select all
`+---+---+---+|764|895|...||389|712|...||215|463|...|+---+---+---+|472|...|518||596|...|342||138|...|697|+---+---+---+|...|951|874||...|387|265||...|246|931|+---+---+---+`

Puzzles from this pattern with 3 empty boxes wont have clues in the equiv. ofb B1B5B6B8B9
Last edited by coloin on Sat Sep 13, 2008 3:18 am, edited 1 time in total.
coloin

Posts: 1864
Joined: 05 May 2005

im sugesting a single sets of possible template(grid)
for only a single digit as the evaluation.

i used that same number only. and place it 1- 9 times.

im looking at conflict created by each template as a whole/ partial.

compared to a possible/valid template of a second clue. this could possibly be my link to a conflict of placements algorithim.

as it creates a 0 solution grid, or perhaps a mutilple solution grid where as the vaildidty of both clues remain selectable in the same space. it also keenly expresses where the singles are. no choice left but a single clue when comparing completed or partial second digit hueristical grids.

is your tallying for each given set of clues in combination take into accout variance of each quad?

i didn't when i manually combined each grid once then removed all duplicated notations.

say the ability for the for each 1 to appear 1^9 places per quad.

then each significant space is reduced buy 9 + 12 squares notes if the successive placement falls into its line of sight, ajasent 4 quads.

i have this mapped out in excelle - noting mathmatical errors due to overlapping line of sites if i assign that basic evaluation of limitations noted above (9+12) to to each successive clue.

since i have that mapped out already i didn't bother to check for variance in a single quad.

thats how i derived out 36 possible ways to mutate the first 2 digist.

which means when slecting 7 numbers u have the reverse logic 2 spaces left unselected out of 9 quads = 36 ways

same effect happens with 4 and 5 numbers = 125 combinations
and 3 / 6 = 84 combinations

1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+3
2+4
2+6
2+7
2+8
2+9
3+4
3+5
3+6
3+7
3+8
3+9
4+5
4+6
4+7
4+8
4+9
5+6
5+7
5+8
5+9
6+7
6+8
6+9
7+8
7+9
8+9
= 36

I manually mapped out each squance for each given number uptill the 4th number then realised the not selecting 4 numbers is the same selecting 4 numbers and created a curve for all combinations of single spaceing. and lastly i ignored the permuation of each indivudual quad ,

as the line of site created a complex problem where given 4 cells hold line of sites and 4 didn't. so i didn;t want to manual tally that permuations count.
Last edited by StrmCkr on Tue Dec 18, 2007 6:09 am, edited 1 time in total.
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1088
Joined: 05 September 2006

Code: Select all
`curious to your numbers.`

I believe this the way to count them.

Starting from a single solution grid........

Ways to pick one clue value = 9 [but they are all the same] ~1
Ways to pick two clue values = 9*8/2 = 36
Ways to pick three clue values = 9*8*7 / 3*2 = 84
Ways to pick four clue values = 9*8*7*6 / 4*3*2 = 126 [not 125]
Ways to pick five clue values = 9*8*7*6*5 / 2*3*4*5 = 126 [not 125]
Ways to pick six clue values = 84
Ways to pick seven clue values = 36
Ways to pick eight clue values = 9

The nine clues of one value "single digit" has been termed a 1-rookery

Code: Select all
`is your tallying for each given set of clues in combination take into accout variance of each quad? `

Yes
In my previous post I counted up the 181 different ways a 1-rookery can combine with another 1-rookery [to make a 2-rookery] A bit of a reduction from 838501632.

I think you have only 34 in your list [1+8,1+9 missing]

There are 36 2-rookeries in a complete grid
There are 84 3-rookeries in a complete grid
there are 126 4-rookeries in a complete grid......etc

It would be a more prolonged exercise to count the total number of distinct 3-rookeries however.

Code: Select all
`basic evaluation of limitations noted above (9+12) to to each successive clue.`

Is this the same as the pencil mark reductions by a single clue ?
This is maximally 9-within a 9x9 box, and 2x6 -for the line interactions in the horizontal and vertical chutes.

So if you have 16 clues you just cant do what we know we can do with 17 clues.........
So, how does 17 clues manage it then !!! .
coloin

Posts: 1864
Joined: 05 May 2005

coloin wrote:It would be a more prolonged exercise to count the total number of distinct 3-rookeries however.
Are you asking about the number of isomorphism classes among the T(3) = 5196557037312 3-rookeries? If so, the answer's 259272. Or, to use slightly more familiar language, there are 259272 essentially-different 3-rookeries.

(I only know this result but I used it way back when I was coding up my first unbiased solution grid generator.)
Red Ed

Posts: 633
Joined: 06 June 2005

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