## Extra feature

Everything about Sudoku that doesn't fit in one of the other sections
is it soo important to know how many essentially different
solutions there are ?
Doesn't Condor already think there are many ?
If you insist, I could examine these 2000+ sets ....
but I hope, it's not necessary/important
dukuso

Posts: 479
Joined: 25 June 2005

I thought Condor thinks that the grid, if you leave out the 8th and 9th number, is essentially unique. But you suggested it's not. I just want to know which is true. If that's unique then I'll save it as one of the most treasured discovery of sudoku maths. If it's not unique then please just pick any one random grid in your 2000+ sets and show us (I hope it's not too much of a trouble). I don't really need to know how many essentially different solutions there are, as long as it's not ONE (or TWO, maybe THREE...)! ^_^
udosuk

Posts: 2698
Joined: 17 July 2005

udosuk wrote:I thought Condor thinks that the grid, if you leave out the 8th and 9th number, is essentially unique. But you suggested it's not. I just want to know which is true. If that's unique then I'll save it as one of the most treasured discovery of sudoku maths. If it's not unique then please just pick any one random grid in your 2000+ sets and show us (I hope it's not too much of a trouble). I don't really need to know how many essentially different solutions there are, as long as it's not ONE (or TWO, maybe THREE...)! ^_^

OK, I can do that. Wait a moment...
Although I have to modify the program - usually it won't output
the sets - there are just too many. And then only the indices
of the sets are given.
I would be surprised if all the 2000+ sets were "essentially" the
same.
Hmm, you can often take the 18 cells of 2 symbols and rearrange them.
Let's see. You posted:

+ 2 3 4 8 7 6 + 9
4 + 6 2 9 3 8 7 +
7 8 9 + + 6 4 3 2
9 3 + 8 7 + 2 6 4
6 4 + 3 2 9 + 8 7
8 7 2 + 6 4 + 9 3
2 9 8 7 + + 3 4 6
+ 6 4 9 3 2 7 + 8
3 + 7 6 4 8 9 2 +

well, I can't spot any. Let me look up the 2000+ solutions ....
dukuso

Posts: 479
Joined: 25 June 2005

When I wrote the hypothesis, suddenly I remembered this Sudoku Queen and I thought the hypothesis must be wrong. Then I came to look at it again very carefully and found there were more than 2 chains. I really don't mean to make any joke about it. It is a great effort and I like the Sudoku Queen. Otherwise, I will give a different example.

Posts: 62
Joined: 15 July 2005

I redid the calculation and now I only have 24 sets,
from 4 symmetry-classes.
I don't know, what exactly I calculated with the 2916 above,
but the maximum was also 7 disjoint sets.
I guess, Udusuk knew all the time that it was wrong ;-)
(and Condor too ?!)

1 2 3 4 5 . 6 7 .
4 . 7 6 1 3 . 5 2
5 6 . . 7 2 1 3 4
. 7 2 5 3 4 . 6 1
3 . 4 1 2 6 5 . 7
6 1 5 . . 7 4 2 3
2 4 6 7 . 5 3 1 .
. 3 . 2 6 1 7 4 5
7 5 1 3 4 . 2 . 6

1 . 2 . 3 4 5 6 7
6 3 7 5 1 2 . 4 .
. 5 4 6 . 7 1 3 2
4 2 3 7 . . 6 5 1
7 . 6 1 2 5 3 . 4
5 1 . 3 4 6 2 7 .
3 4 5 2 7 . . 1 6
2 6 . 4 5 1 7 . 3
. 7 1 . 6 3 4 2 5

1 . 2 3 4 5 6 7 .
. 3 4 6 1 7 2 . 5
7 6 5 2 . . 1 4 3
5 2 7 . 3 4 . 6 1
3 4 . 1 7 6 . 5 2
6 1 . 5 2 . 7 3 4
2 5 6 . . 3 4 1 7
4 . 3 7 6 1 5 2 .
. 7 1 4 5 2 3 . 6

1 . 2 3 4 . 5 6 7
5 4 6 7 . 1 3 . 2
7 . 3 5 6 2 . 1 4
4 2 1 . 7 5 . 3 6
6 3 . . 2 4 1 7 5
. 7 5 1 3 6 2 4 .
2 1 4 6 5 . 7 . 3
. 6 7 2 . 3 4 5 1
3 5 . 4 1 7 6 2 .

1 2 3 . 4 5 6 . 7
6 . 5 7 . 1 2 4 3
4 7 . 6 2 3 5 . 1
2 5 . 3 1 . 7 6 4
3 1 7 4 6 . . 5 2
. 4 6 2 5 7 3 1 .
5 . 1 . 3 2 4 7 6
7 3 4 5 . 6 1 2 .
. 6 2 1 7 4 . 3 5

1 2 . 3 4 5 6 7 .
. 6 5 7 . 1 3 2 4
7 4 3 6 2 . 5 . 1
. 5 2 4 1 6 7 3 .
6 1 . . 7 3 4 5 2
3 7 4 . 5 2 . 1 6
5 . 1 2 6 7 . 4 3
2 3 6 5 . 4 1 . 7
4 . 7 1 3 . 2 6 5

1 . 2 3 4 5 6 7 .
. 4 5 6 7 1 3 . 2
6 7 3 2 . . 5 4 1
4 5 6 . 1 3 . 2 7
7 1 . 4 2 6 . 5 3
3 2 . 7 5 . 4 1 6
5 6 1 . . 2 7 3 4
2 . 7 5 3 4 1 6 .
. 3 4 1 6 7 2 . 5

1 2 3 4 5 . 6 . 7
. 4 . 6 7 3 1 5 2
6 5 7 1 . 2 4 3 .
7 3 2 . . 1 5 6 4
4 . 5 3 6 7 2 . 1
. 1 6 2 4 5 . 7 3
2 7 . . 1 6 3 4 5
5 . 1 7 3 4 . 2 6
3 6 4 5 2 . 7 1 .

1 2 3 4 5 6 7 . .
7 . 6 . 2 3 1 4 5
4 5 . 1 . 7 6 2 3
2 7 . 6 4 5 3 1 .
6 3 1 . 7 . 4 5 2
. 4 5 3 1 2 . 7 6
5 6 2 7 . 4 . 3 1
3 1 4 5 6 . 2 . 7
. . 7 2 3 1 5 6 4

1 2 3 4 . 5 6 . 7
. 7 5 6 2 . 1 4 3
4 6 . 1 7 3 . 2 5
2 3 . 5 4 . 7 1 6
5 . 1 7 3 6 4 . 2
7 4 6 . 1 2 3 5 .
3 5 2 . 6 4 . 7 1
. 1 4 3 5 7 2 6 .
6 . 7 2 . 1 5 3 4

1 2 3 4 . 5 6 . 7
. 7 5 6 3 2 1 4 .
4 6 . 1 7 . 5 3 2
. 3 2 5 4 . 7 1 6
5 . 1 7 2 6 4 . 3
7 4 6 . 1 3 . 2 5
3 5 . 2 6 4 . 7 1
2 1 4 . 5 7 3 6 .
6 . 7 3 . 1 2 5 4

1 2 3 4 5 . 6 . 7
. 7 5 6 . 3 1 4 2
4 6 . 1 7 2 . 5 3
3 . 2 . 4 5 7 1 6
. 5 1 7 2 6 4 3 .
7 4 6 3 1 . . 2 5
2 3 . . 6 4 5 7 1
5 1 4 2 . 7 3 6 .
6 . 7 5 3 1 2 . 4

1 . 2 3 4 5 6 . 7
. 7 4 6 . 2 1 3 5
3 6 5 1 7 . . 4 2
5 2 . . 3 4 7 1 6
. 4 1 7 2 6 3 5 .
7 3 6 5 1 . 2 . 4
4 5 . 2 6 3 . 7 1
2 1 3 4 . 7 5 6 .
6 . 7 . 5 1 4 2 3

1 2 3 4 5 . 6 7 .
4 . 6 2 7 3 5 1 .
7 5 . . 1 6 4 3 2
3 . 1 5 . 7 2 6 4
6 4 . 3 2 1 . 5 7
5 7 2 6 . 4 3 . 1
2 1 5 7 3 . . 4 6
. 3 4 1 6 2 7 . 5
. 6 7 . 4 5 1 2 3

1 . 2 3 4 5 6 . 7
3 5 7 6 . 2 4 1 .
6 4 . . 1 7 3 2 5
2 7 1 4 5 . . 6 3
. 3 5 2 6 1 7 4 .
4 . 6 . 7 3 2 5 1
7 1 . 5 2 6 . 3 4
. 2 3 1 . 4 5 7 6
5 6 4 7 3 . 1 . 2

1 . 2 3 . 4 5 6 7
5 4 7 . 6 2 3 1 .
3 6 . 5 1 7 . 2 4
2 7 1 . 4 3 . 5 6
6 . 4 2 5 1 7 . 3
. 3 5 6 7 . 2 4 1
7 1 . 4 2 . 6 3 5
. 2 6 1 3 5 4 7 .
4 5 3 7 . 6 1 . 2

1 . 2 3 . 4 5 6 7
. 4 7 6 5 2 3 1 .
6 5 3 . 1 7 . 2 4
2 7 1 . 4 3 6 5 .
5 . 4 2 6 1 7 . 3
3 6 . 5 7 . 2 4 1
7 1 . 4 2 6 . 3 5
. 2 5 1 3 . 4 7 6
4 3 6 7 . 5 1 . 2

1 . 2 . 3 4 5 6 7
6 4 7 5 . 2 3 1 .
5 3 . 6 1 7 . 2 4
2 7 1 3 4 . 6 . 5
. 5 4 2 6 1 7 3 .
3 6 . . 7 5 2 4 1
7 1 3 4 2 . . 5 6
. 2 5 1 . 6 4 7 3
4 . 6 7 5 3 1 . 2

. 1 2 3 4 5 6 . 7
3 . 6 1 7 2 4 5 .
5 4 7 . . 6 3 2 1
7 2 . 4 5 . 1 6 3
6 3 . 2 1 7 . 4 5
4 5 1 . 6 3 . 7 2
1 7 4 5 . . 2 3 6
. 6 3 7 2 1 5 . 4
2 . 5 6 3 4 7 1 .

. 1 2 3 4 5 6 . 7
6 . 5 7 1 2 3 4 .
7 4 3 . . 6 5 1 2
1 6 . 5 7 . 2 3 4
5 3 . 2 6 4 . 7 1
2 7 4 . 3 1 . 6 5
4 5 1 6 . . 7 2 3
. 2 7 4 5 3 1 . 6
3 . 6 1 2 7 4 5 .

. 1 2 . 3 4 5 6 7
. 7 3 5 1 6 2 . 4
6 5 4 2 7 . . 3 1
4 2 6 1 . 3 7 . 5
1 3 . 7 6 5 . 4 2
7 . 5 4 . 2 6 1 3
2 4 . . 5 1 3 7 6
3 . 1 6 2 7 4 5 .
5 6 7 3 4 . 1 2 .

. 1 2 3 4 5 . 6 7
4 6 5 7 . 1 3 2 .
7 . 3 . 6 2 5 4 1
. 5 1 2 7 4 6 3 .
6 3 4 5 1 . . 7 2
2 7 . 6 3 . 4 1 5
5 4 . 1 2 6 7 . 3
1 . 7 4 . 3 2 5 6
3 2 6 . 5 7 1 . 4

. 1 2 . 3 4 5 6 7
6 3 . 5 7 2 1 . 4
4 5 7 6 1 . . 2 3
7 2 . 4 5 3 6 1 .
1 . 3 2 6 7 4 . 5
5 6 4 1 . . 3 7 2
. 7 5 3 . 1 2 4 6
3 4 1 7 2 6 . 5 .
2 . 6 . 4 5 7 3 1

. . 1 2 3 4 5 6 7
3 4 7 5 6 . 2 . 1
5 6 2 1 . 7 . 3 4
. 7 5 3 4 2 . 1 6
6 3 4 . 1 . 7 5 2
2 1 . 6 7 5 3 4 .
7 5 . 4 . 1 6 2 3
1 . 6 . 2 3 4 7 5
4 2 3 7 5 6 1 . .
Last edited by dukuso on Fri Jul 22, 2005 2:55 pm, edited 1 time in total.
dukuso

Posts: 479
Joined: 25 June 2005

Well done! Truly thanks for your effort! Now your results are coherent with Condor's. He claimed there are only 4 essential classes of grids and only 1 of them could be filled with the remaining 2 digits validly. I'm sure with some effort I could verify that fact from your 24 sets!

So this remarkable discovery does most likely hold up. Right up there with Gfoyle's 16-clue 7/9 puzzle and Red Ed's super 3-in-1 magic sudoku...

P.S. I just thought one of you and Condor must be wrong here... I was inclined to believe you a little bit more but turns out Condor has been right all the way. Sorry for being susceptible...
udosuk

Posts: 2698
Joined: 17 July 2005

Okay, confirmed that the 24 sets can be grouped into 4 essentially identical classes (2 classes of 8, 2 classes of 4). Of these 4, only 1 can be filled into a valid sudoku in 2 different ways (4 if you include the exchange of the last 2 digits). Thus the suitably named "Sudoku Queen" are actually twin sisters... maybe "Sudoku Queen Twins"?

I have an idea about "norming" every sudoku grid. First, substitute digits so that the top line reads "123456789" (there are 8 ways of doing this by rotation and reflection). Of each of the 8 ways, read the 2nd line, and pick the grid with the smallest 9-digit number. If we have a tie, then compare the 3rd line... etc. That way every essentially identical sudoku grid can be transformed to the same representation. For example, the "Sudoku Queen Twins" can be normed to:

Code: Select all
`1 2 3 4 5 6 7 8 94 8 7 2 9 3 5 6 16 5 9 1 8 7 4 3 29 3 8 5 6 1 2 7 47 4 1 3 2 9 8 5 65 6 2 8 7 4 1 9 32 9 5 6 1 8 3 4 78 7 4 9 3 2 6 1 53 1 6 7 4 5 9 2 81 2 3 4 5 6 7 8 94 8 7 2 9 3 5 6 16 5 9 8 1 7 4 3 29 3 1 5 6 8 2 7 47 4 8 3 2 9 1 5 65 6 2 1 7 4 8 9 32 9 5 6 8 1 3 4 78 7 4 9 3 2 6 1 53 1 6 7 4 5 9 2 8`

Probably some others programming gurus here have used this method before me. It does save us a lot of confusion...
udosuk

Posts: 2698
Joined: 17 July 2005

udosuk wrote:Okay, confirmed that the 24 sets can be grouped into 4 essentially identical classes (2 classes of 8, 2 classes of 4). Of these 4, only 1 can be filled into a valid sudoku in 2 different ways (4 if you include the exchange of the last 2 digits). Thus the suitably named "Sudoku Queen" are actually twin sisters... maybe "Sudoku Queen Twins"?

I have an idea about "norming" every sudoku grid. First, substitute digits so that the top line reads "123456789" (there are 8 ways of doing this by rotation and reflection). Of each of the 8 ways, read the 2nd line, and pick the grid with the smallest 9-digit number. If we have a tie, then compare the 3rd line... etc. That way every essentially identical sudoku grid can be transformed to the same representation. For example, the "Sudoku Queen Twins" can be normed to:

Code: Select all
`1 2 3 4 5 6 7 8 94 8 7 2 9 3 5 6 16 5 9 1 8 7 4 3 29 3 8 5 6 1 2 7 47 4 1 3 2 9 8 5 65 6 2 8 7 4 1 9 32 9 5 6 1 8 3 4 78 7 4 9 3 2 6 1 53 1 6 7 4 5 9 2 81 2 3 4 5 6 7 8 94 8 7 2 9 3 5 6 16 5 9 8 1 7 4 3 29 3 1 5 6 8 2 7 47 4 8 3 2 9 1 5 65 6 2 1 7 4 8 9 32 9 5 6 8 1 3 4 78 7 4 9 3 2 6 1 53 1 6 7 4 5 9 2 8`

Probably some others programming gurus here have used this method before me. It does save us a lot of confusion...

this method doesn't work for normal sudokus, where we have much
more "symmetries" than 8
dukuso

Posts: 479
Joined: 25 June 2005

this method doesn't work for normal sudokus, where we have much more "symmetries" than 8

Okay, I understand it now. I forgot about rows, columns (single line or blocks) exchanging.

For the Sudoku Queen Twins or sudokus with embeded magic squares, you don't need to consider them because they'll likely destroy the extra properties. But for the normal sudokus... you need a program to do it. From the other thread I know you've written a program to analyse the chain-structure... So have you got anything or any ideas that can "sort" the rows and columns out to create a unique representation for each class? I will try to think about it...

Another variation on the "Chess Queen" problem is, trying to limit the total number of duplications instead. For example, if we have 3 identical digits on the same diagonal, we count it as 2 duplications... Then the "Sudoku Queen Twins" both have 8 duplications. Can we do better? I assume this time you can use my algorithm to shorten the resulting list...
udosuk

Posts: 2698
Joined: 17 July 2005

I added some more clues to the puzzle at
http://magictour.free.fr/sudo12.JPG

fill in letters A..L into the grid, such that none of the 12 rows,
12 columns, 2*23 diagonals contains a symbol more than once.
dukuso

Posts: 479
Joined: 25 June 2005

Is it possible to have a sudoku with all the pandiagonals (meaning the diagonals like this)

Code: Select all
`. . . . . * . . . . . . . * . . . .. . . * . . . . .. . * . . . . . .. * . . . . . . .* . . . . . . . . . . . . . . . . *. . . . . . . * .. . . . . . * . .`

having each digit exactly once?

I'm including diagonals going / (bottom left to top right) and diagonals going \ (top left to bottom right).

What if we only use diagonals going / ?
Moschopulus

Posts: 256
Joined: 16 July 2005

not possible.

n*n pandiagonal latin squares exist iff n is coprime to 6.
For the queens-squares see back in the thread or
go directly to:
http://www.cs.concordia.ca/~chvatal/queengraphs.html
dukuso

Posts: 479
Joined: 25 June 2005

So maybe it could be done for 25x25 sudoku.

For 9x9 sudoku, what is the largest k such that k of the pandiagonals have each digit exactly once?
And can we create a puzzle that can only be solved with this info. etc.
Moschopulus

Posts: 256
Joined: 16 July 2005

Not a Sudoku but just to show that 11 "Queens" can be on a 11* 11

00 01 02 03 04 05 06 07 08 09 10
09 10 00 01 02 03 04 05 06 07 08
07 08 09 10 00 01 02 03 04 05 06
05 06 07 08 09 10 00 01 02 03 04
03 04 05 06 07 08 09 10 00 01 02
01 02 03 04 05 06 07 08 09 10 00
10 00 01 02 03 04 05 06 07 08 09
08 09 10 00 01 02 03 04 05 06 07
06 07 08 09 10 00 01 02 03 04 05
04 05 06 07 08 09 10 00 01 02 03
02 03 04 05 06 07 08 09 10 00 01
Frenchy Pilou

Posts: 1
Joined: 02 December 2005

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