## Puzzle families

Programs which generate, solve, and analyze Sudoku puzzles

### Puzzle families

The digits in the puzzles have no meaning execpt from having 9 different values. We could use the nine planets, nine colours of the rainbow or even Snow White's nine dwarfs :-) . Therefore we can swap the digit values around in a puzzle without changing the solution, so long as we swap ALL occurences of each digit. Thus comes a revelation... puzzles form families. In each family, each member puzzle can be generated from any other member just by swapping pairs of digits. In fact in each family there is only one puzzle, each member just makes it look different at first glance. In each family I propose to consider one member as representative of the family as a whole, I propose calling it the "Generator" and the other members "Children". I propose that the Generator puzzle in a family be that puzzle with the top row set to 1,2,3,4,5,6,7,8,9. There are 9! (9*8*7*6*5*4*3*2*1 = 362880) members in each family. This will simplify describing solutions. I am going to add a feature to my solver program to find the Generator puzzle for any puzzle where the top row is known. Comments?
howshaw

Posts: 9
Joined: 13 March 2005

Further to your analysis:

For each of your families, there are 7 other families that are symmetrical, only differing by rotation, reflection, or both.

How would you lump them together?
Guest

Posts: 312
Joined: 25 November 2005

Are there only 8 symmetries? Centainly you can rotate and reflect the puzzle but can you swap the boxes around? If you think not of the physical representation of the puzzle, but of the units as mathematical entities then the symmetries are just permutations of the units. Swapping row and column units around is easy to visualise as rotations and reflections (because any row has each cell shared with a different column and any column has each cell shared with a different row, its possible to express the permutations as reordering the row/column indexes), but the boxes complicate the symmetries because any box does not share cells with every row and/or every column. You should be able to permute the boxes but I'm not sure how to express it. If each cell is a point in an integral 9x9x9 3D space with row-number, column-number and box-number axes then of the 729 points in the space, only 81 are allowed. Try making the space in out of sheets of clear plastic and you will see allowed paths and forbidden regions. Then perform axis transformations of the space.
Anyway its all getting away from the puzzle and, to be honest, I'm moving on to other things. I'm now going to stop watching these forums.
Good Luck.
howshaw

Posts: 9
Joined: 13 March 2005

### Re: Puzzle families

howshaw wrote:This will simplify describing solutions. I am going to add a feature to my solver program to find the Generator puzzle for any puzzle where the top row is known. Comments?

I agree totally - as well as the rotations and reflections also proposed in the next post, we also need to consider that a puzzle does not change if we cut and slice it.

If we ignore box 1 since the single

123
456
789

can be transformed into any other.

For box 2 we can interchange the columns in 3! = 6 ways without altering the validity of the solution, similar for box 3.

For boxes 4, and 7 we can do the same with the rows. Not worked out the rules yet for 5,6,8,9 - perhaps there are none left since we already have control over the rows, and columns.

Havinving finished at the level of rows/columns, we can of course interchange the whole column of Boxes

1 2 3 , 2 1 3 , 3 1 2 etc for all 3!=6 combinations
4 5 6 , 5 4 6 , 6 4 5
7 8 9 , 8 7 9 , 9 7 8

but some of these will be reflections about the vertical axis

AND then repeat again for the horizontal axis.

Needs developing more, but your concept of a Generator Puzzle that can be transformed into all it's choldren must be the right way to go.

Also must not forget that we can use the diagonal axis as a point of refletion also..
Hope this helps ..... the are more ideas along these lines in the Generla/Puzzle --> Su-Doku's Maths.
Tony Williams

Posts: 18
Joined: 02 April 2005

Tony, these are the Sudoku Players forums. The present forum (Solver programs) is justified on the basis that some players use solver programs to help with the puzzles they are working on.

Discussions on generating puzzles are beyond the scope of this forum. If you need to discuss that subject, you should post in the Sudoku Programmers forums: see http://forum.enjoysudoku.com/viewtopic.php?t=96 Thanks.

- Wayne.
Pappocom

Posts: 599
Joined: 05 March 2005

Pappocom wrote:
Discussions on generating puzzles are beyond the scope of this forum. If you need to discuss that subject, you should post in the Sudoku Programmers forums: see http://forum.enjoysudoku.com/viewtopic.php?t=96 Thanks.

- Wayne.

Hi Wayne,
I hadn't really seen this as part of puzzle generation - I was responding to the early 'posts' . that having found a solution to a given Puzzle, how could it be 'described' by transforming it to its 'Generator' Puzzle - clearly the two topics are closely linked. Pease feel free to transfer these Posts to the other Topic, whcih I have also been viewing.

Many congratulations on Generating such an intriguing set of Puzzles, the maths and logic to solve them is facinating
Tony Williams

Posts: 18
Joined: 02 April 2005

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