Twin Equivalent Sudokus

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Twin Equivalent Sudokus

Postby Shintaro » Sun Nov 19, 2006 11:17 am

:DI found a new Sudoku variant on the Yahoogroup Sudokuworld site http://games.groups.yahoo.com/group/sudokuworld/. If you are not a member of Sudokuworld, click on the button "Join This Group!" as member to access the file.

If you are already a member of Sudokuworld, sign in by clicking the button "Sign in to Yahoo!". After that, you can download the file "Twin Equivalent Sudokus" from the folder "Equivalent Sudokus" under the file section.
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Postby Shintaro » Tue Nov 21, 2006 1:58 am

To make it easy for users to access the puzzle, my friend asked me to post it in text format.

Twin A
Code: Select all
. . .|2 . .|4 . 1
. 5 .|. 4 .|. . 7
. . 1|. . .|. 5 .
-----+-----+-----
. . 7|. 8 .|9 . 4
. 3 .|. 9 .|1 . .
8 . .|1 . .|. 2 . 
-----+-----+-----
4 . .|. . .|. . .
. 2 .|. . .|7 1 .
. 7 .|3 . 8|. . 2



Twin B
Code: Select all
.  .  .|.  6 2|.  . .
9  .  8|.  . .|.  7 .
5  .  .|1  . .|4  6 .
-----+-----+----------
.  5  .|.  . .|8  . .
1  .  .|8 .  .|.  . .
.  .  4|.  9 .|.  3 . 
-----+-----+----------
.  .  9|.  5 .|3  . .
6  .  .|.  . .|.  . 9
.  .  .|.  . 8|.  . 5



Source: Yahoogroup Sudokuworld http://games.groups.yahoo.com/group/sudokuworld/

Rules of Twin Equivalent Sudokus (devised by Henry Kwok)

This Sudoku variant consists of a pair of linked standard Sudoku puzzles each with some starting digits. As usual, the object of this Sudoku variant is to fill in the whole of each 9x9 grid with digits 1 through 9 so that each row, each column and each block contain all the digits 1 through 9.

Twin B is related to twin A in the following ways:

Given that twin A is the original puzzle, twin B is obtained
1. by rearranging all the digits 1 through 9,
2. by exchanging a row (column) of blocks with another row (column) of blocks.

By such transformation, twin A and twin B are essentially the same or equivalent sudoku puzzle.

In addition to the usual strategies, we have to locate the row (column) of blocks that has been exchanged with another row (column) of blocks before we can determine how the digits 1 through 9 are rearranged in twin B. By this way the twin puzzles can be solved.:D
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Postby tarek » Tue Nov 21, 2006 2:19 am

A quick question....

In generating such twins.... Is there more to it than cutting a valid puzle into 2 & then scrambling the second ???

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Postby Shintaro » Tue Nov 21, 2006 3:27 am

I have asked my friend. He said his puzzles were all handcrafted, not generated by software. There is no cutting or division of any puzzle into two here. It is a matter of simple mathematical transformation, and in the end both puzzles are essentially the same.

You can experiment on your own by shfting a column of 3 blocks in a Sudoku puzzle sideway to the left or right to create a second puzzle. Similarly, you can also shift a row of 3 blocks upwards or downwards to a new position to create a new puzzle. The final step is, of course, selecting the right number of digits in the original puzzle and the new puzzle to create the twin puzzles. Finding the "right number" of digits to make a Sudoku puzzle with a single valid solution can be a painstaking process if done manually. The task can be made easier if you can do it with the help of software.

You can increase the difficulty of the twin puzzles by changing the value of the digits in the new puzzle so that each digit in the new puzzle corresponds one to one to the digits in the original puzzle before shifting the columns or rows of blocks to a new position.

Any puzzle created by such transformation is actually equivalent or the same as the original puzzle. I hope my answer is sufficed for you.:D
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Postby RW » Tue Nov 21, 2006 8:38 am

Shintaro wrote:Given that twin A is the original puzzle, twin B is obtained
1. by rearranging all the digits 1 through 9,
2. by exchanging a row (column) of blocks with another row (column) of blocks.

By such transformation, twin A and twin B are essentially the same or equivalent sudoku puzzle.


These two puzzles are not equivalent through any transformation. I think we need some specification. I guess they should be transformed to two puzzles from the same grid. Should they have overlapping givens? Should they be all disjoint? Are all givens in Twin B a subset of the givens in Twin A? What happened to the third valid method of scrambling, swapping rows/columns within the chutes, is that not to be used?

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Postby ronk » Tue Nov 21, 2006 1:14 pm

RW wrote:These two puzzles are not equivalent through any transformation.

I think Shintaro means the solutions are isomorphic.
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Postby tarek » Tue Nov 21, 2006 2:27 pm

I thought that each puzzle can't be solved alone......

It is a cool idea.....

One valid puzzle & a scrambled version of the same puzzle ........ you can then remove any number of clues from both puzzle rendering each unsolvable alone.... making sure that each original clue in the valid puzzle is present somewhere in either twin.......

The name also seems fitting .... however I would suggest the name to change into DIZYGOTIC TWINS ....... Genetic material from both siblings can be used to reconstruct the PARENT puzzle......

Any idea how to achieve that ????

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Postby RW » Tue Nov 21, 2006 4:43 pm

tarek wrote:One valid puzzle & a scrambled version of the same puzzle ........ you can then remove any number of clues from both puzzle rendering each unsolvable alone.... making sure that each original clue in the valid puzzle is present somewhere in either twin.......


I'm afraid that doesn't work. Easy to prove:
Code: Select all
Original:
 *-----------*
 |.96|.4.|...|
 |3.1|...|9..|
 |...|..7|3..|
 |---+---+---|
 |...|6..|.3.|
 |24.|3.9|.51|
 |.5.|..8|...|
 |---+---+---|
 |..4|9..|...|
 |..5|...|4.8|
 |...|.6.|19.|
 *-----------*

Unscrambled twin 1:
 *-----------*
 |.96|.4.|...|
 |3.1|...|9..|
 |...|..7|3..|
 |---+---+---|
 |...|6..|.3.|
 |24.|3.9|.51|
 |...|..8|...|
 |---+---+---|
 |..4|9..|...|
 |..5|...|4.8|
 |...|.6.|19.|
 *-----------*

Scrambled twin 2:
 *-----------*
 |...|...|...|
 |...|...|...|
 |...|...|...|
 |---+---+---|
 |...|...|...|
 |...|.8.|...|
 |...|...|...|
 |---+---+---|
 |...|...|...|
 |...|...|...|
 |...|...|...|
 *-----------*

All original clues present... I'm not sure how many of the original clues must be present in both puzzles to ensure uniqueness. But it's not enough to have all clues appear in either twin.

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Postby Shintaro » Tue Nov 21, 2006 7:58 pm

Hi tarek, ronk and RW,

It is a good sign that all of you are discussing the puzzle here. It shows that all of you are interested in the puzzle.

First, I like to find out whether anyone of you has managed to get the solution. Secondly, I also want to find out whether any other forum users have found the solution. If any forum users has obtained the solution, I would like to hear from him.

Thirdly, I think we need to agree on the definition of equivalent puzzles before we start solving the puzzle. Do you all agree with me if we rearrange all the digits in a given puzzle, we should get two equivalent puzzles, that is, both the original puzzle and the new one?

Lastly, do you all agree with me if we rearrange the blocks (that is, boxes of 3x3 cells) of columns (rows) by exchanging the position of one block of columns(rows) with another block of columns(rows), we should also get two equivalent puzzles, that is, both the original puzzle and the new one?

Once we can come to an agreement on the definition of equivalent puzzles, everything will be easy, and we can start solving the puzzle.

If nobody can get the solution, it means that there is a flaw in the design of the puzzle, and more starting digits are required for both grids.

My friend has told me that we should try to solve each twin by the usual Sudoku strategy first to generate more digits in each grid. When we manage to get enough digits in both twins by the usual Sudoku solving techniques, we shall be able to identify at least two blocks in twin B that are equivalent to two blocks in the original puzzle twin A.

In other words, we must try to locate at least two equivalent blocks that have been shifted to their new position in twin B. Once we succeed in doing that, we can subsititute the equivalent values of the digits from the two equivalent blocks in twin A to the two equivalent blocks in twin B to solve the puzzle.
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Postby RW » Tue Nov 21, 2006 9:16 pm

Shintaro wrote:I think we need to agree on the definition of equivalent puzzles before we start solving the puzzle. Do you all agree with me...

Yes, of course, that is part of the basic definition of equivalent puzzles. As I said you may also permutate the grid by swapping the rows/columns within the bands/stacks, but apparently that's not part of this game?

Shintaro wrote:If nobody can get the solution, it means that there is a flaw in the design of the puzzle, and more starting digits are required for both grids.

The puzzle is valid, here's the solution in canonical form:
123457689456189327789623415264538791537916842918742563371264958695871234842395176

It's the only solution that satisfies both grids. This was found by comparing all solutions of both puzzles, which of course isn't a very neat way of doing it. I'm not sure if I understand your instructions here:

Shintaro wrote:In other words, we must try to locate at least two equivalent blocks that have been shifted to their new position in twin B.

Under the basic permutation rules, any block can be equivalent to any other block. Also any two blocks can be equivalent to any other two blocks, if they don't share a band or stack, so the two blocks you are talking about should be in the same chute, or? I couldn't see any such relationship between two blocks, but maybe someone else can...

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Postby tarek » Wed Nov 22, 2006 10:54 am

Thanx for the example RW, it clearly shows that the twins should have just about the same genetic material (your example reminds me of the film "TWINS" .... guess who's Danny DeVito:D ?"

If full permutation is allowed (digit,line & band/Chute) then I think it is too complicated......

We would probably need another clue like the total number of the PARENT's clues (chromosemes:D ).... you can then work out how many are missing from each twin & then try to construct a pattern....

if we don't know the parent's number of clues the I think that permutations should be limited......

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re: "Twin Equivalent Sudoku"

Postby Pat » Wed Nov 22, 2006 11:38 am

tarek wrote:If full permutation is allowed (digit,line & band/Chute)
then I think it is too complicated......

for Twin Equivalent Sudoku,
digits are swapped,
chutes too,
but not lines within a chute.


Shintaro wrote:
A. [ 26 clues ]

Code: Select all
 . . . | 2 . . | 4 . 1
 . 5 . | . 4 . | . . 7
 . . 1 | . . . | . 5 .
-------+-------+------
 . . 7 | . 8 . | 9 . 4
 . 3 . | . 9 . | 1 . .
 8 . . | 1 . . | . 2 .
-------+-------+------
 4 . . | . . . | . . .
 . 2 . | . . . | 7 1 .
 . 7 . | 3 . 8 | . . 2



B. [ 23 clues ]

Code: Select all
 . . . | . 6 2 | . . .
 9 . 8 | . . . | . 7 .
 5 . . | 1 . . | 4 6 .
-------+-------+------
 . 5 . | . . . | 8 . .
 1 . . | 8 . . | . . .
 . . 4 | . 9 . | . 3 .
-------+-------+------
 . . 9 | . 5 . | 3 . .
 6 . . | . . . | . . 9
 . . . | . . 8 | . . 5


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Postby tarek » Wed Nov 22, 2006 1:12 pm

Thanx Pat,

That lowers the difficulty from "too complicated" to "complicated".....

The overlap looks minimal which means that the parent puzzle has too many clues........ (is the parent puzzle minimal ??!!)

I would commit my self only to single in both twins & then look for boxes which if superimposed will have too much overlap..... permute digits & se if it works

Remeber.....The box always keeps its neighbours in the band/chute wherever it moves .... if there is no line permutation then the boxes internal configuration should not change that is why knowing the number of clues in the parent puzzle would make it IMO more enjoyable to solve.....

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Postby RW » Wed Nov 22, 2006 3:49 pm

Pat wrote:for Twin Equivalent Sudoku,
digits are swapped,
chutes too,
but not lines within a chute.

Yes, I can confirm that this is the case for this puzzle. However, even when knowing how the chutes should be swapped, I don't see the two equivalent boxes that Shintaro mentioned earlier. Only after swapping the digits the relationship becomes clear...

tarek wrote:The overlap looks minimal which means that the parent puzzle has too many clues........ (is the parent puzzle minimal ??!!)

The two twins have only 9 clues in common, the parent puzzle has 40 clues and is far from minimal. These are the 9 clues from the first grid that also appear in the second:
Code: Select all
 *-----------*
 |...|2..|...|
 |...|...|...|
 |..1|...|.5.|
 |---+---+---|
 |...|...|..4|
 |...|.9.|...|
 |8..|1..|...|
 |---+---+---|
 |...|...|...|
 |...|...|...|
 |...|..8|..2|
 *-----------*


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Postby Shintaro » Wed Nov 22, 2006 3:52 pm

Pat wrote:
for Twin Equivalent Sudoku,
digits are swapped,
chutes too,
but not lines within a chute.


Pat, you have hit the nail on the head!

You are very close to solving the twin puzzles. You get full marks for finding all the available digits in twin B. However, for twin A, you are still short of 17 available digits!

Once you manage to find all the available digits in both grids, the solution will be staring at you in the face. Can you post both grids with all the available digits obtained by the usual sudoku strategy again? You can even post both grids with the full solution if you have succeeded to find all the answers by now.

Let us see if anybody can do it before Pat.

My friend has asked me to explain this particular puzzle and the definition of equivalent puzzles in more details later on.:)
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