Twin Equivalent Sudokus

For fans of Killer Sudoku, Samurai Sudoku and other variants

Postby RW » Wed Nov 22, 2006 4:11 pm

Shintaro, I posted the solution to both puzzles already yesterday, I just posted it in a canonical form not to spoil the fun for other solvers. If you want the real solutions to both, here they are in tiny text:

789253461653841297241976853167582934532694178894137625415729386328465719976318542
413762958968435172572189463357214896196873524824596731249651387685347219731928645


No need to explain the concept of equivalent puzzles, that is common knowledge around here. But I'd be happy to see the explanation to the "solution staring at you in the face"-part.

Shintaro wrote:Pat ... 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!

As far as I can see Pat only reposted the two grids from your original post...

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Postby tarek » Wed Nov 22, 2006 4:49 pm

40 clues it was RW...
here is the Parent in its canonical form
Code: Select all
 1 . 3 | . . 7 | 6 8 . 
 4 . . | 1 8 9 | 3 . . 
 7 . . | . 2 . | 4 1 5 
-------+-------+------
 2 . . | . 3 . | . 9 . 
 5 . . | 9 . 6 | 8 . 2 
 . . 8 | 7 . . | 5 . 3 
-------+-------+------
 . 7 1 | 2 . . | . . 8 
 . 9 . | 8 . 1 | 2 3 . 
 . . 2 | . 9 5 | . 7 . 

 1 2 3 | 4 5 7 | 6 8 9 
 4 5 6 | 1 8 9 | 3 2 7 
 7 8 9 | 6 2 3 | 4 1 5 
-------+-------+------
 2 6 4 | 5 3 8 | 7 9 1 
 5 3 7 | 9 1 6 | 8 4 2 
 9 1 8 | 7 4 2 | 5 6 3 
-------+-------+------
 3 7 1 | 2 6 4 | 9 5 8 
 6 9 5 | 8 7 1 | 2 3 4 
 8 4 2 | 3 9 5 | 1 7 6 


I solved by finding the more likely overlaps then permuted digits.....the best combination left me with 2 unknown digits....which was easy to resolve with singles

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Postby Shintaro » Wed Nov 22, 2006 7:02 pm

RW wrote:
No need to explain the concept of equivalent puzzles, that is common knowledge around here. But I'd be happy to see the explanation to the "solution staring at you in the face"-part.


Hi RW,
My friend expected the solution to the twin puzzles to be filled in his two grids posted on this forum, not to be posted in the canonical form. Hence it will be much appreciated if you or tarek or anybody else can post the the two grids of twin A (with 40 clues) and twin B (with 23 clues) on this forum.

Once this is done, we can discuss how to continue from there to find the solution. By comparing the two grids, we can identify or locate the chutes or blocks of rows (columns) that have changed position in twin B.

My friend welcomes anyone to come forward to explain how the solution is obtained in this way.:D
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Postby RW » Wed Nov 22, 2006 7:54 pm

Shintaro wrote:My friend expected the solution to the twin puzzles to be filled in his two grids posted on this forum, not to be posted in the canonical form. Hence it will be much appreciated if you or tarek or anybody else can post the the two grids of twin A (with 40 clues) and twin B (with 23 clues) on this forum.

I posted the solutions to the twin puzzles, filled in the original grids, in my last post. Why do you want twin A with 40 clues and twin B with 23 clues? Here they are with 43 and 36 clues:
Code: Select all
 *-----------*
 |.8.|2..|4.1|
 |.5.|.41|..7|
 |.41|...|.5.|
 |---+---+---|
 |167|582|934|
 |.3.|.9.|178|
 |894|1..|.2.|
 |---+---+---|
 |41.|.2.|...|
 |.28|...|71.|
 |.7.|318|.42|
 *-----------*

 *-----------*
 |...|762|958|
 |968|...|.7.|
 |5..|189|463|
 |---+---+---|
 |.5.|...|89.|
 |19.|8..|...|
 |..4|.9.|.3.|
 |---+---+---|
 |..9|.5.|3..|
 |6.5|...|..9|
 |...|9.8|6.5|
 *-----------*


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. 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.

Swap bands 1 and 2 and stacks 2 and 3, then relabel the digits to find the solution. But I still don't see the two equivalent blocks you're talking about...

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Postby tarek » Thu Nov 23, 2006 12:49 am

I would suggest for a good twins puzzle according to the terms posted that a similar box would be revealed as you start solving each puzzle ............ so let's say you place 7 singles in twin A & then 5 in Twin B .... compare ....

Then you will find the linking box & the overlap is do-able ..... That would be a good twin....

Even better...... the Parent puzzle turns out to be a HARD minimal puzzle .....

Shintaro....Could you tell "YOUR FRIEND" that we need another one & quick:D

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Postby Shintaro » Thu Nov 23, 2006 5:10 am

RW asked:
Why do you want twin A with 40 clues and twin B with 23 clues?

RW, your solution for the twin puzzles is absolutely correct! Thanks for having taken all the trouble to work out all the digits in both grids.

The reason for requesting the posting of both grids with all the clues is to make it easier for everybody to find out how the blocks of columns(rows) have been interchanged so as to spot any pair of equivalent chutes or boxes in both grids for solving the puzzle. It may also help to get more people to join in the discussion to share their experience in solving the puzzle.

For those who are still not clear about the concept of equivalent puzzles, let me give an example of how the blocks of columns(rows) are interchanged with each other to create equivalent puzzles from the original puzzle.

Normally, we label the top row of boxes as box 1, box 2 and box 3; the middle row of boxes as box 4, box 5 and box 6; and the bottom row of boxes as box 7, box 8 and box 9. The position of the boxes in the grid is represented by numbers below:

Puzzle A
1 2 3
4 5 6
7 8 9

As an example, let us interchange the middle block of rows with the bottom block of rows to produce an equivalent puzzle B from the original puzzle A:

Puzzle B
1 2 3
7 8 9
4 5 6

To raise the difficulty of the twin puzzle, we can have a second swap by interchanging the first block of columns with the middle block of columns in puzzle B to produce an equivalent puzzle C:

Puzzle C
2 1 3
8 7 9
5 4 6

My friend and I do not want to spoil the fun of others in their attempt to locate the equivalent chutes or boxes. So let us hear from anybody if he has already identified any pairs of equivalent blocks of columns(rows) or any pair of equivalent boxes in both grids. If there is no response, I shall have to provide more hints about the location of the equivalent chutes or boxes.

As to tarek's request, my friend is glad to post another puzzle. But all his puzzles are handcrafted, so it may not be posted here so soon.

Furthermore, some questions about the present puzzle have yet to be resolved. Hence he has to be particularly careful in his design of the next puzzle so as not to make any embarassing mistake and waste the time of others. In the meantime, let us compare both grids and see if we can locate any link between the digits in solving the twin puzzles.:)
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re(2): "Twin Equivalent Sudoku"

Postby Pat » Thu Nov 23, 2006 12:45 pm

RW wrote:
Shintaro wrote:
Pat wrote:
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.


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!


Shintaro,
As far as I can see Pat only reposted the two grids from your original post...


yes, RW, you are absolutely right.

somehow my post gave Shintaro the idea that i had made some progress in solving the puzzle; sorry about the confusion.

in fact i was merely clarifying the rules in response to tarek's question.

i was able to obtain access to Twin Equivalent Sudokus on Yahoo sudokuworld, to read the original description there;
and also copied the 2 components {A,B} from that source.



clearly each of the 2 components can be partly-solved on its own;
but i made no progress in identifying any matching bands or matching stacks.

perhaps those who have solved it could give some detail on the logic of matching the bands or stacks.

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Postby RW » Thu Nov 23, 2006 6:04 pm

Pat wrote:perhaps those who have solved it could give some detail on the logic of matching the bands or stacks.

Looking at the puzzle again, it's actually quite simple. I believe we have just been misled by the "two equivalent boxes" talk, it's much easier by looking at the individual digits. After singles:
Code: Select all
Twin A:
 *-----------*
 |.8.|2..|4.1|
 |.5.|.41|..7|
 |.41|...|.5.|
 |---+---+---|
 |167|582|934|
 |.3.|.9.|178|
 |894|1..|.2.|
 |---+---+---|
 |41.|.2.|...|
 |.28|...|71.|
 |.7.|318|.42|
 *-----------*

Twin B:
 *-----------*
 |...|762|958|
 |968|...|.7.|
 |5..|189|463|
 |---+---+---|
 |.5.|...|89.|
 |19.|8..|...|
 |..4|.9.|.3.|
 |---+---+---|
 |..9|.5.|3..|
 |6.5|...|..9|
 |...|9.8|6.5|
 *-----------*

As the rows/columns are not swapped within the chutes, the column/row relationship for each digit within boxes of each chute will remain unchanged, iow. if one band in one of the twins has digit A in the upper left corner of one box, the middle cell of another and the lower left corner of the third, then it's equivalent digit in the other twin will have one band with the same positionings within the three boxes.

In twin B, all digits 9 are already solved. Starting from row 1, their column position is band 1: L, L, R, band 2: C, C, C, band 3: R, R, L. Now we can start eliminating candidate equivalent digits from twin A.

Digit 1 has a band with R, R, R => 1<>9
Digit 2 has a band with R, ?, C => 2<>9
Digit 3 could fit these column positions, but only if the grid wasn't permutated at all, but we know that it is (Twin A has the same digit in r1c7 and r2c5, twin B doesn't) => 3<>9
Digit 4 has a band with L, C, C => 4<>9
Digit 6 has a band with C, L/R, L/R => 6<>9
Digit 7 has a band with R, C, ? => 7<>9
Digit 8 has a band with C, R, L => 8<>9
Digit 9 has a band with L, C, C => 9<>9

This leaves one possibility: 5 in twin A = 9 in twin B. Now you only need to swap the chutes so that the 9s in the center cells of twin B correspond with the 5s in the center cells of twin A, and so that the 9 in the lower left corner of it's box in twin B corresponds with the 5 in the lower left corner of it's box in twin A and the puzzle is solved.

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Postby Shintaro » Thu Nov 23, 2006 6:15 pm

One way of finding a box in one grid that is the same as (or equivalent to) another box in the second grid is through the laborious process of mapping or matching the digits in one box of the original puzzle to the digits in another box of the equivalent puzzle.

For example, let us find out which box in twin B is equivalent to box 7 of the original puzzle (twin A).

First let us map box 7 (twin A) to box 1 (twin B). Then we shall get a one-to-one mapping between the digits of the two boxes.

1. Mapping box 7 (twin A) to box 1 (twin B)

2 --> 6
8 --> 8

What is left is mapping for box 8 and box 9 (twin A).

Just find the mapping for box 8 (twin A) is enough.

box 8 (twin A) --> box 2 (twin B)
3 --> 1
1 --> 8
8 --> 9
2 --> 6

Box 8 (twin A) is not equivalent to box 2 (twin B).

As the above mapping is invalid, let us map box 8 (twin A) to box 3 (twin B)
1 --> 6
3 --> 4
8 --> 3
2 --> 5

Box 8 (twin A) is not equivalent to box 3 (twin B).

Therefore, box 7 (twin A) is not equivalent to box 1 (twin B).

2. Mapping box 7 (twin A) to box 2 (twin B)

7 --> 8
4 --> 7
1 --> 6

What is left is mapping for box 8 and box 9 (twin A).

Just find the mapping for box 9 (twin A) is enough.

box 9 (twin A) --> box 1 (twin B)

7 --> 9
1 --> 6

Box 9 (twin A) is not equivalent to box 1 (twin B).

As the above mapping is invalid, let us map box 9 (twin A) to box 3 (twin B)

4 --> 6
2 --> 3
1 --> 7

Box 9 (twin A) is not equivalent to box 3 (twin B).

Therefore, box 7 (twin A) is not equivalent to box 2 (twin B).

3. Mapping box 7 (twin A) to box 3 (twin B)

7 --> 6
2 --> 7
4 --> 9
1 --> 5

What is left is mapping for box 8 and box 9 (twin A).

Just find the mapping for box 9 (twin A) is enough.

box 9 (twin A) --> box 1 (twin B)

7 --> 9
1 --> 6

Box 9 (twin A) is not equivalent to box 1 (twin B).

As the above mapping is invalid, let us map box 9 (twin A) to box 2 (twin B)

4 --> 8
2 --> 9

Box 9 (twin A) is not equivalent to box 2 (twin B).

Therefore, box 7 (twin A) is not equivalent to box 3 (twin B).

Hence the bottom block of rows in twin A is not equivalent to the top block of rows in twin B.

In other words, the bottom block of rows has not interchanged position with the top block of rows in the same grid to form an equivalent puzzle.

And in other words, the bottom block of rows may have interchanged position with the middle block of rows in the same grid to form the equivalent puzzle.

The bottom block of rows may even has not shifted at all!

The next step is to continue the laborious process of mapping box 7 (twin A) to box 4. box 5, box 6, box 7, box 8 and box 9 of twin B.

Though the process is laborious, it is the sure way of finding the equivalent boxes and, of course, the solution. Note that this process involves logical deduction. I leave it with anyone who is interested to continue the laborious process on their own.

The second method is to try to spot any digit that has appeared twice or thrice in a block of rows (columns) in a grid and then map them to the other digits that have appeared in corresponding cells in a block of rows(columns) in the other grid.

For example, in the middle block of rows in twin A, the digit 8 appears in all the three boxes. This is a trial and error method, but you may be able to locate the equivalent boxes very fast if you are lucky to choose the right boxes for comparison in both grids.:)
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Postby Shintaro » Thu Nov 23, 2006 6:33 pm

Hi RW,

It seems that we were posting messages almost at the same time, so I had not have the chance to read your message before posting mine.

Your approach is also correct.

My friend is planning to post another example as requested by tarek.

However, he has to be "extra careful" in handcrafting this type of complicated puzzle so as not to make any embarrassing mistake that could waste the time of everybody. So please be patient to wait for the next puzzle.:D
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Postby Shintaro » Fri Nov 24, 2006 6:54 pm

The following twin puzzles "Twin Equivalent Sudokus" are devised by my friend Henry Kwok at the request of tarek.

Twin A
Image

Twin B
Image

Hope all of you in this forum enjoy solving the puzzle this time.:D
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Postby RW » Sat Nov 25, 2006 1:14 am

I like the fact that this time the parent puzzle is minimal. I won't say with how many clues, such hints are not needed.

My problem with most sudoku variants is that the first puzzle is interesting, as that involves coming up with new logical techniques to solve it. After that, the next puzzles are just mechanical application of the already invented logic. In this case the new puzzle solved in a couple of minutes with the technique I explained in my last post. I almost hoped that the rows/columns within the chutes had been swapped also, that way my technique wouldn't have worked and the puzzle would have been more challenging.

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Postby tarek » Sat Nov 25, 2006 2:16 am

Thanx Shintaro & to your friend.....

because the parent is now minimal, I'm afraid that it is now much easier to solve than the previous twins....... the previous twins I had to T&E to match boxes.... here ... well it is quite evident ......... I do not need to solve any individual twin & I can permute .... superimpose & solve the parent.........


I am with RW on this... if the parent is minimal... then probably line permutation within the box should be allowed.....


cheers

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Postby RW » Sat Nov 25, 2006 9:20 am

tarek wrote:because the parent is now minimal, I'm afraid that it is now much easier to solve than the previous twins.......

Aha, but the rules don't say it is minimal... You only know it because it was proposed by some crazy Finn, who by that time on friday night most likely already had spent several hours in the sauna drinking two bottles of kossu. Is that source really reliable enough, so that you can take it as a given that you base your solving techniques on?:D

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Postby tarek » Sat Nov 25, 2006 10:27 am

kossu or no kossu, it is minimal, how many shots were in those 2 bottles, 26:D ?

The rules don't say that it should be minimal....but if it were, then that means if a certain parent clue is missing from one of the twins .... then you can't re-create it .... So if you were to avoid T&E minimal parental clues should be present in both twins........

But if the parent is non minimal........then the redundant clues POSSIBLY can be recreated if they are missing in one of the twins.......

Some ideas for the next genetically-engineered twins:D ?

This one is better Shintaro (although easier) because no T&E was required....the reward should be a tough parent puzzle....

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