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| ChPicard
| | Joined: 12 Mar 2008 | | Posts: 82 | | : | | Location: Montreal, Canada |
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 Posted: Sat Jan 17, 2009 3:07 pm Post subject: Canonicalize again and again! |
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| Lummox JR wrote: | I think I understand gsf's algorithm from the source code, now, well enough to explain it here.
Let's start with terminology. The sudoku is NxN in size, and PQ=N, where P>=Q, but P and Q are as close to each other as possible. If N is a square number, like in a standard 9x9 sudoku, P=Q. The boxes are arranged such that each is P columns by Q rows.
Each column or row of entire boxes we'll call a band. The P-bands will be P boxes stacked to PxN in size, and the Q-bands are Q boxes stacked to QxN in size. If you prefer, think of these as towers and slabs, respectively.
Step 1: "Loop A". Loop through each box, and each orientation (original or transposed). |
Loop A: What for and how?
| Lummox JR wrote: | | This is 2N iterations if P=Q (that is, 18 for a 9x9 puzzle), or N iterations otherwise; ignore transposition if P=Q. Move the chosen box to the box 1 position by transposing the puzzle if necessary, then swapping P- and Q-bands till they reach the desired position. |
How to choose the good box?
| Lummox JR wrote: | | Step 2: "Loop B". Loop through each permutation of the columns and rows in box 1. This is P!Q! iterations, which in a 9x9 puzzle is 3!3!=36. |
Loop B: What for?
Thank you very much for explanations. |
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| ChPicard
| | Joined: 12 Mar 2008 | | Posts: 82 | | : | | Location: Montreal, Canada |
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| Posted: Sat Jan 17, 2009 3:10 pm Post subject: Canonization again and again! |
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| Lummox JR wrote: | I think I understand gsf's algorithm from the source code, now, well enough to explain it here.
Let's start with terminology. The sudoku is NxN in size, and PQ=N, where P>=Q, but P and Q are as close to each other as possible. If N is a square number, like in a standard 9x9 sudoku, P=Q. The boxes are arranged such that each is P columns by Q rows.
Each column or row of entire boxes we'll call a band. The P-bands will be P boxes stacked to PxN in size, and the Q-bands are Q boxes stacked to QxN in size. If you prefer, think of these as towers and slabs, respectively.
Step 1: "Loop A". Loop through each box, and each orientation (original or transposed). |
Loop A: What for and how?
| Lummox JR wrote: | | This is 2N iterations if P=Q (that is, 18 for a 9x9 puzzle), or N iterations otherwise; ignore transposition if P=Q. Move the chosen box to the box 1 position by transposing the puzzle if necessary, then swapping P- and Q-bands till they reach the desired position. |
How to choose the good box?
| Lummox JR wrote: | | Step 2: "Loop B". Loop through each permutation of the columns and rows in box 1. This is P!Q! iterations, which in a 9x9 puzzle is 3!3!=36. |
Loop B: What for?
Thank you very much for explanations. |
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| ChPicard
| | Joined: 12 Mar 2008 | | Posts: 82 | | : | | Location: Montreal, Canada |
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| Posted: Sat Jan 17, 2009 3:12 pm Post subject: Canonization again and again! |
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| Lummox JR wrote: | I think I understand gsf's algorithm from the source code, now, well enough to explain it here.
Let's start with terminology. The sudoku is NxN in size, and PQ=N, where P>=Q, but P and Q are as close to each other as possible. If N is a square number, like in a standard 9x9 sudoku, P=Q. The boxes are arranged such that each is P columns by Q rows.
Each column or row of entire boxes we'll call a band. The P-bands will be P boxes stacked to PxN in size, and the Q-bands are Q boxes stacked to QxN in size. If you prefer, think of these as towers and slabs, respectively.
Step 1: "Loop A". Loop through each box, and each orientation (original or transposed). |
Loop A: What for and how?
| Lummox JR wrote: | | This is 2N iterations if P=Q (that is, 18 for a 9x9 puzzle), or N iterations otherwise; ignore transposition if P=Q. Move the chosen box to the box 1 position by transposing the puzzle if necessary, then swapping P- and Q-bands till they reach the desired position. |
How to choose the good box?
| Lummox JR wrote: | | Step 2: "Loop B". Loop through each permutation of the columns and rows in box 1. This is P!Q! iterations, which in a 9x9 puzzle is 3!3!=36. |
Loop B: What for?
Thank you very much for explanations. |
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| ChPicard
| | Joined: 12 Mar 2008 | | Posts: 82 | | : | | Location: Montreal, Canada |
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| Posted: Sat Jan 17, 2009 5:13 pm Post subject: Canonicalization again and again! |
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| Lummox JR wrote: | I think I understand gsf's algorithm from the source code, now, well enough to explain it here.
Let's start with terminology. The sudoku is NxN in size, and PQ=N, where P>=Q, but P and Q are as close to each other as possible. If N is a square number, like in a standard 9x9 sudoku, P=Q. The boxes are arranged such that each is P columns by Q rows.
Each column or row of entire boxes we'll call a band. The P-bands will be P boxes stacked to PxN in size, and the Q-bands are Q boxes stacked to QxN in size. If you prefer, think of these as towers and slabs, respectively.
Step 1: "Loop A". Loop through each box, and each orientation (original or transposed). |
Loop A: What for and how?
| Lummox JR wrote: | | This is 2N iterations if P=Q (that is, 18 for a 9x9 puzzle), or N iterations otherwise; ignore transposition if P=Q. Move the chosen box to the box 1 position by transposing the puzzle if necessary, then swapping P- and Q-bands till they reach the desired position. |
How to choose the good box?
| Lummox JR wrote: | | Step 2: "Loop B". Loop through each permutation of the columns and rows in box 1. This is P!Q! iterations, which in a 9x9 puzzle is 3!3!=36. |
Loop B: What for?
Thank you very much for explanations. |
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| gsf
| | Joined: 18 Aug 2005 | | Posts: 408 | | : | | Location: NJ USA |
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| Posted: Sat Jan 17, 2009 11:45 pm Post subject: Re: Canonicalization again and again! |
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[ flurry of multiple duplicate posts across the forum ]
recently when I submit a post I get a debug screen that says the post failed
when in fact the post worked
I suspect others have experienced the same
re lummoxjr's comments
read further on in the thread
we determined that the box-by-box alg could be modified to row-by-row without loss of efficiency
row-by-row has the benefit of producing an easy to describe and easy
to understand canonical form: row order minlex w.r.t the solution grid |
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| Adak
| | Joined: 27 Feb 2008 | | Posts: 87 | | : | |
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| Posted: Sun Jan 18, 2009 7:02 am Post subject: |
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| Code: |
===*===*====
000|000|000
000|000|001
000|001|000
---+---+---
000|000|000
000|000|002
001|000|000
---+---+---
000|000|000
000|002|134
123|456|789
===+===+===
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Is this the lowest row min grid possible for 17 givens that could be listed, before any solution testing was done?
Taking into account the requirements for a valid Sudoku grid:
1) No stack or band with 2 empty boxes
2) No two empty columns or rows within any stack/band
3) A valid grid must have at least 8 of the 9 digits as givens
I've read that up to 4 empty boxes (houses), can be allowed and still have a valid Sudoku puzzle. If so, that violates #1, above. |
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| m_b_metcalf
| | Joined: 13 Mar 2006 | | Posts: 210 | | : | | Location: Berlin |
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| Posted: Mon Jan 19, 2009 8:35 am Post subject: |
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| Adak wrote: |
1) No stack or band with 2 empty boxes
...
I've read that up to 4 empty boxes (houses), can be allowed and still have a valid Sudoku puzzle. If so, that violates #1, above. |
A discussion on empty boxes can be found here. That allows 4 empty boxes altogether, and such beasts exist.
Regards,
Mike Metcalf |
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| ChPicard
| | Joined: 12 Mar 2008 | | Posts: 82 | | : | | Location: Montreal, Canada |
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| Posted: Mon Jan 19, 2009 7:02 pm Post subject: Canonicalization again and again! |
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| Lummox JR wrote: | I think I understand gsf's algorithm from the source code, now, well enough to explain it here.
Let's start with terminology. The sudoku is NxN in size, and PQ=N, where P>=Q, but P and Q are as close to each other as possible. If N is a square number, like in a standard 9x9 sudoku, P=Q. The boxes are arranged such that each is P columns by Q rows.
Each column or row of entire boxes we'll call a band. The P-bands will be P boxes stacked to PxN in size, and the Q-bands are Q boxes stacked to QxN in size. If you prefer, think of these as towers and slabs, respectively.
Step 1: "Loop A". Loop through each box, and each orientation (original or transposed). |
What is it done in Loop A and how?
| Lummox JR wrote: | | This is 2N iterations if P=Q (that is, 18 for a 9x9 puzzle), or N iterations otherwise; ignore transposition if P=Q. Move the chosen box to the box 1 position by transposing the puzzle if necessary, then swapping P- and Q-bands till they reach the desired position. |
How to choose the good box?
| Lummox JR wrote: | | Step 2: "Loop B". Loop through each permutation of the columns and rows in box 1. This is P!Q! iterations, which in a 9x9 puzzle is 3!3!=36. |
Loop B: What for?
Thank you very much for explanations. |
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| tarek
| | Joined: 31 Dec 2005 | | Posts: 153 | | : | | Location: London, UK |
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| Posted: Mon Jan 19, 2009 9:01 pm Post subject: |
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ChPicard,
Why are you repeating the same post again & again, you don't need to shout out loud to get noticed.
If somebody ignored you the 1st time, the constant shouting will not get you anywhere. Be patient.
tarek |
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| ChPicard
| | Joined: 12 Mar 2008 | | Posts: 82 | | : | | Location: Montreal, Canada |
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| Posted: Tue Jan 20, 2009 12:59 pm Post subject: Sorry but I was told ERROR |
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| tarek wrote: | ChPicard,
Why are you repeating the same post again & again, you don't need to shout out loud to get noticed.
If somebody ignored you the 1st time, the constant shouting will not get you anywhere. Be patient.
tarek |
Sorry but I was told ERROR |
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| ChPicard
| | Joined: 12 Mar 2008 | | Posts: 82 | | : | | Location: Montreal, Canada |
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| Posted: Fri Jan 23, 2009 11:37 pm Post subject: Find the smallest digit in row 1 that is not in box 1 !!! |
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Lummox Jr wrote:
| Quote: | | Step 4: Find the smallest digit in row 1 that is not in box 1. |
Think about it one minute.
How could it be possible? Because in each box, there are all digits from 1 to 9.
Thanks Lummox Jr for the effort but it is impossible to understand your explanation. The word LOOP has many meanings, the word ITERATE too. |
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| gsf
| | Joined: 18 Aug 2005 | | Posts: 408 | | : | | Location: NJ USA |
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| Posted: Sat Jan 24, 2009 4:51 pm Post subject: Re: Find the smallest digit in row 1 that is not in box 1 !! |
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| ChPicard wrote: | Lummox Jr wrote:
| Quote: | | Step 4: Find the smallest digit in row 1 that is not in box 1. |
Think about it one minute.
How could it be possible? Because in each box, there are all digits from 1 to 9.
Thanks Lummox Jr for the effort but it is impossible to understand your explanation. The word LOOP has many meanings, the word ITERATE too. |
lummoxjr is referring to source code that I either posted on the forum or on a web site
if you find the reference to that code I will help decipher |
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| ThemePark
| | Joined: 22 Mar 2009 | | Posts: 17 | | : | |
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| Posted: Sun Mar 22, 2009 2:03 pm Post subject: |
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Since this thread has been brought back from the dead by someone else, I suppose there is no harm in posting here to ask some questions.
Specifically, I am curious about this algorithm that Red Ed mentions. How can it be possible that there are at most 648 transformations of a solution grid, when I've read elsewhere that there are 9!*3!*3!*3*3!*3!*3*2 transformation. Or does the number 648 only refer to the automorphic solution grids? |
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| JPF
| | Joined: 05 Dec 2005 | | Posts: 29 | | : | | Location: Paris |
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| Posted: Tue Mar 24, 2009 5:47 pm Post subject: |
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648 is the maximum number of automorphisms of a solution grid.
There is only one grid (up to isomorphism) having this property, named the Most Canonical grid (MC)
Here is the MC grid :
| Code: | *-----------*
|123|456|789|
|456|789|123|
|789|123|456|
|---+---+---|
|231|564|897|
|564|897|231|
|897|231|564|
|---+---+---|
|312|645|978|
|645|978|312|
|978|312|645|
*-----------* |
For more informations on automorphisms, you may have a look at :
JPF |
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