Special solution

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Special solution

Postby m_b_metcalf » Tue Jun 20, 2006 1:17 pm

The following puzzle is special, not for the puzzle itself, but for its solution. Can anyone spot what it is, working by hand?

Regards,

Mike Metcalf

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




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Postby tso » Tue Jun 20, 2006 3:35 pm

I can see the answer without bothering to solve it. Answer in tiny text below, followed by another example.

The clues are in sequential order. Each row is 123456789, shifted as required shifted as required.

Code: Select all
+-------+-------+-------+
| . . 3 | 4 . 6 | 7 . . |
| . 8 . | 1 . 3 | . 5 . |
| . 5 . | . . . | . 2 . |
+-------+-------+-------+
| . 1 . | 3 . 5 | . 7 . |
| . 7 . | . . . | . 4 . |
| . . 5 | 6 . 8 | 9 . . |
+-------+-------+-------+
| . . 1 | 2 . 4 | 5 . . |
| . 6 . | 8 . 1 | . 3 . |
| 2 . . | . 6 . | . . 1 |
+-------+-------+-------+
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Postby JPF » Tue Jun 20, 2006 3:57 pm

An other one,
22 clues, minimal with 180-degree rotational symmetry :

Code: Select all
 . . . | x . . | . . x
 x . . | . x x | . x .
 . . . | . . . | x x .
-------+-------+-------
 . . . | . . x | . x .
 . . x | . . . | x . .
 . x . | x . . | . . .
-------+-------+-------
 . x x | . . . | . . .
 . x . | x x . | . . x
 x . . | . . x | . . .


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Postby m_b_metcalf » Tue Jun 20, 2006 4:06 pm

tso wrote:I can see the answer without bothering to solve it. Answer in tiny text below, followed by another example.



Right. It's perhaps less obvious with a random start:

Code: Select all
  2  6  8  5  1  9  7  3  4
  7  3  4  2  6  8  5  1  9
  5  1  9  7  3  4  2  6  8
  4  2  6  8  5  1  9  7  3
  9  7  3  4  2  6  8  5  1
  8  5  1  9  7  3  4  2  6
  3  4  2  6  8  5  1  9  7
  1  9  7  3  4  2  6  8  5
  6  8  5  1  9  7  3  4  2
   


Is this a well-known structure?

Regards,

Mike Metcalf
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Postby frazer » Tue Jun 20, 2006 4:32 pm

This structure came up right at the start of the very long Mathematics of Sudoku thread; there was, briefly, a hope that all sudoku grids would be equivalent to this structure.
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Postby ab » Tue Jun 20, 2006 5:32 pm

I have created a similar puzzle, but you have to know something about juggling notation to understand why this puzzle is special:
http://uk.geocities.com/aidan_001/sudo.html
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Postby m_b_metcalf » Tue Jun 20, 2006 5:42 pm

frazer wrote:This structure came up right at the start of the very long Mathematics of Sudoku thread; there was, briefly, a hope that all sudoku grids would be equivalent to this structure.


I'd looked for some such thread, but to no avail.

It's obviously easy to perform various random permutations to get a large number of variations. This is useful if you want to generate 1000 x 1000 grids, but who is capable of making puzzles out of them? (My best was 144 x 144.)

For even N there is an alternative construction. I posted details in a Mathematics of Sudoku thread some time ago.

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Postby frazer » Wed Jun 21, 2006 12:18 am

Sorry - I should have been more specific. The mathematics of Sudoku thread was actually called "Su-doku's maths", and got up to 600 posts before people started separating such questions into smaller threads...! The thread hasn't been active for several months now.

Try http://forum.enjoysudoku.com/viewtopic.php?t=44 - the post with a similar grid to yours is actually the second in the 600, so you don't have to go too far.
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Postby m_b_metcalf » Wed Jun 21, 2006 12:33 am

frazer wrote:Sorry - I should have been more specific. The mathematics of Sudoku thread was actually called "Su-doku's maths", and got up to 600 posts before people started separating such questions into smaller threads...! The thread hasn't been active for several months now.

Try http://forum.enjoysudoku.com/viewtopic.php?t=44 - the post with a similar grid to yours is actually the second in the 600, so you don't have to go too far.


Many thanks. Just what I'd hoped to see. I've been posting in the Sudoku Programmers' Forum http://www.setbb.com/sudoku/viewforum.php?f=4&mforum=sudoku and never get a reply.

I'll read it through and see what nuggets of information I find.

Regards,

Mike Metcalf
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Postby coloin » Thu Jun 22, 2006 2:07 pm

I think these grids are what has been termed canonical grids

Code: Select all
"Most" canonical
+---+---+---+
|123|456|789|
|789|123|456|
|456|789|123|
+---+---+---+
|231|564|897|
|564|897|231|
|897|231|564|
+---+---+---+
|312|645|978|
|645|978|312|
|978|312|645|
+---+---+---+

canonical "varient" - no 19
+---+---+---+
|123|789|456|
|456|123|789|
|789|456|123|
+---+---+---+
|231|564|897|
|564|897|231|
|897|231|564|
+---+---+---+
|312|645|978|
|645|978|312|
|978|312|645|
+---+---+---+


Subtly different but one has many 19s [648 ?]the other has none.

The above grids need two clues per 3-clue latin sub-square......and they have up to 648 isomorphic puzzles,see Red Eds post

Yours are slightly different but nevertheless they are throughly unpleasant and confusing grids !

C
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Postby m_b_metcalf » Thu Jun 22, 2006 9:53 pm

coloin wrote:I think these grids are what has been termed canonical grids

[snip]
Yours are slightly different but nevertheless they are throughly unpleasant and confusing grids !

C


Thanks for the further information. There seem to be many variant canonical forms, depending on the shifting beyond the first horizontal chute. I devised one for even N that starts with the second row the reverse of the first. In fact, for even N, you can generate grids with random odd rows and each even row the reverse of the preceding odd row. I append one below. But it looks as though this topic has been thoroughly explored already.

As you point out, they're unpleasant, but if you randomize the first row and use a canonical form for N > 7 (where my generator breaks down) you can generate puzzles up to N=12 (after which my solver becomes too slow).

Regards,

Mike Metcalf

Code: Select all
  1  2  5  6 11 10 16 15  3 14  9 13  4 12  8  7
  7  8 12  4 13  9 14  3 15 16 10 11  6  5  2  1
 16 13 10 11  5  1  4  6  7  2  8 12  3 15  9 14
 14  9 15  3 12  8  2  7  6  4  1  5 11 10 13 16
  4  7  1 13  9 11 10 16  2 15 12  8 14  3  6  5
  5  6  3 14  8 12 15  2 16 10 11  9 13  1  7  4
 10 11  8  9  6  5 13  4  1  7  3 14  2 16 12 15
 15 12 16  2 14  3  7  1  4 13  5  6  9  8 11 10
  2 10 11  5 16  6  9 13 14  1 15  3  8  7  4 12
 12  4  7  8  3 15  1 14 13  9  6 16  5 11 10  2
  9 14  6 15  7  4 11  5  8 12  2 10 16 13  1  3
  3  1 13 16 10  2 12  8  5 11  4  7 15  6 14  9
  6  3  4 12  1 14  8  9 11  5  7 15 10  2 16 13
 13 16  2 10 15  7  5 11  9  8 14  1 12  4  3  6
  8 15 14  1  4 16  3 12 10  6 13  2  7  9  5 11
 11  5  9  7  2 13  6 10 12  3 16  4  1 14 15  8
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