Symmetric clue values

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Symmetric clue values

Postby m_b_metcalf » Tue May 15, 2007 2:56 pm

We have seen how some of the most difficult puzzles have a degree of symmetry not only in the positions of the clues but also in their values. The following puzzle:
Code: Select all
 0 0 0 . . . . . .
 0 0 2 3 . 6 7 0 .
 3 1 0 4 . . 0 8 5
 . 6 5 . . 2 . 9 .
 . . . 9 . 5 . . .
 . 9 . 8 . . 5 6 .
 5 8 0 . . 4 0 1 3
 . 0 7 6 . 3 2 0 0
 . . . . . . 0 0 0   SE 7.1

is rotationally symmetric both in the positions of the clues and, with the exception of those in the central box, also in their values. What's more, the empty cells marked with 0s also have position and value symmetry (thus, boxes 1 and 9 of the solution are completely symmetric to one another).

This is perhaps just a curiousity, but is it possible to construct valid puzzles whose clues are all symmetric with respect to both position and value? (Sorry if this has been discussed before, but I don't recall it.)

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Re: Symmetric clue values

Postby Ocean » Tue May 15, 2007 3:50 pm

m_b_metcalf wrote:This is perhaps just a curiousity, but is it possible to construct valid puzzles whose clues are all symmetric with respect to both position and value? (Sorry if this has been discussed before, but I don't recall it.)


I think such puzzles either have zero, two or more solutions. JPF showed some of these a while ago.
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Re: Symmetric clue values

Postby m_b_metcalf » Tue May 15, 2007 4:59 pm

Ocean wrote:I think such puzzles either have zero, two or more solutions. JPF showed some of these a while ago.

Yes, when you think about it. If a puzzle with 180-degree rotational symmetry has clues with symmetric values and has one solution, then you can rotate the puzzle and it must yield the same solution. But that can't be true because box 5 cannot be symmetric with respect to itself. Therefore, the puzzle cannot have only one solution.

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Postby JPF » Tue May 15, 2007 6:13 pm

Here
I wrote:Here's an absolutely symmetrical puzzle :

39 clues, diagonal symmetry.
Code: Select all
 1 . . | . . 9 | 7 5 6
 . 5 . | . . 1 | 9 3 4
 . . 9 | . 5 . | 1 2 8
-------+-------+-------
 . . . | 7 . . | . . 9
 . . 5 | . 9 . | 3 4 .
 9 1 . | . . 3 | . . .
-------+-------+-------
 7 9 1 | . 3 . | 4 . .
 5 3 2 | . 4 . | . 9 .
 6 4 8 | 9 . . | . . 3       2 solutions


124389756856271934379654128483765219265198347917423865791536482532847691648912573
183429756257861934469357128326714589875692341914583672791238465532146897648975213


  • no cell can be solved ; the 2 solutions don't have any common digits except the given clues
  • in each empty cell, only 2 digits are possible to make a valid puzzle.
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Postby Smythe Dakota » Thu May 17, 2007 4:34 am

You might be able to achieve symmetry with a unique solution if you throw in mirror-image digits: 1 converts to 9, 2 converts to 8, etc.

In other words, if there is an N in row R, column C, then there is a 10-N in row 10-R, column 10-C.

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Postby m_b_metcalf » Thu May 17, 2007 1:26 pm

Smythe Dakota wrote:You might be able to achieve symmetry with a unique solution if you throw in mirror-image digits: 1 converts to 9, 2 converts to 8, etc.

In other words, if there is an N in row R, column C, then there is a 10-N in row 10-R, column 10-C.

Well, my first attempt has many clues but still two solutions:(:!:
Code: Select all
 7 3 8 5 2 4 . . .
 5 6 2 7 9 1 3 8 4
 4 1 9 3 6 8 7 5 2
 . 8 5 4 3 2 . . 7
 2 7 6 1 5 9 4 3 8
 3 . . 8 7 6 5 2 .
 8 5 3 2 4 7 1 9 6
 6 2 7 9 1 3 8 4 5
 . . . 6 8 5 2 7 3

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Postby underquark » Thu May 17, 2007 2:16 pm

This from an earlier thread?
Code: Select all
*-----------------------*
| . . . | . . . | 6 . 5 |
| . . . | 3 . . | . 9 . |
| . 8 . | . . 4 | . . 1 |
|-------+-------+-------|
| . 4 . | . 2 . | 9 7 . |
| . . . | . . . | . . . |
| . 3 1 | . 8 . | . 6 . |
|-------+-------+-------|
| 9 . . | 6 . . | . 2 . |
| . 1 . | . . 7 | . . . |
| 5 . 4 | . . . | . . . |
*-----------------------*
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Postby m_b_metcalf » Thu May 17, 2007 2:31 pm

underquark wrote:This from an earlier thread?

Nice. With a fully symmetric solution grid. (I was just about to try to generate one, but now I don't need to bother.)

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Postby Red Ed » Thu May 17, 2007 6:23 pm

A much prettier one:
Code: Select all
 . 1 . 5 . . . . .
 . . 2 3 . . . . 4
 . . 8 . . . 3 7 .
 . . . . 1 . . 2 5
 . . . 6 . 4 . . .
 5 8 . . 9 . . . .
 . 3 7 . . . 2 . .
 6 . . . . 7 8 . .
 . . . . . 5 . 9 .
:D
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Postby m_b_metcalf » Thu May 17, 2007 7:24 pm

Red Ed wrote:A much prettier one:

And a cousin:
Code: Select all
 . . . . . . . . .
 . . 5 . 3 . 7 . .
 7 . 2 9 . 4 3 . 6
 8 . 9 7 . 2 4 . 5
 . . . . . . . . .
 5 . 6 8 . 3 1 . 2
 4 . 7 6 . 1 8 . 3
 . . 3 . 7 . 5 . .
 . . . . . . . . .           SE 7.1

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Postby udosuk » Fri May 18, 2007 12:29 pm

Well, it surprises me nobody here notices that this concept has been thoroughly discussed 7 months before, in the following thread:

http://forum.enjoysudoku.com/viewtopic.php?t=4931&start=1
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