## Symmetric clue values

Everything about Sudoku that doesn't fit in one of the other sections

### Symmetric clue values

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.)

Regards,

Mike Metcalf

m_b_metcalf
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### Re: Symmetric clue values

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.
Ocean

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### Re: Symmetric clue values

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.

Regards,

Mike Metcalf

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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 solutions124389756856271934379654128483765219265198347917423865791536482532847691648912573183429756257861934469357128326714589875692341914583672791238465532146897648975213`

• 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.
JPF
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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.

Bill Smythe
Smythe Dakota

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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`

Regards,

Mike Metcalf

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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 | . . . | . . . |*-----------------------* `
underquark

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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.)

Thanks,

Mike Metcalf

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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 .`
Red Ed

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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`

Regards,

Mike Metcalf

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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
udosuk

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