Symmetry as a constraint

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Symmetry as a constraint

Postby Smythe Dakota » Sun Jul 27, 2008 11:26 am

What if the following constraint is added to the vanilla Sudoku rules:

In the final solution (not just the initial clues), each digit's opposite counterpart must appear in the opposite location. That is, if the digit D appears in row R, column C, then the digit 10-D must appear in row 10-R, column 10-C. For example, if there is a 4 in r2c3, there must be a 6 in r8c7.

Can anybody come up with an interesting puzzle along these lines?

What about minimality of the initial clues? Obviously, each clue will automatically generate its counterpart, so I'm inclined to believe that a minimal clue set could contain as few as 8 clues. (Note that the digit in r5c5 must always be 5, so it is never necessary to provide that as a clue.)

Of course, a minimal clue set cannot also be symmetric. In fact, it must be anti-symmetric, in the sense that if there is an initial clue in row R, column C, then there cannot also be an initial clue in row 10-R, column 10-C. Thus, a minimal symmetric clue set (minimal among symmetric clue sets) will have exactly twice as many clues as its minimal non-symmetric counterparts.

Any thoughts?

Bill Smythe
Smythe Dakota
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Postby udosuk » Sun Jul 27, 2008 1:17 pm

This has been discussed before, the name of this variant is Emerald, coined by gurth:

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Postby Smythe Dakota » Mon Jul 28, 2008 7:39 am

Thank you for the two links. But the first seems to contain nothing but the basic definition, and is buried within a discussion of various other precious gems. And the second seems to bog down into a heated political debate of some sort (I lost interest after the first page).

I was hoping somebody would simply post an interesting Emerald puzzle, one whose difficulty is not great but which requires new and interesting solving techniques.

Bill Smythe
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Postby HATMAN » Thu Aug 07, 2008 6:44 pm


I've done a few killers, about a year ago on DJApe's site, which Matt pointed out had this property, which if assumed significantly simplified the solution path.

Given that the solution is rotationaly symmetrical, surely you can have left right symetrical clues, for example?

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Re: Symmetry as a constraint

Postby dyitto » Mon Dec 13, 2010 9:39 pm

Here's two Emeralds:
Code: Select all

Code: Select all
evert on the crashed forum
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