## Symmetry as a constraint

For fans of Killer Sudoku, Samurai Sudoku and other variants

### Symmetry as a constraint

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

Posts: 555
Joined: 11 February 2006

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

http://forum.enjoysudoku.com/viewtopic.php?t=4912

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

udosuk

Posts: 2698
Joined: 17 July 2005

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
Smythe Dakota

Posts: 555
Joined: 11 February 2006

Bill

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?

Maurice
HATMAN

Posts: 221
Joined: 25 February 2006

### Re: Symmetry as a constraint

Here's two Emeralds:
Code: Select all
`...|...|......|...|.25.3.|...|6.4---+---+---2..|...|......|...|..9.53|...|...---+---+---...|5..|......|..1|4.....|8..|...`

Code: Select all
`...|8..|.3....|...|.5...8|3..|...---+---+---...|...|..7...|...|61....|...|...---+---+---..1|...|.6.4..|.1.|..9...|...|...`
evert on the crashed forum

dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam