## A Similar Variant ?

For fans of Killer Sudoku, Samurai Sudoku and other variants

### A Similar Variant ?

If the cells marked asterisk (*) were required to contain the digits 1..9, which Sudoku variant would it come closest to matching? Would the extra constraint allow for unique puzzles with less than 17 clues?

Code: Select all
` +-----------------------+ | * . . | . . . | . . . | | . . . | * . . | . . . | | . . . | . . . | * . . | |-------+-------+-------| | . * . | . . . | . . . | | . . . | . * . | . . . | | . . . | . . . | . * . | |-------+-------+-------| | . . * | . . . | . . . | | . . . | . . * | . . . | | . . . | . . . | . . * | +-----------------------+`

Note: This constraint spans every row, column, and box.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

No idea.

Long diagonals is another constraint. Did you omit it on purpose?
wintder

Posts: 297
Joined: 24 April 2007

Asterisk Sudoku is quite similar. Asterisk can contain as few as 15 givens, so there are chances yours could have less than 17 clues. I'll try to generate some.
Jean-Christophe

Posts: 149
Joined: 22 January 2006

asterisk, centre dot & girandola are similar sudoku variants...

what sets this DIAMOND variant is the fact that members of the diamond group do not share any other sector

This also would make it fun if one of those (jumping animals) constraints is added

tarek

tarek

Posts: 2647
Joined: 05 January 2006

### Re: A Similar Variant ?

daj95376 wrote:.... Would the extra constraint allow for unique puzzles with less than 17 clues? ....

Almost certainly. You now have 28 constraints instead of 27 (9 rows, 9 columns, 9 boxes, and your new one).

Why not go whole-hog and introduce 9 new constraints instead of just 1? In addition to the one you depicted, you could, for example, generate another one by moving each asterisk one cell to the right (or two cells to the left, to keep it in the same box).

This would give nine new constraints, pairwise disjoint (as are rows, columns, and boxes), whose union is the entire grid. And each of the nine would share the property you mentioned, of intersecting each row, column, and box exactly once.

The nine new constraints would be:

A. the one you proposed
B. move each asterisk 1 cell to the right (or 2 to the left)
C. move each asterisk 1 cell to the left (or 2 to the right)
D. move each asterisk 1 cell down (or 2 up)
E. move each asterisk 1 cell up (or 2 down)
F. combine B and D
G. combine B and E
H. combine C and D
I. combine C and E

I'd bet the minimum number of initial givens would be a lot less than 17.

Bill Smythe
Smythe Dakota

Posts: 543
Joined: 11 February 2006

apart from the diamond, you can have many variations:
Code: Select all
` Zorro +-----------------------+  | * . . | . . . | . . . |  | . . . | * . . | . . . |  | . . . | . . . | * . . |  |-------+-------+-------|  | . . . | . . . | . * . |  | . . . | . * . | . . . |  | . * . | . . . | . . . |  |-------+-------+-------|  | . . * | . . . | . . . |  | . . . | . . * | . . . |  | . . . | . . . | . . * |  +-----------------------+ `
tarek

tarek

Posts: 2647
Joined: 05 January 2006

here are 2 Diamond Windokus, advanced techniques needed
Code: Select all
`1 . 6|. . .|3 . 4. . .|. 1 .|. . .. . .|. . .|. . 2-----+-----+-----. . .|. . .|. . .. 9 .|. . .|. 7 .. . .|. . .|. . .-----+-----+-----2 . .|. . .|. . .. . .|. 4 .|. . .5 . 1|. . .|2 . 9. 6 .|. . .|. 3 .4 . 1|. . .|5 . 2. 3 .|. . .|. 8 .-----+-----+-----. . .|. . .|. . .. . .|. . .|. . .. . .|. . .|. . .-----+-----+-----. 5 .|. . .|. 4 .1 . 3|. . .|2 . 8. 9 .|. . .|. 1 .`
tarek

tarek

Posts: 2647
Joined: 05 January 2006

This diamond constraint is found in several "restrained prototypes" which HATMAN and I have been studying. It reminds me of the "DG diagonals" that HATMAN introduced to me many months ago.

Just to recap, these are the normal diagonals:
Code: Select all
`A..|...|..B.A.|...|.B...A|...|B..---+---+---...|A.B|......|.X.|......|B.A|...---+---+---..B|...|A...B.|...|.A.B..|...|..A`

These are the DG diagonals:
Code: Select all
`A..|.A.|..A...|...|.....B|.B.|B..---+---+---...|...|...A.B|.X.|B.A...|...|...---+---+---..B|.B.|B.....|...|...A..|.A.|..A`

If you read the cells row by row and list out the column indices, a leading diagonal would have the list [123456789] and a non-leading one would have [987654321]. As for the DG ones, you're reading the cells box by box and list out the "position indices" (within each box), and the lists are also [123456789] & [987654321] respectively.

Now look at the "diamond":
Code: Select all
`A..|...|......|A..|......|...|A..---+---+---.A.|...|......|.A.|......|...|.A.---+---+---..A|...|......|..A|......|...|..A`

If you list out the column indices row by row, the list is [147258369]. Now if you list out the position indices box by box, it is [147258369] again! And for completeness we should have a "non-leading diamond" to complement this "leading diamond", with the list [963852741]:
Code: Select all
`...|...|..B...|..B|.....B|...|...---+---+---...|...|.B....|.B.|....B.|...|...---+---+---...|...|B.....|B..|...B..|...|...`

So, for the sake of elegancy we should focus on puzzles with both "diamonds" instead of those with only one, just as we usually deal with X puzzles instead of \ or / puzzles.

Now here is a solution grid of a puzzle with both diamonds, both diagonals, both DG diagonals, the DG property and the additional properties that each mini-row sums to 15 and each mini-column being a set of 3 consecutive digits (and it even includes all 18 "generalised diamonds" as proposed by Bill):
159 672 834
267 483 915
348 591 726

726 348 591
834 159 672
915 267 483

483 915 267
591 726 348
672 834 159

Of all the "restrained prototypes" I know, only some NNC (kNight Non-Consecutive) grids has both diamonds, e.g.:
231 564 897
456 789 123
978 312 645

897 231 564
123 456 789
645 978 312

564 897 231
789 123 456
312 645 978

Also, in "NC AK AN" and "FNC AK AN" grids, one of the diamonds has 1..9 while the other is solely occupied by a single digit. In "FNC CNC" grids, both diamonds are occupied by 3 digits 3 times each. In "FNC AK X" grids, none of the diamonds are neatly occupied (e.g. one has [117654399], one has [438555276]).

PS:
tarek, don't think your "Zorro" constraint is of any particular significance other than fictional value, just like we don't study this constraint much:
Code: Select all
`A..|...|....A.|...|.....A|...|...---+---+---...|..A|......|.A.|......|A..|...---+---+---...|...|A.....|...|.A....|...|..A`

Last edited by udosuk on Wed Jun 18, 2008 10:43 am, edited 2 times in total.
udosuk

Posts: 2698
Joined: 17 July 2005

Out of the 46656 [6^6] ways to have these disjoint 1-rookerys - or non attacking rook [NAR] patterns, in an average completed grid there are on average 20 of these equivalent templates. In the MC grid there are considerably more. So for a given template e.g diamond, there are 20/46656 less total [6*10^21] grids. There are also less unavoidable sets in each grid.

The minimum will be whatever it will be, but if Tarek's windokus are valid at least 14 !

C

Code: Select all
`+---+---+---+|1..|...|..8||...|2.7|...||6..|...|3..|+---+---+---+|.4.|...|.1.||...|.5.|...||.9.|...|.6.|+---+---+---+|..7|...|..4||...|3.8|...||2..|...|..9|+---+---+---+`

This double diamond grid has relativly few completions, so real scope for clues less than ?12

C
Last edited by coloin on Wed Jun 18, 2008 10:54 am, edited 1 time in total.
coloin

Posts: 1702
Joined: 05 May 2005

coloin wrote:The minimum will be whatever it will be, but if Tarek's windokus are valid at least 14 !

Well, if you guys are so keen on the minimum, the answer is ONE.

With both diamonds, as well as kNight Non-Consecutive and Quad-kNight Non-Consecutive, a single clue of 8 @ r3c4 gives a unique solution!

PS:
Since coloin mentioned the MC grid, it also has 1..9 in one diamond and 9 occurences of a single digit in another, much like "NC AK AN" & "FNC AK AN".
123 456 789
456 789 123
789 123 456

231 564 897
564 897 231
897 231 564

312 645 978
645 978 312
978 312 645

udosuk

Posts: 2698
Joined: 17 July 2005

I was just curious if adding this single constraint that spanned all units would result in some interesting puzzles and solutions. I like the nickname Diamond pattern.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

Code: Select all
`159672834267483915348591726726348591834159672915267483483915267591726348672834159231564897456789123978312645897231564123456789645978312564897231789123456312645978123456789456789123789123456231564897564897231897231564312645978645978312978312645`

=
Code: Select all
`C:\>sudoku -qFN -f'%#0c' test.txt'123456789456789123789123456231564897564897231897231564312645978645978312978312645''123456789456789123789123456231564897564897231897231564312645978645978312978312645''123456789456789123789123456231564897564897231897231564312645978645978312978312645'`

All MC grid isomorphs, I suspected this.....
C
coloin

Posts: 1702
Joined: 05 May 2005

No Zorro moves then

Here is a double diamond X
Code: Select all
`6 . 8|. 7 .|2 . 3. . .|. . .|. . .2 . .|. . .|. . 4-----+-----+-----. . .|7 . 5|. . .1 . .|. . .|. . .. . .|6 . 1|. . .-----+-----+-----7 . .|. . .|. . 5. . .|. . .|. . .4 . 3|. . .|6 . 2`

tarek

tarek

Posts: 2647
Joined: 05 January 2006

Here are two with 16 clues:

Code: Select all
`.2.8...........92..1.6.5.........7.....5.....5.9..........9..4.....7..1...8.........5...9.2.4.............1.3.....4.98....3.........2.......4.........58...6..1...`

The first one isn't hard. The second one is harder.
Jean-Christophe

Posts: 149
Joined: 22 January 2006

Thanks everyone for the puzzles and info

My solver can't handle these puzzles right now, but my puzzle generator could easily be modified to generate them. I just didn't know if generating Diamond puzzles would be more interesting than standard Sudoku puzzles.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

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