Sudoku-V (or did this already exist?)

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Sudoku-V (or did this already exist?)

Postby evert » Sun Aug 24, 2008 7:16 pm

The cells forming a V-shape are a constraint - i.e. must contain all digits 1-9.

Code: Select all
X . . | . . . | . . X
. . . | . . . | . . .
. X . | . . . | . X .
------+-------+-------
. . . | . . . | . . .
. . X | . . . | X . .
. . . | . . . | . . .
------+-------+-------
. . . | X . X | . . .
. . . | . . . | . . .
. . . | . X . | . . .


The consecutive cells are separated by a knight's move.

Also, the rotated/mirrored V-shapes are constraints:

Code: Select all
. . . | . X . | . . .
. . . | . . . | . . .
. . . | X . X | . . .
------+-------+-------
. . . | . . . | . . .
. . X | . . . | X . .
. . . | . . . | . . .
------+-------+-------
. X . | . . . | . X .
. . . | . . . | . . .
X . . | . . . | . . X

X . . | . . . | . . .
. . X | . . . | . . .
. . . | . X . | . . .
------+-------+-------
. . . | . . . | X . .
. . . | . . . | . . X
. . . | . . . | X . .
------+-------+-------
. . . | . X . | . . .
. . X | . . . | . . .
X . . | . . . | . . .


. . . | . . . | . . X
. . . | . . . | X . .
. . . | . X . | . . .
------+-------+-------
. . X | . . . | . . .
X . . | . . . | . . .
. . X | . . . | . . .
------+-------+-------
. . . | . X . | . . .
. . . | . . . | X . .
. . . | . . . | . . X


Here's a puzzle with this constraint:

Code: Select all
000000020
000002750
070830100
000003200
040000070
500000000
000000040
000006082
005000000
evert
 
Posts: 186
Joined: 26 August 2005

Postby Smythe Dakota » Sun Aug 24, 2008 8:39 pm

I assume all four of the V-constraints apply simultaneously.

Bill Smythe
Smythe Dakota
 
Posts: 533
Joined: 11 February 2006

Postby Glyn » Sun Aug 24, 2008 9:33 pm

Bill I found a solution using three constraints v,^,>. The first two were not sufficient to solve the puzzle.
To solve with singles all four constraints were required.
Glyn
 
Posts: 357
Joined: 26 April 2007

Postby evert » Mon Aug 25, 2008 2:45 am

Interesting thought that in certain cases not all constraints are actually used.

Here's one combined with X:

Code: Select all
000405000
000900083
000030000
002000000
040000000
000000040
300050000
070890001
000000000


(so < > ^ V and two main diagonals)
evert
 
Posts: 186
Joined: 26 August 2005

Postby evert » Wed Sep 03, 2008 5:21 pm

This 10-clue one should be a little bit more fun:
Code: Select all
500000006
300000000
000106000
000000000
090200000
018000000
000007000
000000000
000000000

It's all V-constraints and also both main diagonals - let's say Sudoku V-X.

I also found some situations with very few clues and no solutions.
For example:
Code: Select all
100000000
007000000
000050000
000000600
000000002
000000300
050090000
004000000
800000000


has no solutions (with or without X).

Either windoku-V has no solutions AT ALL.
evert
 
Posts: 186
Joined: 26 August 2005

Postby HATMAN » Wed Sep 03, 2008 6:09 pm

Evert

Nice one - Matt and I have been collecting interacting constaints that either have no solutions or more importantly minimal solutions.

Can I clarify: with windoku how many Vs cause failure?
HATMAN
 
Posts: 203
Joined: 25 February 2006

Postby udosuk » Fri Sep 05, 2008 1:18 pm

Nice puzzles. Enjoyed all 3 of them!:)

evert wrote:I also found some situations with very few clues and no solutions.
For example:
Code: Select all
100000000
007000000
000050000
000000600
000000002
000000300
050090000
004000000
800000000


has no solutions (with or without X).

This shouldn't come as any surprise, because you have r3c5=5, which rules out 5 from r1c9 (<), r3c28 (r3), & r9c5 (c5). As a result, 5 @ V is locked @ r5c37+r7c46, which has a pointing elimination @ r7c2 (Λ & r7). So you can't have r7c2=5.:idea:

evert wrote:Either windoku-V has no solutions AT ALL.

Again this is easy to prove:
Code: Select all
 V  .  . | .  Λ  . | .  .  V
 .  .  . | .  .  . | .  .  .
 .  .  . | .  .  . | .  .  .
---------+---------+---------
 .  .  . | .  .  . | .  .  .
 R  .  * | .  R  . | .  .  R
 .  .  . | .  .  . | .  .  .
---------+---------+---------
 .  .  . | .  .  . | .  .  .
 .  .  . | .  .  . | .  .  .
 Λ  .  . | .  V  . | .  .  Λ

Consider r5c3. It eliminates all identical candidates @ r159c159 (through V, Λ or r5). But in windoku this is a hidden group of 9 different values, so a clear contradiction there. Therefore no windoku is possible as long as there are 2 opposite V's existing.:idea:
udosuk
 
Posts: 2698
Joined: 17 July 2005

Postby evert » Fri Sep 05, 2008 5:37 pm

Tx udosuk!

With only two and non-opposite V-groups, Windoku-V is possible.
Following puzzle has constraints (>,V) and windoku:
Code: Select all
000000000
000000200
004070000
000000000
003000010
000000050
000000300
000000820
090050600


Having both main diagonals (X) and windoku (NRC) and two non-opposite V-groups - there are zero solutions.
evert
 
Posts: 186
Joined: 26 August 2005

Postby udosuk » Sat Sep 06, 2008 2:49 am

evert wrote:Having both main diagonals (X) and windoku (NRC) and two non-opposite V-groups - there are zero solutions.

Yep that's true, but it's very difficult to prove explicitly. I'm running out for free time this weekend so it has to be put into the waiting list (for me)...:(

At the mean time, here is a side product - I found that with X, V, >, 2 upper windows (r234c234+r234c678) and the "outside dots" (r159c159) you get a puzzle with an essentially unique solution (barring digit mapping). In other words, if you fill in 8 cells of any established constraint group (the 9th cell can be forced as a naked single) you get a valid puzzle. Note there are a total of 36 constraint groups here: 9 rows, 9 columns, 9 boxes, 2 diagonals, 2 V's, 2 upper windows, 1 outside dots, 2 hidden windows (r234c159, r678c159).

Or, if you ditch the outside dots, you can have a valid 8-clue puzzle like this:

Code: Select all
Sudoku X-V->-UW (Upper Windows: r234c234, r234c678)

 1 . . | . 2 . | . . 3
 . . . | . . . | . . .
 . . . | . . . | . . .
-------+-------+-------
 . . . | . . . | . . .
 4 . . | . 5 . | . . 6
 . . . | . . . | . . .
-------+-------+-------
 . . . | . . . | . . .
 . . . | . . . | . . .
 . . . | . 8 . | . . 9

Note because I didn't specify the r159c159 group so r9c1 could be 2 or 7 at the start. It's not easy to eliminate the 2 to confirm r9c1=7, and even then it's still diabolical to solve.

But I like the elegancy of the extra constraints here: 2 diagonals, 2 non-opposite V's and 2 non-opposite windows.:)

There are also 2 alternative (slightly easier) 8-clue versions of this:
Code: Select all
 1 . . | . 2 . | . . 3
 . . . | . . . | . . .
 . . . | . . . | . . .
-------+-------+-------
 . . . | . . . | . . .
 4 . . | . . . | . . 6
 . . . | . . . | . . .
-------+-------+-------
 . . . | . . . | . . .
 . . . | . . . | . . .
 7 . . | . 8 . | . . 9

 . . . | . 2 . | . . 3
 . . . | . . . | . . .
 . . . | . . . | . . .
-------+-------+-------
 . . . | . . . | . . .
 4 . . | . 5 . | . . 6
 . . . | . . . | . . .
-------+-------+-------
 . . . | . . . | . . .
 . . . | . . . | . . .
 7 . . | . 8 . | . . 9

:idea:
udosuk
 
Posts: 2698
Joined: 17 July 2005

Postby evert » Mon Sep 15, 2008 3:32 pm

I've tried to add some more interactions between the V-groups and the other groups:

Here's three Sudoku V:
Code: Select all
006000090
000010000
010080000
000300002
930000000
000070900
000040000
000000000
065000000

040003000
090002000
000670000
006009000
000000000
080700000
020106000
000000000
000800091

003000000
000800074
000000000
000080607
100060908
000200000
000005000
050000000
092000000


Here's three Sudoku VX:
Code: Select all
000040006
000000000
000000003
000000060
900000000
020560090
008000010
000080002
000100000

000000000
000500000
000009001
002000009
100000000
000006082
000000000
000000000
400030000

050030000
010000020
000010000
000000000
000070002
080040600
000000000
004050000
000000000
evert
 
Posts: 186
Joined: 26 August 2005

Postby evert » Mon Nov 17, 2008 1:47 pm

What am I missing in the following puzzle? It's Sudoku VX: V^<>/ \
Code: Select all
...|..1|..3
.49|...|...
8..|...|...
---+---+---
...|...|372
..8|...|...
7..|...|...
---+---+---
..2|...|...
...|...|5..
...|...|...
I completed it until:
Code: Select all
2      | 7     | 56    | 45689 | 4569  | 1     | 4689  | 45689 | 3
1356   | 4     | 9     | 368   | 7     | 3568  | 2     | 568   | 168
8      | 156   | 35    | 234569| 46    | 23469 | 7     | 1569  | 1469
-------+-------+-------+-------+-------+-------+-------+-------+-------
456    | 56    | 146   | 1689  | 1469  | 4689  | 3     | 7     | 2
9      | 23    | 8     | 237   | 16    | 237   | 46    | 146   | 5
7      | 23    | 146   | 456   | 23    | 56    | 16    | 89    | 89
-------+-------+-------+-------+-------+-------+-------+-------+-------
3456   | 1569  | 2     | 145679| 8     | 45679 | 169   | 1369  | 1469
1346   | 1689  | 7     | 123469| 23    | 2346  | 5     | 3689  | 14689
146    | 15689 | 3456  | 134569| 1569  | 34569 | 4689  | 2     | 7
I expect some pointing pair...
evert
 
Posts: 186
Joined: 26 August 2005

Postby udosuk » Sat Nov 22, 2008 3:24 am

evert you're indeed missing some pointing eliminations.

There is an easy pair of pointing cells of 8 @ r24c6 (pointing @ r2c8) but it isn't critical to solve the puzzle.

Here is one possible short path to solve it from that position:

4 @ ">" locked @ r3c5+r9c1 (pointing @ r1c5+r3c46)
4 @ "^" locked @ r5c7+r9c1 (pointing @ r9c7)
4 @ b9 locked @ r78c9 (not elsewhere @ c9)
Hidden single @ r3: r3c5=4 (not elsewhere @ "<",">")
Hidden single @ "^": r5c7=4 (not elsewhere @ "V")
Hidden single @ r6: r6c4=4
Hidden single @ r6: r6c6=5 (not elsewhere @ d\)
Naked single: r3c3=3 (not elsewhere @ d\)
Naked pair: r5c58={16} (pointing @ r28c8)
Naked pair: r68c8={89} (not elsewhere @ c8)

All singles from here.

Another random observation: from your position it only takes X^ or V^ to give a unique solution, but you probably need wings/fishes to crack it.:idea:
udosuk
 
Posts: 2698
Joined: 17 July 2005


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