double lozenge tiling with inner patterns 3.8

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double lozenge tiling with inner patterns 3.8

Postby denis_berthier » Sun Dec 06, 2020 7:56 am

Let me introduce the double lozenge tiling with alternating inner lozenges and octagons

The pattern is as follows:

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 . ! . X . !
   +-------+-------+-------+
29 givens


When the pattern is repeated horizontally and vertically, it gives a semi-regular tiling of the plane (with lozenges of two sizes), each tile having an inner pattern: an octagon for the larger tiles and a lozenge for the smaller tiles
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 . . 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 . 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 . . . 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 . . 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 . . 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 . 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 . . . 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 . . 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 . . 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 . . 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 . . 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
 . 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 . . 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 . . 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 . . 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 X . . X . X . . X
 . X . . X . . X . . X . . X . . X . . X . . X . . X . . X . . X . . X .


Generating minimal puzzles with this pattern is not easy. (It took me almost a full day to generate 1200 of them with gsf's program.)
The first puzzle I'll propose is an easy one for fishermen:


Code: Select all
   +-------+-------+-------+
   ! . 2 . ! . 5 . ! . 6 . !
   ! 3 . . ! 7 . 1 ! . . 2 !
   ! . . 4 ! . . . ! 1 . . !
   +-------+-------+-------+
   ! . 7 . ! . 8 . ! . 3 . !
   ! 2 . . ! 5 . 3 ! . . 9 !
   ! . 3 . ! . 9 . ! . 2 . !
   +-------+-------+-------+
   ! . . 7 ! . . . ! 6 . . !
   ! 9 . . ! 6 . 5 ! . . 7 !
   ! . 4 . ! . 7 . ! . 5 . !
   +-------+-------+-------+

.2..5..6.3..7.1..2..4...1...7..8..3.2..5.3..9.3..9..2...7...6..9..6.5..7.4..7..5. #  2697 FNBTHW C28.m/S8.f
29 givens, SER = 3.8, S = 3, W = 3
Last edited by denis_berthier on Sun Dec 06, 2020 9:28 am, edited 1 time in total.
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Re: double lozenge tiling with inner patterns 3.8

Postby m_b_metcalf » Sun Dec 06, 2020 8:44 am

denis_berthier wrote:
Generating puzzles with this pattern is not easy. (It took me almost a full day to generate 1200 of them with gsf's program.)


On the contrary, it's trivial. My program poured them out by the thousand in a few seconds. Among them were these minimal(!) puzzles with symmetry of givens:
Code: Select all
.1..2..3.2..4.5..6..5...4...7..1..6.5..2.7..1.8..5..2...3...1..8..1.3..7.4..7..5.   ED=1.5/1.2/1.2   
.1..2..3.4..5.1..6..7...2...6..5..4.7..6.2..8.5..4..2...6...8..2..9.4..5.3..6..9.   ED=4.2/1.2/1.2   
.1..2..3.4..5.6..7..7...2...8..1..6.7..6.3..1.3..7..4...5...1..1..3.2..8.6..5..7.   ED=4.5/1.2/1.2   
.1..2..3.4..1.5..6..5...4...6..4..7.3..2.1..4.5..3..8...3...7..8..7.2..3.4..1..2.   ED=6.6/1.2/1.2   
.1..2..3.4..5.6..2..2...7...4..6..2.1..8.4..7.5..9..8...1...5..5..9.2..8.3..5..7.   ED=7.6/1.2/1.2   
.1..2..3.3..4.5..1..2...6...3..6..1.2..3.7..8.5..4..7...4...8..5..1.6..7.7..8..5.   ED=8.4/1.2/1.2 

.9..3..6.3..1.8..5..5...3...5..2..9.8..9.1..2.1..8..4...7...4..4..2.9..7.6..7..1.   ED=8.3/8.3/3.4   
.2..6..5.4..1.2..3..6...2...3..1..7.1..7.3..9.7..9..3...8...4..7..8.9..6.5..4..8.   ED=8.5/8.5/3.8   
.1..9..7.4..1.7..2..2...9...6..7..8.3..9.1..7.2..3..5...1...8..8..3.9..4.3..1..9.   ED=8.6/8.6/3.8   
.6..1..5.8..9.6..4..7...6...4..8..3.1..7.3..9.7..2..6...4...3..6..4.1..2.5..9..4.   ED=8.9/8.9/3.0   
.5..4..9.1..8.9..3..9...8...9..8..3.4..3.7..5.7..2..1...2...1..7..1.2..9.1..5..4.   ED=9.0/8.5/2.6   
.4..2..5.9..8.3..4..2...3...2..5..1.6..2.8..5.9..6..8...7...8..4..7.2..1.6..8..4.   ED=9.1/9.1/2.6   



Regards,

Mike
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Re: double lozenge tiling with inner patterns 3.8

Postby denis_berthier » Sun Dec 06, 2020 9:28 am

m_b_metcalf wrote:
denis_berthier wrote:
Generating puzzles with this pattern is not easy. (It took me almost a full day to generate 1200 of them with gsf's program.)


On the contrary, it's trivial. My program poured them out by the thousand in a few seconds. Among them were these minimal(!) puzzles with symmetry of givens:
Code: Select all
.1..2..3.2..4.5..6..5...4...7..1..6.5..2.7..1.8..5..2...3...1..8..1.3..7.4..7..5.   ED=1.5/1.2/1.2   
.1..2..3.4..5.1..6..7...2...6..5..4.7..6.2..8.5..4..2...6...8..2..9.4..5.3..6..9.   ED=4.2/1.2/1.2   
.1..2..3.4..5.6..7..7...2...8..1..6.7..6.3..1.3..7..4...5...1..1..3.2..8.6..5..7.   ED=4.5/1.2/1.2   
.1..2..3.4..1.5..6..5...4...6..4..7.3..2.1..4.5..3..8...3...7..8..7.2..3.4..1..2.   ED=6.6/1.2/1.2   
.1..2..3.4..5.6..2..2...7...4..6..2.1..8.4..7.5..9..8...1...5..5..9.2..8.3..5..7.   ED=7.6/1.2/1.2   
.1..2..3.3..4.5..1..2...6...3..6..1.2..3.7..8.5..4..7...4...8..5..1.6..7.7..8..5.   ED=8.4/1.2/1.2 

.9..3..6.3..1.8..5..5...3...5..2..9.8..9.1..2.1..8..4...7...4..4..2.9..7.6..7..1.   ED=8.3/8.3/3.4   
.2..6..5.4..1.2..3..6...2...3..1..7.1..7.3..9.7..9..3...8...4..7..8.9..6.5..4..8.   ED=8.5/8.5/3.8   
.1..9..7.4..1.7..2..2...9...6..7..8.3..9.1..7.2..3..5...1...8..8..3.9..4.3..1..9.   ED=8.6/8.6/3.8   
.6..1..5.8..9.6..4..7...6...4..8..3.1..7.3..9.7..2..6...4...3..6..4.1..2.5..9..4.   ED=8.9/8.9/3.0   
.5..4..9.1..8.9..3..9...8...9..8..3.4..3.7..5.7..2..1...2...1..7..1.2..9.1..5..4.   ED=9.0/8.5/2.6   
.4..2..5.9..8.3..4..2...3...2..5..1.6..2.8..5.9..6..8...7...8..4..7.2..1.6..8..4.   ED=9.1/9.1/2.6   



It's great for your program. But for gsf's it takes much longer than other patterns. All the puzzles I generated were minimal. I'll state it in my first post.
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Re: double lozenge tiling with inner patterns 3.8

Postby Leren » Sun Dec 06, 2020 9:34 am

Hi Mike, all those puzzles have 180 rotational symmetry, which makes for relatively easy solutions, except this one, I think.

Code: Select all
.2..6..5.4..1.2..3..6...2...3..1..7.1..7.3..9.7..9..3...8...4..7..8.9..6.5..4..8.

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Re: double lozenge tiling with inner patterns 3.8

Postby m_b_metcalf » Sun Dec 06, 2020 9:39 am

denis_berthier wrote: All the puzzles I generated were minimal. I'll state it in my first post.

Ah, minimal or not minimal, that's a huge difference with these high clue counts. M
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Re: double lozenge tiling with inner patterns 3.8

Postby denis_berthier » Sun Dec 06, 2020 9:41 am

Leren wrote:Hi Mike, all those puzzles have 180 rotational symmetry, which makes for relatively easy solutions, except this one, I think.
Code: Select all
.2..6..5.4..1.2..3..6...2...3..1..7.1..7.3..9.7..9..3...8...4..7..8.9..6.5..4..8.

SER = 8.5, Z = 7, W = 5. not trivial. But easier than the hardest I found.
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Re: double lozenge tiling with inner patterns 3.8

Postby denis_berthier » Sun Dec 06, 2020 9:45 am

m_b_metcalf wrote:
denis_berthier wrote: All the puzzles I generated were minimal. I'll state it in my first post.

Ah, minimal or not minimal, that's a huge difference with these high clue counts. M

Sure.
When I find a nice looking pattern, I first try to generate minimal puzzles for it. If I can't after one hour (of gsf's program), and only then, I try non minimal ones.
In any case, I think your program is faster than gsf's.
BTW, could you try it for minimal puzzles on the spiral pattern? I couldn't find any.
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Re: double lozenge tiling with inner patterns 3.8

Postby m_b_metcalf » Sun Dec 06, 2020 10:04 am

denis_berthier wrote:BTW, could you try it for minimal puzzles on the spiral pattern? I couldn't find any.

A quick look yields nothing so far. I'll post one if one turns up. In the meantime, a gem:
Code: Select all
 . 1 . . 2 . . 3 .
 4 . . 3 . 5 . . 2
 . . 2 . . . 6 . .
 . 5 . . 7 . . 1 .
 7 . . 8 . 1 . . 3
 . 8 . . 3 . . 4 .
 . . 6 . . . 9 . .   A non-minimal diamond.
 9 . . 4 . 7 . . 5
 . 7 . . 9 . . 8 .   ED=8.3/8.3/8.3
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Re: double lozenge tiling with inner patterns 3.8

Postby RSW » Sun Dec 06, 2020 10:13 am

denis_berthier wrote:The first puzzle I'll propose is an easy one for fishermen:


Code: Select all
 
.2..5..6.3..7.1..2..4...1...7..8..3.2..5.3..9.3..9..2...7...6..9..6.5..7.4..7..5. #  2697 FNBTHW C28.m/S8.f
29 givens, SER = 3.8, S = 3, W = 3

I don't like to fish, so:
Code: Select all
 +----------------+------------------+-------------------+
 | 178  2    18   | 3489   5    489  | 37-489 6    348   |
 | 3    5689 568  | 7      46   1    | 4589   489  2     |
 | 578  5689 4    | 2389   236  2689 | 1      789  358   |
 +----------------+------------------+-------------------+
 | 145  7    9    | 124    8    246  | 45     3    1456  |
 | 2    168  168  | 5      146  3    | 7-48   1478 9     |
 | 1458 3    1568 | 14     9    7    | 458    2    14568 |
 +----------------+------------------+-------------------+
 | 158  158  7    | 123489 1234 2489 | 6      1489 1348  |
 | 9    18  *23   | 6      134  5    |*248-3  148  7     |
 | 6    4   *23   | 1389   7    89   |*289-3  5    138   |
 +----------------+------------------+-------------------+

Unique Rectangle UR+2 (2/3)r89c37 with UR+ candidates in c7 forms a locked set with 4589r246c7 => -48r5c7 -489r1c7, and with bilocal digit 2 in column 7 => -3r89c7, btte
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Re: double lozenge tiling with inner patterns 3.8

Postby denis_berthier » Sun Dec 06, 2020 10:32 am

RSW wrote:
denis_berthier wrote:The first puzzle I'll propose is an easy one for fishermen:
Code: Select all
  .2..5..6.3..7.1..2..4...1...7..8..3.2..5.3..9.3..9..2...7...6..9..6.5..7.4..7..5. #  2697 FNBTHW C28.m/S8.f
29 givens, SER = 3.8, S = 3, W = 3
I don't like to fish,

Maybe you prefer fish already prepared as sushi. Sorry, no sushi here. And no sashimi either.
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Re: double lozenge tiling with inner patterns 3.8

Postby m_b_metcalf » Sat Dec 19, 2020 1:12 pm

denis_berthier wrote:Let me introduce the double lozenge tiling with alternating inner lozenges and octagons

The pattern is as follows:

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 . ! . X . !
   +-------+-------+-------+
29 givens

Denis, You might be interested to look at this variant.

Regards,

Mike
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Re: double lozenge tiling with inner patterns 3.8

Postby denis_berthier » Sat Dec 19, 2020 3:42 pm

m_b_metcalf wrote:Denis, You might be interested to look at this variant

Mike, Thanks
It's a very nice pattern also.

And the first puzzle by JPF with this pattern is extremely interesting also:
Code: Select all
 1 . . | . 2 . | . . 3
 . 4 . | 5 . 6 | . 1 .
 . . 7 | . . . | 5 . .
-------+-------+-------
 . 8 . | . 4 . | . 5 .
 5 . . | 9 . 1 | . . 8
 . 3 . | . 5 . | . 9 .
-------+-------+-------
 . . 3 | . . . | 2 . .
 . 1 . | 2 . 3 | . 4 .
 4 . . | . 9 . | . . 6    ED=9.3/9.3/9.3


SER 9.3 is normally beyond a manual solver's abilities.

It is easy to check that this puzzle is in T&E(1). But, it has no easy solution whip braids, let alone with whips.
However, it is a champion of g-whips. it is very rare to see so many g-whips in a resolution path.
Notice that the first elimination is done by a g-whip[4] and the puzzle is solved in gW10, but if g-whips are not activated, the first elimination is done by a whip[12] or a braid[11].

Hidden Text: Show
***********************************************************************************************
*** SudoRules 20.1.s based on CSP-Rules 2.1.s, config = gW
*** Using CLIPS 6.32-r773
***********************************************************************************************
183 candidates, 1024 csp-links and 1024 links. Density = 6.15%
140 g-candidates, 858 csp-glinks and 519 non-csp glinks
g-whip[4]: b2n7{r2c5 r1c456} - c8n7{r1 r789} - r8n7{c9 c1} - b4n7{r4c1 .} ==> r5c5 ≠ 7
whip[6]: r7c8{n8 n7} - r8c7{n7 n9} - r2c7{n9 n7} - r5n7{c7 c2} - c1n7{r6 r8} - c5n7{r8 .} ==> r9c7 ≠ 8
g-whip[7]: c4n6{r7 r456} - r5c5{n6 n3} - b6n3{r5c7 r4c7} - c4n3{r4 r3} - c4n1{r3 r9} - r9c7{n1 n7} - r7c8{n7 .} ==> r7c4 ≠ 8
g-whip[8]: c6n5{r9 r7} - r7n4{c6 c4} - c4n6{r7 r456} - r5c5{n6 n3} - r4n3{c4 c7} - r9c7{n3 n1} - c4n1{r9 r3} - c4n3{r3 .} ==> r9c6 ≠ 7
g-whip[9]: r7n4{c6 c4} - b8n6{r7c4 r789c5} - r5c5{n6 n3} - c8n3{r5 r9} - c7n3{r9 r4} - c4n3{r4 r3} - c4n1{r3 r9} - r9c7{n1 n7} - r7c8{n7 .} ==> r7c6 ≠ 8

whip[10]: c8n2{r5 r3} - r2n2{c9 c1} - r2n3{c1 c5} - r5c5{n3 n6} - c4n6{r6 r7} - r7n4{c4 c6} - c6n5{r7 r9} - r9c3{n5 n8} - c1n8{r8 r3} - r3n3{c1 .} ==> r5c3 ≠ 2
whip[6]: r5n2{c8 c2} - r5n7{c2 c7} - r5n3{c7 c5} - r2n3{c5 c1} - c1n2{r2 r3} - c8n2{r3 .} ==> r5c8 ≠ 6
whip[1]: b6n6{r6c7 .} ==> r1c7 ≠ 6
whip[7]: c8n3{r9 r5} - r5c5{n3 n6} - b8n6{r7c5 r7c4} - r7n4{c4 c6} - c6n5{r7 r9} - r9c2{n5 n2} - r5n2{c2 .} ==> r9c8 ≠ 7
whip[8]: r5n7{c8 c2} - r9n7{c2 c4} - r9n1{c4 c7} - c7n3{r9 r5} - c7n6{r5 r6} - r6c4{n6 n8} - r1c4{n8 n4} - c7n4{r1 .} ==> r4c7 ≠ 7
whip[9]: r5c5{n6 n3} - b2n3{r2c5 r3c4} - r4n3{c4 c7} - b6n6{r4c7 r6c7} - c7n1{r6 r9} - c4n1{r9 r7} - c4n4{r7 r1} - c7n4{r1 r5} - r5c3{n4 .} ==> r5c2 ≠ 6
whip[9]: r4c6{n7 n2} - r4c9{n2 n1} - c7n1{r6 r9} - r9n7{c7 c2} - r5c2{n7 n2} - c8n2{r5 r3} - c1n2{r3 r2} - r2n3{c1 c5} - b5n3{r5c5 .} ==> r4c4 ≠ 7
whip[2]: r4c4{n6 n3} - r5c5{n3 .} ==> r6c4 ≠ 6
whip[7]: r5n7{c8 c2} - r9n7{c2 c4} - r6c4{n7 n8} - r1c4{n8 n4} - c7n4{r1 r5} - r5c3{n4 n6} - r6n6{c1 .} ==> r6c7 ≠ 7
g-whip[8]: r2n3{c1 c5} - r3n3{c5 c1} - c1n2{r3 r456} - r5c2{n2 n7} - b6n7{r5c8 r456c9} - r2n7{c9 c7} - r9n7{c7 c4} - c5n7{r7 .} ==> r2c1 ≠ 8, r2c1 ≠ 9
whip[9]: r4c4{n6 n3} - r4c7{n3 n1} - b9n1{r9c7 r7c9} - c9n5{r7 r8} - b9n9{r8c9 r8c7} - r8c3{n9 n8} - c1n8{r8 r3} - r3n3{c1 c5} - c5n1{r3 .} ==> r4c3 ≠ 6
whip[9]: r5c2{n7 n2} - r9c2{n2 n5} - c6n5{r9 r7} - r7n4{c6 c4} - c4n6{r7 r4} - c4n3{r4 r3} - c1n3{r3 r2} - c1n2{r2 r3} - c8n2{r3 .} ==> r7c2 ≠ 7
whip[9]: c4n6{r7 r4} - r4n3{c4 c7} - r5n3{c7 c5} - b2n3{r2c5 r3c4} - c4n1{r3 r9} - c7n1{r9 r6} - c7n6{r6 r5} - c7n4{r5 r1} - c4n4{r1 .} ==> r7c4 ≠ 7
whip[9]: c4n4{r3 r7} - c4n6{r7 r4} - r4n3{c4 c7} - r5n3{c7 c5} - b2n3{r2c5 r3c4} - c4n1{r3 r9} - c7n1{r9 r6} - c7n4{r6 r5} - c7n6{r5 .} ==> r1c6 ≠ 4
g-whip[9]: r9c6{n5 n8} - r6n8{c6 c4} - b5n7{r6c4 r456c6} - r1c6{n7 n9} - r3c6{n9 n4} - r1c4{n4 n7} - r9n7{c4 c7} - c2n7{r9 r5} - c8n7{r5 .} ==> r9c2 ≠ 5
whip[2]: r9c2{n2 n7} - r5c2{n7 .} ==> r3c2 ≠ 2
whip[7]: r5n2{c2 c8} - r3n2{c8 c9} - c9n4{r3 r6} - r6n2{c9 c6} - r4c6{n2 n7} - b6n7{r4c9 r5c7} - r5c2{n7 .} ==> r4c1 ≠ 2
whip[9]: c1n8{r8 r3} - c1n3{r3 r2} - c1n2{r2 r6} - r5n2{c2 c8} - c8n3{r5 r9} - b9n8{r9c8 r7c8} - c8n7{r7 r1} - b2n7{r1c4 r2c5} - c5n8{r2 .} ==> r8c3 ≠ 8
g-whip[8]: b8n8{r8c5 r9c456} - b7n8{r9c3 r789c1} - r3n8{c1 c8} - r7c8{n8 n7} - c5n7{r7 r8} - c1n7{r8 r456} - r5c2{n7 n2} - c8n2{r5 .} ==> r2c5 ≠ 8
whip[3]: b9n8{r9c8 r8c7} - c1n8{r8 r7} - c5n8{r7 .} ==> r3c8 ≠ 8
whip[4]: c1n3{r3 r2} - c1n2{r2 r6} - r5n2{c2 c8} - r3c8{n2 .} ==> r3c1 ≠ 6
whip[5]: r3c2{n9 n6} - r3c8{n6 n2} - r5n2{c8 c2} - c1n2{r6 r2} - c1n3{r2 .} ==> r3c1 ≠ 9
whip[5]: c5n8{r8 r3} - r3n1{c5 c4} - r9c4{n1 n7} - c5n7{r7 r2} - b2n3{r2c5 .} ==> r9c6 ≠ 8
naked-single ==> r9c6 = 5
whip[4]: r9c3{n2 n8} - b1n8{r1c3 r3c1} - c1n2{r3 r2} - c1n3{r2 .} ==> r4c3 ≠ 2
whip[4]: b5n7{r6c6 r4c6} - b4n7{r4c1 r5c2} - r5n2{c2 c8} - r4n2{c9 .} ==> r6c9 ≠ 7
whip[4]: c8n2{r3 r5} - b4n2{r5c2 r6c3} - r9c3{n2 n8} - b1n8{r1c3 .} ==> r3c1 ≠ 2
whip[1]: r3n2{c9 .} ==> r2c9 ≠ 2
whip[4]: r9c3{n2 n8} - b1n8{r1c3 r3c1} - c1n3{r3 r2} - r2n2{c1 .} ==> r6c3 ≠ 2
whip[5]: c9n1{r6 r7} - r4n1{c9 c3} - r4n9{c3 c1} - r7n9{c1 c2} - r7n5{c2 .} ==> r6c7 ≠ 1
whip[4]: c9n4{r3 r6} - r6n1{c9 c3} - r4c3{n1 n9} - r2n9{c3 .} ==> r3c9 ≠ 9
whip[5]: c4n6{r7 r4} - r4n3{c4 c7} - c7n1{r4 r9} - c4n1{r9 r3} - c4n3{r3 .} ==> r7c4 ≠ 4
hidden-single-in-a-block ==> r7c6 = 4
whip[4]: c2n7{r9 r5} - c8n7{r5 r1} - b2n7{r1c4 r2c5} - b8n7{r8c5 .} ==> r9c7 ≠ 7
whip[4]: b2n4{r1c4 r3c4} - r3n8{c4 c1} - r3n3{c1 c5} - r3n1{c5 .} ==> r1c4 ≠ 8
whip[4]: b3n6{r1c8 r3c8} - b3n2{r3c8 r3c9} - b3n4{r3c9 r1c7} - r1c4{n4 .} ==> r1c8 ≠ 7
whip[2]: c8n7{r7 r5} - c2n7{r5 .} ==> r7c1 ≠ 7
whip[5]: c4n6{r7 r4} - c4n3{r4 r3} - r3n4{c4 c9} - r3n2{c9 c8} - r3n6{c8 .} ==> r7c2 ≠ 6
whip[1]: c2n6{r3 .} ==> r1c3 ≠ 6
whip[5]: r5n2{c2 c8} - c9n2{r6 r3} - r3n4{c9 c4} - r1c4{n4 n7} - r9n7{c4 .} ==> r9c2 ≠ 2
stte

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P.S.: here is the semi-regular tiling produced by repeating the pattern (with two types of lozenge tiles with different inner patterns).
There's also a stereographic effect.

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denis_berthier
2010 Supporter
 
Posts: 1983
Joined: 19 June 2007
Location: Paris


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