Toroidal Anti Chess (Formerly The Touchless Wrapper)

For fans of Killer Sudoku, Samurai Sudoku and other variants

Re: Update about corkscrew variant

Postby tarek » Fri Oct 13, 2017 3:35 pm

The paper lists many :D

I'm in the final phase before implementing the corkscrew variant

Tarek
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Re: 25x25 Anti-Root325 puzzle Difficult Machine solvable

Postby tarek » Sat Oct 14, 2017 7:31 pm

Anti-Root325 puzzle

The following puzzle is to demonstrate a double-orthogonally-symmetrically-minimal puzzle. This happens to target machine solvers only. Don't attempt solving

ROOT325_25x25_Puzzle.png
ROOT325_25x25_Puzzle.png (124.88 KiB) Viewed 151 times
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Re: continuity planes explained - The toroid

Postby tarek » Sun Oct 15, 2017 9:13 pm

When we discuss the continuity planes it is easier to understand them when you display your grid as a block of 4

Your starting Grid being the bottom right. You then display how the grid would appear if you moved up, if you moved left or if you moved diagonally up & left ( left and up achieves the same). You would need then to imagine that this block of 4 is endlessly surrounded by similar blocks of 4 from every direction.

Here is the Toroid made up from a 9x9 Sudoku (81 cells):
Continuity_Planes_Toroid.png
Continuity_Planes_Toroid.png (97.41 KiB) Viewed 143 times

Look at Square 1 (R1C1). You can see what happens when a king leaps and where it can potentially travel to: Orthogonally to Squraes 9,2,73 & 10 & diagonally to squares 81,74, 18 & 11. You can now confirm that leaping 9 squares up from Square 12 will land you on the same square. The same fate would happen had you leaped 18 squares up!! This is Easy to see when you imagine the grid as a 3D structure (Halo or doughnut)

This makes the toroid easy to understand. It is nothing more than a side to side wrap (Horizontal cylinder) combined with an up down wrap (vertical cylinder)

[Edit: corrected a small mistake]
Last edited by tarek on Sun Oct 15, 2017 11:11 pm, edited 1 time in total.
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Re: continuity planes explained - The projective plane

Postby tarek » Sun Oct 15, 2017 9:40 pm

It is more difficult to explain the projective plane because it is difficult to visualize as a 3d structure. The difficulty arises from the fact that it extends the ordinary concept of Euclidean plane.

On a Sudoku Anti chess variant the Projective plane is achieved by having a moebius side to side wrap and a moebius up down wrap. Here is how our block of 4 visualization (see the above post) appears:
Continuity_Planes_PP.png
Continuity_Planes_PP.png (98.32 KiB) Viewed 127 times


Here, a king on Square 1 (r1c1) will leap orthogonally to Squares 81 (from 2 directions), 10 & 2. Diagonally it leaps to Squares 80, 11,72 and 1. This means that the King can leap to the same square. This unfortunately means that the King is incompatible with this type of continuity plane puzzles. Other Fairy chess pieces that are not compatible include the Camel (1,3) and Tripper (3,3). A knight leaping from squar 1 by 2 squares up and 1 square left would land on square 10. This makes the Projective plane challenging to visualize and more challenging to crack in a puzzle.

Also with the block of 4 visualization you can confirm that leaping 9 squares up from square 12 would land on square 16. Leaping 18 squares up from square 12, however, would return to the same square.

Note that the continuity plane rules apply to leaper only not to the Latin Square or Sudoku rules.
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Re: Corkscrew contiuity plane

Postby tarek » Wed Oct 18, 2017 11:11 pm

this comes from the twisted mind of Smythe Dakota.

With a cylinder or toroid, your grid edges are wrapped so that each line end is connected to the beginning of the same line. It is easy to visualise and it preserves the rules of Latin square/Sudoku as orthogonal travel on a line will come across the same cells you would encounter on a vanilla type board.

If the wrapping, however, manages to stick each line's ending with its neighbouring line's beginning then the appearance would be slightly different
spiral_small.png
spiral_small.png (79.85 KiB) Viewed 105 times


This skew can be controlled as the 1st column may attach to any line on the grid. The rest of the lines will skew in the same way to follow suite. if the skew gets bigger then our corscew will appear like this
spiral_Skewed2_small.png
spiral_Skewed2_small.png (92.63 KiB) Viewed 67 times


when you join the ends of the corkscrew then you get the torus with a cylinder continuity plane in one direction and the corkscrew in another

Here is an anti-king (AK) puzzle with a corkscrew that has a skew of 2 (bottom of line 1 joins line 3, bottom of 2 joins top of 4 ...)
mdds_7_9x9_AK_Skew2.png
mdds_7_9x9_AK_Skew2.png (86.95 KiB) Viewed 105 times
Last edited by tarek on Sat Oct 21, 2017 6:43 pm, edited 1 time in total.
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Re: AK Corkscrew Skew 1

Postby tarek » Fri Oct 20, 2017 6:41 pm

Keeping it simple. Here are 3 Anti-King Corkscrew Skew 1 puzzles. Remember that the corkscrew in the way described forces a horizontal cylinder

AK Corkscrew Skew 1 Puzzle 1
7_9x9_AK_Skew1_001.png
7_9x9_AK_Skew1_001.png (71.92 KiB) Viewed 96 times


AK Corkscrew Skew 1 Puzzle 2
7_9x9_AK_Skew1_002.png
7_9x9_AK_Skew1_002.png (72.11 KiB) Viewed 96 times


AK Corkscrew Skew 1 Puzzle 3
7_9x9_AK_Skew1_003.png
7_9x9_AK_Skew1_003.png (72.86 KiB) Viewed 96 times


To prove that it has been a while since I was active on the forum. It appears that I have been forgetting to post the puzzles in line format
Code: Select all
....1......15.98....2...3.....1.2....2.....4....7.8.....8...5....39.42......3....
1..5.3..7.3.....8...9...1..8.......4.........3.......9..3...4...1.....6.2..7.8..5
............723...3..8.1..2..1.5.6...2.....7...8.9.2..2..1.4..9...932............



Enjoy

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Re: Corkscrew contiuity plane

Postby Smythe Dakota » Sat Oct 21, 2017 2:12 am

tarek wrote:this comes from the twisted mind of Smythe Dakota.

Yes, it does. Thank you! :twisted:

.... With a cylinder or toroid, your grid edges are wrapped so that each line end is connected to the beginning of the same line. It is easy to visualise and it preserves the rules of Latin square/Sudoku as orthogonal travel on a line will come across the same cells you would encounter on a vanilla type board.

If the wrapping, however, manages to stick each line's ending with its neighbouring line's beginning then the appearance would be slightly different
spiral_small.png

This skew can be controlled as the 1st column may attach to any line on the grid. The rest of the lines will skew in the same way to follow suite. if the skew gets bigger then our corscew will appear like this
spiral_small_Skewed.png
....

The above two look the same to me. In both cases, each line's ending is glued to its neighbor's beginning. I think it's just the camera angle or something.

If each line's ending is glued to the beginning of itself plus 2 (instead of to itself plus 1), it will still wrap around, and cover the entire torus in 9 trips around the short way. But in so doing there will be 2 trips (instead of 1) around the long way.

If, in place of the bolded 2 in the above paragraph, you substitute one of 1,2,4,5,7,8, then it will still take 9 trips around the short way, but respectively 1,2,4,5,7,8 trips around the long way.

If you use 3 or 6 in place of 2, now you will have three separate helixes intertwined. To get around one of them would take 3 trips around the short way, and 1 or 2 trips around the long way (I think).

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Re: The corkscrew

Postby tarek » Sat Oct 21, 2017 9:56 am

Ah that escaped me. So the 2nd photo is not representative!!! I've had a play with the photo editor & got this close approximation of a skew = 2 (apart from the number of rotations). You can see that the 1,3,5,7,9 (blue numbers) is connected to the 2,4,6,8 spiral (yellow numbers). Red dotted ends stick to each other as do the green dotted ends.
spiral_Skewed2_small.png
spiral_Skewed2_small.png (92.63 KiB) Viewed 66 times


Separate Spirals/Helices will happen when the Skew 3 & 6 for a 9x9 grid. For a 16x16 it would be 2,4,6,8,10,12,14 ...

The situation with non intersecting spirals/helices is interesting from a chess tour point of view. As the orthogonal rider like a rook will need more then 1 move to cover the entire grid

Thanks for clearing that out.

Did you have a chance to go through one of the puzzles? All the anti-king puzzles should be easy

Tarek
[EDIT: Added a Corkscrew image to explain the skew when it is bigger than 1, The same image updates the old bendy corkscrew image a few post above]
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36x36 Anti Root 325 puzzle

Postby tarek » Mon Oct 23, 2017 12:31 am

I've managed to generate a puzzle in this size to demonstrate the Root325 leaper

ROOT325_36x36_demo.png
ROOT325_36x36_demo.png (177.19 KiB) Viewed 60 times


The following is a puzzle with the solution being the demo grid above

ROOT325_35x36_001.png
ROOT325_35x36_001.png (162 KiB) Viewed 60 times


Enjoy

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