## Toroidal Anti Chess (Formerly The Touchless Wrapper)

For fans of Killer Sudoku, Samurai Sudoku and other variants

### Re: Update about corkscrew variant

The paper lists many

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

Tarek

tarek

Posts: 2699
Joined: 05 January 2006

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

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 (124.88 KiB) Viewed 341 times

tarek

Posts: 2699
Joined: 05 January 2006

### Re: continuity planes explained - The toroid

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 (97.41 KiB) Viewed 333 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.

tarek

Posts: 2699
Joined: 05 January 2006

### Re: continuity planes explained - The projective plane

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 (98.32 KiB) Viewed 317 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.

tarek

Posts: 2699
Joined: 05 January 2006

### Re: Corkscrew contiuity plane

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 (79.85 KiB) Viewed 295 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 (92.63 KiB) Viewed 257 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 (86.95 KiB) Viewed 295 times
Last edited by tarek on Sat Oct 21, 2017 6:43 pm, edited 1 time in total.

tarek

Posts: 2699
Joined: 05 January 2006

### Re: AK Corkscrew Skew 1

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 (71.92 KiB) Viewed 286 times

AK Corkscrew Skew 1 Puzzle 2
7_9x9_AK_Skew1_002.png (72.11 KiB) Viewed 286 times

AK Corkscrew Skew 1 Puzzle 3
7_9x9_AK_Skew1_003.png (72.86 KiB) Viewed 286 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

Tarek

tarek

Posts: 2699
Joined: 05 January 2006

### Re: Corkscrew contiuity plane

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

Yes, it does. Thank you!

.... 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).

Bill Smythe
Smythe Dakota

Posts: 546
Joined: 11 February 2006

### Re: The corkscrew

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 (92.63 KiB) Viewed 256 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]

tarek

Posts: 2699
Joined: 05 January 2006

### 36x36 Anti Root 325 puzzle

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

ROOT325_36x36_demo.png (177.19 KiB) Viewed 250 times

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

ROOT325_35x36_001.png (162 KiB) Viewed 250 times

Enjoy

Tarek

tarek

Posts: 2699
Joined: 05 January 2006

### Re: Toroidal Anti Chess (Formerly The Touchless Wrapper)

Question about the "Toroidal" name. I understood "Toroidal" to refer to a form of jigsaw sudoku in which the puzzle pieces (irregular nonets) could "wrap" around to the opposite side of the puzzle, whereas in the "Touchless Wrapper", all the nonets were 3x3 squares (albeit with the addition of the Anti-King constraint). Has that been changed, or is my initial understanding incorrect?

enxio27

Posts: 460
Joined: 13 November 2007

### Re: Toroidal Anti Chess (Formerly The Touchless Wrapper)

enxio27 wrote:Question about the "Toroidal" name. I understood "Toroidal" to refer to a form of jigsaw sudoku in which the puzzle pieces (irregular nonets) could "wrap" around to the opposite side of the puzzle, whereas in the "Touchless Wrapper", all the nonets were 3x3 squares (albeit with the addition of the Anti-King constraint). Has that been changed, or is my initial understanding incorrect?

Hi enxio,

The toroidal term refers to the board. On this doughnut shaped board the jigsaw nonets look like any other jigsaw puzzle. It is when you try to represent this doughnut (torus, toroid) on a 2-d page that some of those nonets get divided.

These toroidal jigsaw puzzles (which is the proper description) were called toroidal only because imh they were presented to the public first and nobody challenged the accuracy of the term.

Tarek

tarek

Posts: 2699
Joined: 05 January 2006

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