## Toroidal Anti Chess (Formerly The Touchless Wrapper)

For fans of Killer Sudoku, Samurai Sudoku and other variants

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

This Puzzle variant which can be described as an anti-king (a cell can't be the same as any of its direct 8 neighbours) with a top-down Rt-Left wrap ... is an easy one

The wrapping will make each cell have 8 cell neighbours including the edge & corner cells.
Here is an example of such a grid with cell r1c1 & its 8 neighbours

Here is the puzzle which I'm providing the solution later

Code: Select all
`8..1.2..................5...1.7.........5....3....4...2.8..................9.8... 8 . . | 1 . 2 | . . .   . . . | . . . | . . .   . . . | . . . | 5 . .  -------+-------+------  . 1 . | 7 . . | . . .   . . . | . 5 . | . . .   3 . . | . . 4 | . . .  -------+-------+------  2 . 8 | . . . | . . .   . . . | . . . | . . .   . . . | 9 . 8 | . . .`

N.B. this has also been posted on http://www.rcbroughton.co.uk/sudoku/forum/viewtopic.php?f=13&t=766

tarek

[EDIT1: added "Anti-King Toroidal" to the title]
[EDIT2: Changed the name of the thread to accommodate more Anti chess variants]
Last edited by tarek on Mon Jul 11, 2011 7:19 pm, edited 2 times in total.

tarek

Posts: 2737
Joined: 05 January 2006

### Re: The Touchless Wrapper

Nice puzzle!

It's just the right degree of difficulty. It does not require so much effort as to make it more work than play -- like filling in all candidates in all cells right from the outset -- but it does take a bit longer than the vanilla puzzles. Or maybe that's because I wasn't accustomed to all the solving techniques available in this variant. Anyway, it's just right.

I think some on this forum might prefer to call it an "anti-King toroidal", which is in line with terminology used elsewhere in these forums.

Thanks again!

Bill Smythe
Smythe Dakota

Posts: 546
Joined: 11 February 2006

Thanks Bill,

I think that the word "toroidal" escaped me completely.

I thought at the time that the concept of "toroidal chess" was fascinating especially if combined with fairy chess. "Touchless" was used to describe Anti-King before & "Wrapper" just gave it a MUSICAL edge .

tarek

tarek

Posts: 2737
Joined: 05 January 2006

### Anti-kNight Toroidal

The Anti-kNight(AN) variant is trickier than Anti-King(AK) because the the cells are further away. The toroidal version is no different.

Here is a solution grid demonstrating the r1c1 relationship with cells that are "a Knight's move away" on the toridal grid.

The next puzzle should be easy to solve

Line: Show
Code: Select all
`....9.......3.5.....3...7...8.....7.2.......6.3.....4...2...1.....8.7.......1....`

solution: Show

tarek

Posts: 2737
Joined: 05 January 2006

### Re: Anti-kNight Toroidal

Thanks -- I'll take a crack at it (I hope I can solve it without looking at your solution).

Is a circular Knight's tour (first and last cells a Knight's move apart) possible on a 9x9 toroidal board? It's not possible on a 9x9 flat board, because the first and last cells would have to be the same color. But on a toroidal board, some pairs of cells a Knight's move apart are the same color.

Bill Smythe
Smythe Dakota

Posts: 546
Joined: 11 February 2006

### Re: Anti-kNight Toroidal

Smythe Dakota wrote:Is a circular Knight's tour (first and last cells a Knight's move apart) possible on a 9x9 toroidal board? It's not possible on a 9x9 flat board, because the first and last cells would have to be the same color. But on a toroidal board, some pairs of cells a Knight's move apart are the same color.

I guess it can ... you have to start from from c1, c2,c7,c8,r1,r2,r7 or r8 to achieve that.

The toroidal 9x9 chess board doesnt look pretty because the "sticky ends" are of similar colour. This is why a 16x16 Sudoku Chess board will look better as a toroid.

I haven't seen the concept of toroidal chess before & I'm not sure if it works. The sudoku toroidal anti chess works though.

You and Hatman are probaly the only 2 who have attempted to solve these anti-chess toroidals .

tarek

tarek

Posts: 2737
Joined: 05 January 2006

### Re: Anti-kNight Toroidal

Smythe Dakota wrote:Is a circular Knight's tour (first and last cells a Knight's move apart) possible on a 9x9 toroidal board? It's not possible on a 9x9 flat board, because the first and last cells would have to be the same color. But on a toroidal board, some pairs of cells a Knight's move apart are the same color.

Example:

Last edited by dyitto on Sun Aug 12, 2012 8:51 am, edited 1 time in total.
evert on the crashed forum

dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam

well done dyitto. That was a quick response. Obviously i wasn't thinking of achieving this in the correct way.

tarek

tarek

Posts: 2737
Joined: 05 January 2006

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

Well, I got the anti-Knight toroidal -- after a few hours. Good puzzle.

Is there such a thing as a pro-King toroidal or a pro-Knight toroidal? The rules could be something along the following lines:

Each digit must be a Knight's move away from at least one other of the same digit.

Or maybe:

The nine occurrences of each digit must form a chain of eight Knight's moves (or perhaps nine, making it circular).

Or maybe:

Each digit, except those in the center cell of each box, must be a King's move away from at least one other of the same digit.

Or something else along similar lines.

Bill Smythe
Smythe Dakota

Posts: 546
Joined: 11 February 2006

### Re: Anti-kNight Toroidal

tarek wrote:You and Hatman are probaly the only 2 who have attempted to solve these anti-chess toroidals .

tarek

Hey! I quietly solve most of the variants...

Got totally sucked into the Windmill Diagonal Windoku
.
BryanL

Posts: 247
Joined: 28 September 2010

### Re: Anti-kNight Toroidal

tarek wrote: .... I haven't seen the concept of toroidal chess before & I'm not sure if it works. ....

If the initial setup of pieces is the same as in regular chess, then of course it's completely impossible. Both kings are checkmated in the initial position, and each is in check from at least four different pieces (king, queen, bishop, knight).

Bill Smythe
Smythe Dakota

Posts: 546
Joined: 11 February 2006

Smythe Dakota wrote:Is there such a thing as a pro-King toroidal or a pro-Knight toroidal? The rules could be something along the following lines:
Each digit must be a Knight's move away from at least one other of the same digit

The above sounds like the most promising ... My thinking cap hurts when attempting to program it though. It should follow:

Code: Select all
`Cell X equals ALoop through all Cells within a Knight's move from Cell X {     Count the number of A(s)}If the number of A(s) equals 0 {     Cell X Can't be A (Return 0)}`

BryanL wrote:Hey! I quietly solve most of the variants...

I'm hopeful that you are then part of a larger silent majority

BryanL wrote:Got totally sucked into the Windmill Diagonal Windoku

They are nice ... I'm not sure if there is any appetite for more though

Smythe Dakota wrote:If the initial setup of pieces is the same as in regular chess, then of course it's completely impossible. Both kings are checkmated in the initial position, and each is in check from at least four different pieces (king, queen, bishop, knight).

Would you believe me if I said that I was talking of different initial setup of pieces ?

tarek

Posts: 2737
Joined: 05 January 2006

### Re:

tarek wrote:
Smythe Dakota wrote:If the initial setup of pieces is the same as in regular chess, then of course it's completely impossible. Both kings are checkmated in the initial position, and each is in check from at least four different pieces (king, queen, bishop, knight).

Would you believe me if I said that I was talking of different initial setup of pieces ?

You'd probably have to put the white pieces on the 2nd and 3rd ranks, and the black on the 6th and 7th, or something similar.

Bill Smythe
Smythe Dakota

Posts: 546
Joined: 11 February 2006

### Re:

tarek wrote:
BryanL wrote:Hey! I quietly solve most of the variants...

I'm hopeful that you are then part of a larger silent majority

BryanL wrote:Got totally sucked into the Windmill Diagonal Windoku

They are nice ... I'm not sure if there is any appetite for more though

There seemed to be a bit of interest.

After finally getting off my butt and implementing a generic Dlx, I am currently working on reducing a 225x225 towards minimal - not that dlx works for large N but it can be used in conjunction with normal methods - and I have a couple of other tricks I want to try to speed it up.

After that I hope to create a Dlx implementation for windmill diagonal windoku so I can generate a few. It will be an exercise getting it right and I am not sure it wont suffer the same time issues spent on fruitless paths as for higher N, but have the feeling that all the extra constraints will reduce that...
.
BryanL

Posts: 247
Joined: 28 September 2010

### Re: Re:

BryanL wrote:After finally getting off my butt and implementing a generic Dlx, I am currently working on reducing a 225x225 towards minimal - not that dlx works for large N but it can be used in conjunction with normal methods - and I have a couple of other tricks I want to try to speed it up.
It won't be easy, I've tried it with the 3D cube & the Windmill Diagonal Kazaguruma ... You need all the tricks up your sleeve.

tarek

tarek

Posts: 2737
Joined: 05 January 2006

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