## Anti Chess, NC and Forbidden Pairs (FP) puzzles

For fans of Killer Sudoku, Samurai Sudoku and other variants

### Anti-Commuter Toroidal & Projective planes

The Commuter is one of the less known & described fairy chess pieces. It leaps 4 squares diagonally in every direction. It can be thought as "Double Alfil".

I've decided to use this piece because it has a reach of over 3 squares therfore taking full advantage of the prospective plane. Here is a demo of a commuter's reach when on r1c1

Here is an Anti-Commuter sudoku puzzle where the commuter can leap into the toroidal or the projective planes. The puzzle is beyond easy.
There will be no advantage if the comuter leaps into the Klein bottle plane

Line: Show
Code: Select all
`...59........4...7.......................1....3...5.2.............3........46....`

Solution: Show
Last edited by tarek on Thu Dec 19, 2019 7:26 pm, edited 2 times in total.

tarek

Posts: 3748
Joined: 05 January 2006

### Re: Anti-Commuter Toroidal & Projective planes

When the starting cell is one of the nine "anchor" cells -- rows 1,5,9 and columns 1,5,9 -- toroidal, projective, and both versions of Klein bottle are all the same to a Commuter. Starting from any other cell, the Commuter has a lot more possible moves, if toroidal and projective are both allowed.

Bill Smythe
Smythe Dakota

Posts: 564
Joined: 11 February 2006

Very limited spare time lately to do any sudoku-related activity ....

Hopefully some new puzzles in the new year ....

tarek

tarek

Posts: 3748
Joined: 05 January 2006

### Re:Toroidal anti Squirrel

The Squirrel is an interesting Piece. It leaps similar to the Alfil, Knight & Dabbaba

ASQ001-demo.jpg (107.11 KiB) Viewed 1367 times

Here is an easy Fully symmetric Toroidal puzzle:

ASQ001-Toroid.jpg (48.07 KiB) Viewed 1367 times

Here is the solution:
Hidden Text: Show
ASQ001-Toroid-Solution.jpg (100.74 KiB) Viewed 1367 times

tarek

tarek

Posts: 3748
Joined: 05 January 2006

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

Oh, no -- this old thread has jumped back to life!

I've aged a bit since 2011. I don't think I can follow my own old logic anymore, let alone yours or anybody else's.

It was amazing, though, to re-read all the weird stuff we talked about six years ago.

Bill Smythe
Smythe Dakota

Posts: 564
Joined: 11 February 2006

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

It was nice though with a programming challenge and ideas and ideas I didn't come across before. I can't even follow some of my posts. I will be - time permitting - visiting some the ideas posted, and post some new puzzles from time to time

tarek

Posts: 3748
Joined: 05 January 2006

### Re:Setting the Rules - The Chess Pieces

Recapping on some issues & ideas here. I'll start by a recap of what chess pieces we may use (additions are most welcome):

Code: Select all
`        Simple leapers:            0           1       2       3           4        0                          1   Wazir       Fers                    2   Dabbaba     Knight  Alfil               3   Threeleaper Camel   Zebra   Tripper         4   Fourleaper  Giraffe Lancer  Antelope    Commuter        5               Ostrich        6               Flamingo                Two Combination leapers        (1,1)   King                    (0,2)   Wazaba  Duke                (1,2)   Emperor Prince  Templar         (2,2)   Caliph  Ferfil  Alibaba Hospitaller                (0,1)   (1,1)   (0,2)   (1,2)        Additional leapers:        Centaur     (0,1) + (1,1) + (1,2)        Squirrel    (0,2) + (1,2) + (2,2)        Root25      (3,4) + (0,5)        Root50      (1,7) + (5,5)        Root100     (6,8) + (0,10)        Root200     (2,14)+ (10,10)        Root225     (9,12)+ (0,15)`

I've added Root100, Root200, Root225 to the list

tarek
[EDIT: Corrected Squirrel entry]
Last edited by tarek on Mon Jan 21, 2019 4:31 pm, edited 1 time in total.

tarek

Posts: 3748
Joined: 05 January 2006

### Re:Setting the Rules - Continuity Planes

What if our Sudoku puzzle is a 2D representation of a 3D object constructed by sticking the edges of the Sudoku board together?

This idea has been explored before with Jigsaw Sudoku variants, Non-consecutive Sudoku variants and anti-chess Sudoku variants ... But Smythe Dakota wanted to expand it further!!

Until this thread explored it. We were dealing mostly with a right-left, top-bottom, or a combination wrapping of the board ... Giving us a cylinder (tape, strip) or a Toriod (doughnut, halo). This was because this continuity still allowed the Latin Square rules to remain intact as each line wraps on itself.

If you, however, twist the board before sticking the edges together, you get a moebius strip which doesn't allow the Latin square rules to be valid anymore. That is why Smythe Dakota suggested exploring these new continuity planes only if we restrict them to the chess pieces only.

At the moment we have the following possible continuity planes explored:
1. Cylinder: Vertical or Horizontal
2. Toroid: Vertical + Horizontal Cylinders
3. Moebius: Vertical or Horizontal
4. Projective Plane: Vertical and Horizontal Moebius
5. Klein bottle: Cylinder in one direction + Moebius in the other perpendicular direction.
(This has also 2 possibilities: Vertical Cylinder + Horizontal Moebius or Horizontal cylinder + Vertical Moebius)

I have posted several puzzles exploring these continuity planes

tarek

tarek

Posts: 3748
Joined: 05 January 2006

### Re: Re:Setting the Rules - Continuity Planes

tarek wrote: .... If you, however, twist the board before sticking the edges together, you get a moebius strip which doesn't allow the Latin square rules to be valid anymore. That is why Smythe Dakota suggested exploring these new continuity planes only if we restrict them to the chess pieces only. ....

Hmm, is THAT what I was suggesting? Maybe so. It all started when I finally realized (duh) that, in a moebius strip or Klein bottle or projective plane, there are actually 18 cells instead of just 9 in at least some of the rows/columns. That fact, of course, blew pretty much everything out of the water in all the ideas we had been talking about for months.

Bill Smythe
Smythe Dakota

Posts: 564
Joined: 11 February 2006

### Re: Re:Setting the Rules - Continuity Planes

tarek wrote: .... At the moment we have the following possible continuity planes explored:
1. Cylinder: Vertical or Horizontal
2. Toroid: Vertical + Horizontal Cylinders
3. Moebius: Vertical or Horizontal
4. Projective Plane: Vertical and Horizontal Moebius
5. Klein bottle: Cylinder in one direction + Moebius in the other perpendicular direction.
....

Here's another possibility. Call it the Moebius Corkscrew (or Wormscrew).

When you get to the top of each column of 9, glue it to the bottom of the next column. Wrap this idea around, so that the top of column 9 also glues to the bottom of column 1.

This gives you a strip 81 cells high by 1 cell wide, with the top and bottom edges glued together, if you look at it vertically, or a 9x9 grid if you look at it horizontally. At least I think that's what it would be.

No repeats allowed in the rows. In each column, any two occurrences of any digit must be at least 7 (or some similar number) positions away from each other, i.e. there must be at least 6 other digits in between any two occurrences of the same digit.

I guess you could also have a Projective Plane Corkscrew, which I can't visualize, where each row would be glued to the next in the same manner as the columns. I guess this would be 81 by 1 if viewed horizontally, or 1 by 81 if viewed vertically. But my mind boggles.

Then you could forget the chess altogether and get back to actual Sudoku (sort of).

Bill Smythe
Smythe Dakota

Posts: 564
Joined: 11 February 2006

### Re: Re:Setting the Rules - Continuity Planes

Smythe Dakota wrote:Here's another possibility. Call it the Moebius Corkscrew (or Wormscrew).

When you get to the top of each column of 9, glue it to the bottom of the next column. Wrap this idea around, so that the top of column 9 also glues to the bottom of column 1.

I managed to visualise this! It is a skewed wrap where you wrap the sudoku board fusing the top border to the bottom one but slightly skewed!

Because you are joining the very end with the very start, you will end up with a toroid but it is a special one. When you move in an up-down direction on this toroid you are actually moving in a spiral which ends where you started having covered all squares. the right -left movement is your standard cylinder.

spiral_small.png (79.85 KiB) Viewed 1319 times

This should be doable if the it is restricted to the leaper only

tarek

tarek

Posts: 3748
Joined: 05 January 2006

### Poisoned Alibaba

Alibaba has been poisoned & his powers have been weakened. Both of his famous leaps are known to go into the projective plane but now only one of his powers can do that. Never to turn down a challenge, he accepted to help solve this puzzle
Anti_Alibaba_Confused.jpg (34.67 KiB) Viewed 1317 times

Can you trace Alibaba's path to success?

tarek

Posts: 3748
Joined: 05 January 2006

### Re:Anti Alibaba Hint

Poisoned Alibaba Hint:

Hidden Text: Show
Alibaba is a combination leaper that could leap 2 squares orthogonally or 2 squares diagonally
It is therefore a combination of Dabbaba & Alfil
In the specific puzzle above, one of these combination leaps can leap into the projective plane while the other is just a vanillaleap that can't go beyond the borders of the Sudoku board
The correct combination hasn't been declared but if known the puzzle can be easily solved. No other way can solve it!
Enjoy

tarek

Posts: 3748
Joined: 05 January 2006

### Root325

an interesting paper investigates integers that can be reached using the sum of 2 squares but in more than 1 way.

This allows us to have many combination leapers like the ROOT50. I have added many of these to my solver but interestingly the 325 has 3 different ways to sum 2 squares (1,18) (6,17) and (10,15). This is demonstrated nicely in this Anti-Root325 25x25 solution grid. Note that the central square cells will not have any squares on the board within reach

ROOT325_25x25.png (113.06 KiB) Viewed 1298 times

tarek

Posts: 3748
Joined: 05 January 2006

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

If 50 is the sum of two squares in 2 ways, and 325 is the sum of two squares in 3 ways, I wonder whether the next one (whatever it is) will be the sum of two squares in 4 ways, the next one after that in 5 ways, etc?

This is the sort of thing math people just love to prove (or disprove).

Bill Smythe
Smythe Dakota

Posts: 564
Joined: 11 February 2006

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