Higher dimensions of chess

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Higher dimensions of chess

Postby wintersolstice » Sun Nov 08, 2009 2:20 pm

recently I've been investigating many things in higher dimensions (chess being one)

This is how moves a king would have (one space in any direction) in different parts of the board:

1D:

2 moves away from the sides
1 moves at the corner

2D:

8 moves away from the sides
5 moves at an edge
3 moves at a corner

3D:

26 moves away from the sides
17 moves at a face
11 moves at an edge
7 moves at a corner

4D:

80 moves away from the sides
53 moves at a cell
35 moves at a face
23 moves at an edge
15 moves at a corner

5D:

242 moves away from the sides
161 moves at a teron
107 moves at a cell
71 moves at a face
47 moves at an edge
31 moves at a corner

6D:

728 moves away from the sides
486 moves at a peton
323 moves at a teron
215 moves at a cell
143 moves at a face
95 moves at an edge
63 moves at a corner

(Hope I've caculated all that right and got the right terminolgy!) btw I could go beyond that!

I've set rules for pieces so that:

An extra piece for each dimension (for each new type of diagonal)
a knight moves (1,2,3,...etc) one square more each direction (or one square orthogonally and diagonly which means the same thing)

Both of these mean that the number of "spaces" each way increases by two:

8X8, 10X10X10, 12X12X12X12 etc.

I having trouble with one aspect right now:

How many of each piece shold each person get! Any ideas?
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Re: Higher dimensions of chess

Postby Keiji » Mon Nov 09, 2009 6:55 am

I don't see why you'd need to increase the number of spaces on the board (other than obviously raising the power of 8).

Remember, in a 3D chessboard there'd already be 8*8*8 = 512 cubes on the board - and half of these would be allocated to the players, 128 each.

Alternatively one could have four-player chess with four lots of 32 cubes in the "corners" of the board. Though this would obviously involve 3 rows of 8 pawns and the original 8 pieces, which wouldn't fit your extra piece per dimension. Also, pawns would need to be able to move in multiple directions.

If we stick with my first idea, one possibility is to only have the center 16 cubes used for non-pawns, with the outer 48 cubes of the non-pawn layer empty. This is exactly double the number of non-pawn pieces of a 2D chessboard, which could be in the following quantities:
64 pawns
4 rooks (move out of the 6 faces of a cube)
4 knights (move 2 cubes in one direction, one cube in all other directions)
4 bishops (move out of the 8 vertices of a cube)
2 of your new piece type (move out of the 12 edges of a cube)
1 queen
1 king

The only problem here is that in order to preserve symmetry the 4 knights would all have to go on one color of square and the 4 bishops on the other.

Alternatively if we really do have 64 non-pawn pieces (which does seem overkill), they could be like this:
64 pawns
16 rooks (at positions 0,0, 0,1 and 2,2, reflected around the layer in full symmetry)
16 knights (at positions 0,2 and 2,3, reflected around the layer in full symmetry)
16 bishops (at positions 0,3 and 1,2, reflected around the layer in full symmetry)
12 of your new piece type (at positions 1,1 and 1,3, reflected around the layer in full symmetry)
3 queens (they would probably have to have a different name !)
1 king

The only remaining problem is that in either situation there would be at least two of some pieces which both start on the same color cube (light or dark) so you couldn't refer to them as "the light bishop" or "the dark rook" etc.
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Re: Higher dimensions of chess

Postby wintersolstice » Mon Nov 09, 2009 11:36 am

if the knight makes 2 squares one way and 1 another it's move remains in a 2D plane so I generalised it.(1 sq one way, 2 sq another way, 3 sq another way etc) the fact that the knight moves one extra square each time means extra squares are needed to it space to move. And one extra pair to accomadate the new piece. (different types of diagonal)
That's why it needs extra spaces each time

It doesn't really matter though, there is no right or wrong way to generalise chess in higher dimensions I don't think(there's probably loads of ways)

This is just how I would do it :D

Maybe other people can share their way of doing it. :D
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Re: Higher dimensions of chess

Postby Keiji » Mon Nov 09, 2009 7:55 pm

It would be very weird to make the knight work like that. With my way the knight moves one space along all axes plus a further one space on one axis. Therefore it is always restricted to a hypercube of side 5 centred on the knight's current space. Also you wouldn't need a bigger board to accommodate the extra pieces as they are already way over-accounted for in the increase in dimensionality of the board.
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Re: Higher dimensions of chess

Postby wintersolstice » Tue Nov 10, 2009 11:56 am

Well you prefer your way, I prefer mine, that's fine :D

basically the reason for an extra two rows (for the extra piece) is based on two idea (my ideas) on how pieces should be arranged.
On a 3D board (just to keep this simple)

on an 8X8X8 board:

16 rooks
16 bishops
16 knights
8 queens
8 kings
84 pawns
for each player! occupying 8 coluhms
but that creates more than one king so

have rings with rooks on the outer ring and 1 king and 3 queens on the inner ring (or block of 4 spaces)
but this creates a situation where a player has a lot of rooks

both these methods require an extra row (an extra coluhm in the former; and extra ring in the latter)

So that's why I needed help on arranging pieces
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Re: Higher dimensions of chess

Postby Keiji » Tue Nov 10, 2009 2:29 pm

You can't have more than 1 king or the entire strategy behind chess falls apart.
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Re: Higher dimensions of chess

Postby wintersolstice » Wed Nov 11, 2009 10:16 am

Keiji wrote:You can't have more than 1 king or the entire strategy behind chess falls apart.

I know that's why I'm having trouble and that's why I came up with the "rings" idea to avoid that :D
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Re: Higher dimensions of chess

Postby PWrong » Sun Nov 15, 2009 11:43 am

I always loved the idea of Cheops (pyramid chess) in Dune. It would be a bit simpler than chess on a cube. Apparently the board has nine levels, which seems odd unless the bottom level is 9x9. Also there is a "dual objective": you have to get the queen to the top and the opponents king in check. I don't know if you have to do both or if the first person to do either wins. It would be hard to do both if both the queens died.
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Re: Higher dimensions of chess

Postby wendy » Mon Nov 16, 2009 7:17 am

Even in the realm of 'fairy chess', the board is not restricted to either 8×8 , one can play great chess on a board of 100 squares, being 10 deep and 12 wide.
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the dream we dream together is reality.
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Re: Higher dimensions of chess

Postby zero » Sat Nov 28, 2009 10:07 pm

wendy wrote:a board of 100 squares, being 10 deep and 12 wide.

Where did the other 20 squares go, or have we expanded on the decimal system, too?
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Re: Higher dimensions of chess

Postby wendy » Sun Nov 29, 2009 7:38 am

long numbers, i am afraid. C = vixx
The dream you dream alone is only a dream
the dream we dream together is reality.
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Re: Higher dimensions of chess

Postby thigle » Wed Jan 13, 2010 1:30 pm

also, there is a possibility of modesty. it is already practiced. there are already spherical and toroidal chess that are being played.
of course, it is not 4d chess, but the 2-dimensionality of the chess-board is taken one level up to 3d, so 8x8squares are folded into torus or sphere.
most of these are played on magnetic boards, gravity has to be counted in when your queen is on the south pole or on the downside of the torus.
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Re: Higher dimensions of chess

Postby Halfbaker » Sat Jun 04, 2011 6:15 am

Klein bottle chess anyone?
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