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I'm workin on a game, in which there is a board on which every player should have one to 3 areas with a total of about 12 (+4) square spaces.
At the moment I don't see a possibilty to make the map equal for all players. No problem for 2 and 4, but it seems impossible for 3. Some neutral spaces would be possible too, but even with that I don't get any solution.
4 players e.g. a a b b c c d d a a b b c c d d a a b b c c d d c c d d a a b b c c d d a a b b c c d d a a b b
or
a a b b c c a a b b c c d d   a a d d   a a b b c c d d b b c c d d
or simply a a a b b b a a a b b b a a a b b b c c c d d d c c c d d d c c c d d d
but how to do it with 3? is there a possibility?
note: I'm aware that it wold be possible with hex spaces, but that would make a 3pgame very different from a 2 and 4 p game, also needing different tiles, so that's not really an acceptible solution. On the other hand I don't see how to make a symmetric board for 2 and 4 p with hexagon spaces, so making it all hexes doens't seem reasonable either.

Carthoris Pyramidos
United States Centennial Colorado

The answer is chess wedges.

Martin Larouche
Canada Longueuil Quebec
Melting souls with cuteness since 2007
Lovin' N16

City of Remnants has a 3 players balanced map.
There's no clear picture of it on BGG, but this shows a bit how it looks like:
They remove the squares that are black and yellow lines from the grid. Those are not spaces to play on.
From then, all players start at an equal distance from each other and have access to the same colors of squares.

Nigel Buckle
United Kingdom Thornton Heath Croydon
Omega Centauri
Published in 2014

If you're making your board up with separate square tiles then you can use the same square tiles to make a hex grid, just offset each row:
a a a a a a .b b b b b c c c c c c
That gives you the same as a hex (ie 6 adjacent rather than 9)

BT Carpenter
United States Reston Virginia

sample 1 is 8x8, which doesn't divide well by 3.
sample 2 is 6x6 (4) which is 32, which also doesn't divide well sample 3 is 6x6 which is 36, which seems like it would divide very well.
So how to divide a 6x6 region into 3 "equal" sets of starting positions?
a a b b c c a a b b c c b b c c a a b b c c a a c c a a b b c c a a b b
Is that 'equal' for your purposes?

Tommy Occhipinti
United States Decorah Iowa
Magic Fanboy

byronczimmer wrote: sample 1 is 8x8, which doesn't divide well by 3. sample 2 is 6x6 (4) which is 32, which also doesn't divide well sample 3 is 6x6 which is 36, which seems like it would divide very well.
So how to divide a 6x6 region into 3 "equal" sets of starting positions?
a a b b c c a a b b c c b b c c a a b b c c a a c c a a b b c c a a b b
Is that 'equal' for your purposes?
That is certainly not symmetric, as c has the center squares, which is certainly either an advantage or disadvantage, unless the board wraps around the edges.
If I phrase mathematically what I think you are looking for, then it is not possible with square grids, though my formalization is quite mathy.
For those curious with a background in abstract algebra, here is my thinking: I'd be looking for an injective homomorphism from the group the S_3 to the automorphism group of the playing area that exchanges the player positions in all 6 possible ways. (You could ask only for a map from Z_3 to the automorphism group of the playing area that does this, but even if you did, it does not change the answer.) This implies that 6 divides the order of the automorphism group of the playing area, but the automorphism group of a square grid is always a subgroup of the Dihedral group with 8 elements, and hence this is not possible.
So generally, in order for this to be possible, you at a minimum need to use a board that has an automorphism of order 3. A few boards that have such an automorphism (and hence admit such arrangements) include:
) a symmetric Hex grid ) a cylindrical board with square cells (where the left side wraps to the right side) where the width has a multiple of 3 number of squares [This is probably the cleanest solution mathematically, but I think it would be awful in practice.] ) a symmetric board with triangular cells ) any number of boards where connections are shown via edges (like Pandemic)

Geoffrey Burrell
United States Cedar Rapids Iowa

Sequence works well with 3 people.

Russ Williams
Poland Wrocław Dolny Śląsk

Wrap around in one dimension? E.g. start like this on a 6x6 board where the north and south rows are also adjacent to each other:
aaaaaa aaaaaa bbbbbb bbbbbb cccccc cccccc

Tommy Occhipinti
United States Decorah Iowa
Magic Fanboy

russ wrote: Wrap around in one dimension? E.g. start like this on a 6x6 board where the north and south rows are also adjacent to each other:
aaaaaa aaaaaa bbbbbb bbbbbb cccccc cccccc
That's what I called a cylindrical board, it certainly works, although I have a feeling that psychologically when playing it that it would not feel symmetric. If it were being played on an iPad where you could just scroll and not see the "seams" then I think this would work perfectly.

United States Riva Maryland

Nexus Ops does it with modular hex tiles. The same board plays 2, 3 or 4. Worth a look.
S.

Tommy Occhipinti
United States Decorah Iowa
Magic Fanboy

One could argue that a hex (or square) grid with four players is not as symmetric as desired, as arrangements tend to produce "near" neighbors and "far" neighbors, so while the player positions might be symmetric, the player relationships are not.
Of course, for many designs this asymmetry is a feature not a flaw.


In order to really get a rigorous answer, you should probably specify exactly what grid properties your game cares about. For instance, the answers may be different for a game with diagonal movement compared to one with only orthogonal movement, or for a game where you're going to have wind that makes some directions not equivalent to others. Maybe there's an arrangement that wouldn't be symmetrical for every possible game but is strategically equivalent within the rules of your particular game.
I was also going to suggest the same board arrangement as Cathoris, which can be used to scale anywhere from 1 to 6 players with everyone in symmetrical positions while keeping the squarestoplayers ratio constant. There's a pretty good description here.
It's got 16 squares per player, which is on the upper end of your requested range, but you could always block some of them (e.g. place walls in the 4 corner spaces of each wedge  or the 4 center spaces).


delirimouse wrote: One could argue that a hex (or square) grid with four players is not as symmetric as desired, as arrangements tend to produce "near" neighbors and "far" neighbors, so while the player positions might be symmetric, the player relationships are not. I suspect you're not going to be able to get isomorphic relationships for 4+ players without using wraparound or other weird topologies.

Tommy Occhipinti
United States Decorah Iowa
Magic Fanboy

Antistone wrote: delirimouse wrote: One could argue that a hex (or square) grid with four players is not as symmetric as desired, as arrangements tend to produce "near" neighbors and "far" neighbors, so while the player positions might be symmetric, the player relationships are not. I suspect you're not going to be able to get isomorphic relationships for 4+ players without using wraparound or other weird topologies.
Your point about diagonal adjacency etc has been on my mind today. Between making lesson plans today I've been trying to decide if the four corners of a chessboard are symmetric if diagonal squares are adjacent.
I had all but convinced myself the answer was yes, but unfortunately, the answer is no. So I'm inclined to agree that if you really want that level of symmetry for 4 players it is going to require something other than a grid that will likely be impractical for gaming purposes.

Chris Williams
Seattle Washington

1) Hex grid (selfexplanatory)
2) Loop
a a a a b c b c b c b c
You could make the loop circular instead of triangular, and you could make the width of the band whatever you wanted, so it's not linear. I think a loop works better, for most purposes, than an open map since it will lessen the chance of kingmaking.

T. Dauphin
Canada Belleville Ontario

How about one board on one side for 2 or 4 players, and another on the other side for 3 players?

Kristian Järventaus
Finland

aaaa
B╔╗c B╚╝c BBcc
Asymmetrical shapes.
aaBB aaBB ║╚cc ╚═cc
Teleporting.
You don't have to have a torus if you pinch the typology inside the board and "remove" one part of it. You could even have slight lines to show where they connect.


