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Stone placement game based on 4 Color Theorem

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ACG
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Joined: 12/31/1969

Well, considering my 3D Go idea "went", what about this? The catch is that it seems so obvious someone must have done it before.

Consider a three-dimensional grid. In a two-player game, each player has a bunch of stones in one of two colors (four colors total). The goal is simple: place stones in the grid so that no adjacent pair of stones (all 6 neighbors) have the same color.

I saw somewhere online that the three-dimensional version of the Four Color Theorem has no limit on the number of colors necessary. So it should be possible to force someone to place two stones of the same color next to each other in a 3D grid with a finite number of colors.

Maybe in the 2 player game each person gets 3 colors, 3 player each person gets 2, stuff like that.

Alternatively, you can have a 2D version where each player gets two colors but the same number of each color (so the Four Color Theorem isn't necessarily going to work with restrictions on the number of stones of each color -- hopefully, the restriction should interfere enough to prevent a successful tiling with four colors).

This must have been done before...has it?

Thanks in advance,

ACG

Zomulgustar
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Joined: 07/31/2008
Stone placement game based on 4 Color Theorem

Unfortunately, I think you misunderstood something...the cubic lattice has a chromatic number of 2 (stack alternately colored chessboards to see what I mean). IIRC, graphs with arbitrarily high chromatic numbers can be realized as incidences of adjacent regions of space, but that's not the same thing. And while related, the four-color theorem doesn't apply directly to competitive graph coloring, though there has been substantial research along those lines as well. It sounds like you might be interested in looking into Nim, Col, Hackenbush and Snort, which John Conway used as examples in establishing some of the foundations of modern combinatorial game theory. Let me know if you have any trouble finding references, but there's a ton of stuff out there. Have fun, and don't get discouraged...if I had a nickel for every time I independently invented somebody else's mechanic, I'd probably need to visit the change machine soon.

CDRodeffer
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Joined: 08/04/2008
Re: Stone placement game based on 4 Color Theorem

ACG wrote:
Well, considering my 3D Go idea "went", what about this? The catch is that it seems so obvious someone must have done it before.

I'm not sure the discussion was ever really finished, especially with respect to the multi-planar type Go. The mechanical and visualization difficulties could be overcome. The board could be virtual, just an image on a screen with pieces placed by coordinates. Or there could be a 3D grid of wires with stones moved in by pullies. Or even an array of LEDs encased in a transparent cube that could be picked up and viewed from any angle, with stone placement controlled by three knobs ala Etch-A-Sketch.... But I digress.

ACG wrote:
Consider a three-dimensional grid. In a two-player game, each player has a bunch of stones in one of two colors (four colors total). The goal is simple: place stones in the grid so that no adjacent pair of stones (all 6 neighbors) have the same color.

As Zomulgustar commented, the four color theorem (or some variant of it) does not (yet) apply to competitive situations. But you may still be able to develop a playable game from variations on your ideas.

Digressing again to your original idea of some sort of 3D Go, the main mechanical problem that was raised was that, disregarding edge and corner effects, a one stone extension in 3D orthogonal space typically gains four liberties instead of only two as in 2D orthogonal space. But what if 3D Go was developed as a four-player game in four independent colors? It would require temporary alliances that could lead to "petty diplomacy" issues, but the three opposing turns between your own subsequent turns could be enough to counteract the extra liberties. And two volumetric "eyes" would still be sufficient to keep a group alive.

All this is, of course, totally untested, so there may be numerous other problems with these ideas. But I don't want you to become so discouraged that you abandon all of your ideas without at least trying variations on them. Games are supposed to be fun, and designing new ones can be a lot of fun, too. So please don't give up!

Clark

ACG
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Joined: 12/31/1969
Stone placement game based on 4 Color Theorem

The idea of 3D Go with multiple players is actually quite interesting. Or even 2D Go.
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If Black and White combine their stones to make a wall/surface surrounding a bunch of Red stones, the Red stones are removed from the board. The territory goes to the player who contributed the most stones to the wall (it can't be split evenly: otherwise Black will wait until White's got virtually a complete ring and throw in the last piece to get half the territory -- great gain for little effort). Ideally Black would get (B/(W+B)) of the territory, but that would throw math into it and make things too compliated.
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You can also do this: two players, but each controls two sets of colors (say Black/White and Red/Green). A Red stone can block the liberties of all other colors, INCLUDING GREEN. A stone completely surrounded by the three other colors gets removed. I wonder what an effect this would have. Each player has an equal number of his two colors to make the choice of color to be played important.
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What do you think of a Set/Go combination? Take a set (er...) of four Set decks and put the pictures on 324 Go stones. Each player gets two decks. Two adjacent stones form a "wall" if they share N attributes. Note that the opponent can contribute his tile to the wall if he isn't careful. Mixed territories go to the person who contributed the most tiles.
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Incidentally, I told a game store about my Five Crowns poker odds. The owner of the store told the game's manufacturer, and they were ecstatic. They want me to give them a PDF of the odds (especially the math of determining the combinatorics) so they can post it on their website (both for poker purposes and teaching purposes).

ACG

Hedge-o-Matic
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Stone placement game based on 4 Color Theorem

I have to chuckle at the image of a Go hybrid where every stone had a Set image on it. Talk about a brain burner!

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