Chroma Cameron Browne Stephen Tavener (c) 2008 Chroma is a simple packing game based on the 4-colour theorem.

Rules

Pieces: Chroma is played with 35 coloured cubes: 9 x Red, 9 x Green, 9 x Blue and 8 x White.

Figure 1.  The four cube colours.

The cubes are arranged in the sequence { R,G,R,B,... } for the first player and { W, B, W, G, ...} for the second player, and hence played in the following order:

R, W, G, B, R, W, B, G, ...

Board: The board is a triangular grid of triangular holes (five per side) such that each hole will snugly hold a cube placed corner-downward.

Play: The board is initially empty. Players take turns adding the next cube in the sequence to the board such that it does not directly sit on top of or visibly share an edge with another cube of the same colour (i.e. same-coloured cubes may not share a face or visible edge, but corners are okay).

Cubes may stack on three existing mutual neighbours provided that all four cubes are of different colour.

Aim: The last player to move wins.

For example, Figure 2 shows a game in progress. No white cubes can be added to this position; the current player would lose the game at this point if they only had white cubes remaining.

Figure 2.  A game in progress (no white placements).

Notes

Chroma is an embodiment of the 4-colour theorem in topology.

All cubes may be packed to completely fill the board such that no two cubes of the same colour are adjacent. Figure 3 shows one such packing; note that the pattern on the board level decides the colours of all cubes stacked above.

Figure 32.  A complete packing.

It is in Player B's interest to avoid such perfect packing patterns, otherwise they are guaranteed to lose when Player A makes the final move that fills the board.

The phased cube ordering means that players must play three intervening colours before playing the same colour again.

Multiplayer Version: The repeated sequences for three players are { R, G, R, W, ...}, { B, R, B, G, ... } and { W, B, W, G, ... } respectively. The repeated sequences for four players are { R, G, R, W, ...}, { W, R, W, B, ... }, { B, W, B, G, ... } and { G, B, G, R, ... } respectively. In both cases, a player wins as soon as the next player has no legal move (this reduces the effectiveness of coalitions and makes games longer).

The maximum number of cubes required for a given board size is given by the tetrahedral number Tn = n (n + 1) (n + 2) / 6.

 Holes per side Total cubes 1 1 2 4 3 10 4 20 5 35 6 56 7 84 8 120 9 165

History

Chroma rules copyright Cameron Browne and Stephen Tavener © 2008.

The name “Chroma” refers to the importance of colour in the game.

The Chroma board with six holes per side is equivalent to the Foursite 3D board available from IQideas Ltd.

Chroma can be played on Richard's PBeM server - check out the help file for more details. Many thanks to the server regulars who helped test the game. Please challenge me (camb) to a game any time.

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Site designed by Cameron Browne © 2008.