Boche Cameron Browne (c) 2008 Boche is 3D connection game played with Truchet cubes.

Rules

Tiles: Two players, White and Blue, share a common pool of 56 Boche cubes. Each cube consists of identical white and blue parts joined togther such that each face forms a duotone Truchet tile.

Figure 1.  Blue and white parts form a cube.

Each Boche cube has eight unique rotations, labelled a to h, when fitted to a cubic lattice and viewed corner-on (Figure 2).

Figure 2.  The eight unique rotations.

The four rotations with the Blue corner uppermost (a, b, c, d) are described as being of Blue parity and the four rotations with the White corner uppermost (e, f, g, h) are described as being of White parity.

Board: The board is a triangular grid of triangular holes (six per side) as shown in Figure 3. Each hole will snugly hold a cube placed corner-downward.

Figure 3.  The Boche board.

Play: The board is initially empty. White places a cube on the hole of their choice, rotated as they wish. Blue may then elect to swap colours in lieu of making the second move (swap option). Players then take turns adding a cube, rotated as they wish.

Each cube may be placed on a board hole or stacked to nestle in the intersection formed by three mutual neighbours. The only constraint is that all cubes on the same level must be of the same parity so that edge colours will always match any neighbouring cubes.

Figure 4.  Opening move, bad parity and good parity.

For example, Figure 4 shows a typical opening move d (left) which sets the board level to Blue parity. Move f (middle) would then be an illegal reply as it's of the wrong parity and does not match edge colours, however move b (right) is a valid reply.

Aim: The game is won by enclosing a group of the opponent's colour of at least size 4. The group must be completely enlosed, that is, it must not connect to any board side. A group's size is given by the number of "blobs" that it contains.

For example, Figure 5 shows a game won by Blue who has completely enclosed a White group containing four blobs. Blue has also surrounded a White group with a single blob (lower left), but this smaller group is ignored and does not have any bearing on the game.

Figure 5. Blue surrounds a White group to win.

If the board fills up before any groups are surrounded, then the game is won by the owner of the path connecting the three sides of the board. For example, Blue wins the game shown in Figure 6.

Figure 6. Blue owns the winning path.

There will always be such a winning path connecting all three sides on a full board; every game will have exactly one winner.

Notes

The "blobs" that make up each group are spheres centred on cube corners.

Boche is functionally equivalent to Y in that ties are not possible; every game will produce a winning path even if the board fills up (Figure 6). The Truchet tile designs on each cube face resolve potential deadlock problems at square (non-trivalent) intersections.

Parity alternates with each level upwards. So if the board level is set to Blue parity, then level 1 will be White parity, level 2 will be Blue parity, level 3 will be White parity, etc.

Beginners may like to play with the following optional rule for the first game or two until they understand the concept of parity: board level cubes (after the first move) must be placed adjacent to at least one existing cube such that edge colours match. This explicitly forces all cubes into the same parity.

The player with the parity on a given level will have the advantage on that level as they'll be able to play the fully connected piece of their colour (a for Blue and e for White) at will. However, the opponent will have the advantage in any stacking move which will be of the opposite parity.

The first player has the advantage of setting the board level to their preferred parity with the opening move, however the second player has the advantage of playing the last move, which may prove decisive. For example, the last piece played on the pyramid's apex in Figure 6 decided the result of that particular game. These two advantages are significant but balance each out over the course of the game; Boche is different to other connection games in this respect.

The number of cubes per layer is given by the triangular number Ln = n (n + 1) / 2. 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 Layer count Total cubes 1 1 1 2 3 4 3 6 10 4 10 20 5 15 35 6 21 56 7 28 84 8 36 120 9 45 165 10 55 220

Board sizes that give an odd number of cubes (1, 5, 9, 13, 17, ...) should be avoided as the opening player would then get the double advantage of having first and last move.

History

Boche rules and design by Cameron Browne and copyright © Cybterite Ltd, 2008.

The name “Boche” is derived from the fact that the game is effectively played with cubed Che tiles within the corner of a box.

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

Boche 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.