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I'm most likely misunderstanding what Tony Paletta means by a three-geometry chessboard game, but I'm going to pretend it means a chessboard with three different types of square. Actually, there's four, look:
<pre>
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| | | | | | | | | |
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| | | | | | | | | |
--/-\----/-\----/-\--
/\/ \/\/ \/\/ \/\
/--\ /--\ /--\ /--\
/ \ / \ / \ / \
\ /-\ /-\ /-\ /
\--/ \--/ \--/ \--/
/ \ / \ / \ / \
/ \ / \ / \ / \
\ /-\ /-\ /-\ /
\--/ \--/ \--/ \--/
\/\ /\/\ /\/\ /\/
/--\-/----\-/----\-/--
| | | | | | | | | |
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</pre>
For the ASCII Art impaired, a verbal description: The top and bottom two ranks have 9 files, and are regular squares. Bordering files a, c, d, f, g, and i are equilateral triangles. Bordering files b, e, and h are regular hexagons. Between the triangles on files c and d, and files f and g, is a triangle facing the opposite way. Two more triangles are placed next to the leftmost and rightmost triangles. These two triangles face opposite to the triangle they are next to (regular trigon tesselation).<p>
Above the row of hexagons and triangles is a crooked row of seven hexagons in regular tesselation. This pattern is half the board. Create a mirror image facing the other way, and join the two halves such that the three soon-to-be-center-most hexagons overlap. Despite the ASCII, all squares are the same size, all hexagons are the same size, and all triangles are the same size.<p>
I can think of tons of variations on this board, mostly by adding, removing, or replacing hexagons with triangles or vice versa.<p>
My question: How would the pieces move? Here's what I think:<p>
The Rook, as its first step, can move to any cell which shares a border with its current cell. Its second and subsequent steps must be to the cell whose border is directly opposite the border it entered from. Triangles don't have an opposite, so they require some obnoxious rules. There are two kinds of triangles: Attacking (with points towards your opponents) and Defending (with points towards yourself). It is the nature of the board and the rook move that it must alternate between attacking and defending triangles, no matter how many cells of other shapes lie in between. Because of this, I will define the step by calling them Odd and Even triangles. The Odd triangle is the *first* triangle you move *to* in a rook move. When entering an Odd triangle, pretend the border you entered on is connected to an Even triangle (even if it is a square or hexagon). The next time you enter an Even triangle, the only way to continue your move (if you wish) is to exit by the same border as the imaginary triangle you exited when you first entered the original Odd triangle. By leaving the Odd triangle, you similarly define the border by which you must exit (if you choose to continue your move) the next Odd triangle you enter.<p>
Bishops. If the cell you are currently in is a Square or Triangle, valid directions for a bishop are those cells which touch your current cell, but a Rook cannot reach (That is, they share a corner). Unfortunately, this would trap the Bishops on their respective sides, because the Rook can reach all the hexagons surrounding a hexagon. I could alleviate this by replacing the three central hexagons with rings of triangles, but then the board 'degenerates' into an 8 rank board with 4 ranks of 9 files on the edges, and 4 ranks of 13 files in the center, with some odd connections at the seam. Instead, I will say (generically) that if two of the cells a rook can reach in a single step share a border, then the bishop can jump to the nearest square in the same direction as the shared border (that is also not one of the two original cells). (There *has* to be a simpler way to say all that). For subsequent steps, if you are on a square or hexagon, you must leave to the first available square in the direction of the corner opposite the one you entered by. Theoretically, similar 'marking' rules as the Rook can be used for the triangles, but in practice, it makes my head hurt.
<p>The queen would simply combine rook and bishop.
<p>The knight would be able to leap to the nearest N squares which cannot be reached by the queen (rook or bishop), and whose manhattan distances all share the same value. For purposes of manhattan distances, the distance between the center of two cells which share a complete border is the same for any combination of cell shapes. (Not that that's possible to *draw* without seriously skewing the board).
<p>After examining some of the possible moves for this type of game, I have decided that it somewhat resembles a game of shifted square chess, which says something, I just don't know what. The triangles have a tendency to skew the movement in unusual directions, especially if a piece can choose to enter more than one triangle in a turn.