Three-player chess

Standard variant

For the "standard" three-player chess, the positions are labeled as follows.

a1 a2 a3 a4 b1 b2 b3 b4 c1 c2 c3 c4 d1 d2 d3 d4 e1 e2 e3 e4 f1 f2 f3 f4 g1 g2 g3 g4 h1 h2 h3 h4 i1 i2 i3 i4 j1 j2 j3 j4 k1 k2 k3 k4 l1 l2 l3 l4 m1 m2 m3 m4 n1 n2 n3 n4 o1 o2 o3 o4 p1 p2 p3 p4 q1 q2 q3 q4 r1 r2 r3 r4 s1 s2 s3 s4 t1 t2 t3 t4 u1 u2 u3 u4 v1 v2 v3 v4 w1 w2 w3 w4 x1 x2 x3 x4

This is the setup of the pieces.

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Cylinder - closed ranks

a1 a2 a3 a4 b1 b2 b3 b4 c1 c2 c3 c4 d1 d2 d3 d4 e1 e2 e3 e4 f1 f2 f3 f4 g1 g2 g3 g4 h1 h2 h3 h4 i1 i2 i3 i4 j1 j2 j3 j4 k1 k2 k3 k4 l1 l2 l3 l4 q1 q2 q3 q4 r1 r2 r3 r4 s1 s2 s3 s4 t1 t2 t3 t4 m1 m2 m3 m4 n1 n2 n3 n4 o1 o2 o3 o4 p1 p2 p3 p4 u1 u2 u3 u4 v1 v2 v3 v4 w1 w2 w3 w4 x1 x2 x3 x4

By gluing the left and ride sides of each player's third of the board, one obtains a cylinder-like shape. Note that the positions a4-h4-i4-p4-q4-x4 meet at a common corner point, just like d4-e4-l4-m4-t4-u4. Each rank forms a closed line containing 8 squares. The board has three circular edges.

x4 x3 x2 x1 a4 i4 i3 i2 i1 h4 h4 h3 h2 h1 i4 q4 q3 q2 q1 p4 p4 p3 p2 p1 q4 a4 a3 a2 a1 x4

The pieces start at the first and second rank as usual. The pieces of each player start along one of the three circular edges.

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Cylinder - closed files

a1 a2 a3 a4 a5 a6 a7 b1 b2 b3 b4 b5 b6 b7 c1 c2 c3 c4 c5 c6 c7 d1 d2 d3 d4 d5 d6 d7 e1 e2 e3 e4 e5 e6 e7 f1 f2 f3 f4 f5 f6 f7 g1 g2 g3 g4 g5 g6 g7 h1 h2 h3 h4 h5 h6 h7 i1 i2 i3 i4 i5 i6 i7 j1 j2 j3 j4 j5 j6 j7 k1 k2 k3 k4 k5 k6 k7 l1 l2 l3 l4 l5 l6 l7 m1 m2 m3 m4 m5 m6 m7 n1 n2 n3 n4 n5 n6 n7 o1 o2 o3 o4 o5 o6 o7 p1 p2 p3 p4 p5 p6 p7 q1 q2 q3 q4 q5 q6 q7 r1 r2 r3 r4 r5 r6 r7 s1 s2 s3 s4 s5 s6 s7 t1 t2 t3 t4 t5 t6 t7 u1 u2 u3 u4 u5 u6 u7 v1 v2 v3 v4 v5 v6 v7 w1 w2 w3 w4 w5 w6 w7 x1 x2 x3 x4 x5 x6 x7

In order to close the files instead of the ranks, the inner squares are doubled. Also this board has three edges, along the files h-i, p-q and x-a.

u1 v1 w1 x1 t1 l1 k1 j1 i1 m1 m1 n1 o1 p1 l1 d1 c1 b1 a1 e1 e1 f1 g1 h1 d1 t1 s1 r1 q1 u1

One may also twist each third-board before gluing the files together. Now the new board has only a single circular edge, similarly to a Möbius strip, but in contrast to the latter it is orientable.

m1 n1 o1 p1 l1 t1 s1 r1 q1 u1 e1 f1 g1 h1 d1 l1 k1 j1 i1 m1 u1 v1 w1 x1 t1 d1 c1 b1 a1 e1

For both geometries, the initial setup of the pieces is the same. The main pieces start on the fourth ranks, guarded by pawns on the third and fifth ranks.

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Torus and Klein bottle type boards

a1 a2 a3 a4 a5 a6 a7 b1 b2 b3 b4 b5 b6 b7 c1 c2 c3 c4 c5 c6 c7 d1 d2 d3 d4 d5 d6 d7 e1 e2 e3 e4 e5 e6 e7 f1 f2 f3 f4 f5 f6 f7 g1 g2 g3 g4 g5 g6 g7 h1 h2 h3 h4 h5 h6 h7 i1 i2 i3 i4 i5 i6 i7 j1 j2 j3 j4 j5 j6 j7 k1 k2 k3 k4 k5 k6 k7 l1 l2 l3 l4 l5 l6 l7 m1 m2 m3 m4 m5 m6 m7 n1 n2 n3 n4 n5 n6 n7 o1 o2 o3 o4 o5 o6 o7 p1 p2 p3 p4 p5 p6 p7 q1 q2 q3 q4 q5 q6 q7 r1 r2 r3 r4 r5 r6 r7 s1 s2 s3 s4 s5 s6 s7 t1 t2 t3 t4 t5 t6 t7 u1 u2 u3 u4 u5 u6 u7 v1 v2 v3 v4 v5 v6 v7 w1 w2 w3 w4 w5 w6 w7 x1 x2 x3 x4 x5 x6 x7

Using a tesselation of the hyperbolic plane, one can glue the board to a genus 2 surface analog of a torus.

l1 k1 j1 i1 m1 p1 a7 a6 a5 a4 a3 a2 a1 x1 x7 q1 u1 v1 w1 x1 t1 q1 h7 h6 h5 h4 h3 h2 h1 i1 i7 p1 d1 c1 b1 a1 e1 h1 q7 q6 q5 q4 q3 q2 q1 p1 p7 i1 m1 n1 o1 p1 l1 i1 x7 x6 x5 x4 x3 x2 x1 a1 a7 h1 t1 s1 r1 q1 u1 x1 i7 i6 i5 i4 i3 i2 i1 h1 h7 a1 e1 f1 g1 h1 d1 a1 p7 p6 p5 p4 p3 p2 p1 q1 q7 x1

In addition, one may also flip each third-board before gluing.

t1 s1 r1 q1 u1 x1 a7 a6 a5 a4 a3 a2 a1 p1 x7 i1 m1 n1 o1 p1 l1 i1 h7 h6 h5 h4 h3 h2 h1 q1 i7 x1 l1 k1 j1 i1 m1 p1 q7 q6 q5 q4 q3 q2 q1 h1 p7 a1 e1 f1 g1 h1 d1 a1 x7 x6 x5 x4 x3 x2 x1 i1 a7 p1 d1 c1 b1 a1 e1 h1 i7 i6 i5 i4 i3 i2 i1 x1 h7 q1 u1 v1 w1 x1 t1 q1 p7 p6 p5 p4 p3 p2 p1 a1 q7 h1

The initial positions of the pieces are as for the file cylinder.

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