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64 cells  .          F.i.d.e.'s going to triangles, troika, would
         ._.         mean brand new, fresh start opening 
 II     ._._. 234    theory with retention of holy of holies 
III    ._._._. 5-9   sixty-four. Study triangles for a twenty-
 IV   ._._._._.      four period and squares are unnatural. In
  V  ._._._._._.     both square and triangle, unlike hexagon,
 VI ._._._._._._.    Rook cannot triangulate. Rook circumnavigating
   ._._._._._._._.   the board goes 50-3-64-51 ending over one cell
  ._._._._._._._._.  from origination whilst not entering #1 --
 (Level VIII 50-64)  contrasted to squares' 4-move circumnavigation. Rook
mandatorily switches Bishop binding precisely from turn to turn, like Knight does in
squares.  Knight's two-step through side or vertex is colourbound, enabling two types, corner and interior, as with Bishops; alternatively let the two Knights have the Pawn-capturing one-step through vertex at option for neater colourswitching availability in one pure Equine piece-type. On standard 64 the corner Bishop reaches
36 spaces and the interior Bishop 28 spaces for necessary magnified value differential
at about 3.5 to 2.5. Strong omni-Pawns work best, one-stepping through
side to move and vertex to capture. Promotion should exclude Queen.  The opening imperative for the first ten moves is tug between Castling King from array 50 and 64 on one hand and control of valuable real estate, the center, triangles 21, 30, 31, 32. Whoso controls those four will have the presumptive edge, other things being equal. Ironically, King Castling to 26 is counterintuitively rather towards that center, but he does not wish to step any closer. Scads of equilateral triangular boards are possible from about 4^4^4 to about 16^16^16 each with its untold THOUSANDS of VARIANTS of core classical Chess on the triadic theme. Use your imagination: two-move, AMAZON, Fischer arrays, NIGHTRIDER, cylindrical, royal Knight, you name it.