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Multi-pathers. Which squares from what positions(on 8x8) are multi-path to the extent of three-path? In other words, when can a square be reached 3 different ways in one move? Some squares of Bishop and Queen are one-path, others two-path, and some three-path, according to position. From about 1950, Billiards Chess by M. Jacques Berthoumeau is not Progressive like present write-up. Joseph Boyer modified Berthoumeau's Rules for Billiards Chess in 1957. Castelli made this Progressive form in 1998, never yet Commented. The Billiards Mutator would go well enough with the Progressive Mutator, instinct says. Better would be Two-Move form, because moves of 3 pieces/Pawns with bouncing off edges would be devastating, and games would end with whoever first gets four or at most five to move at once. In any one separate move(standard Billiards after Berthoumeau or Boyer), pieces(B,Q) can bounce off edges any number of times, but four of course is upper limit on square boards, because of returning to a departure square(illegal). Ralph Betza has 'Three Kinds of Billiards Chess' from 15.January.1996, when I started reading here, exploring more options. Normally, the Billiard effect does not apply to Pawn or Rook or Knight; of course, Betza creatively looks into those also in the above one of the very first Chess Variant Page articles. Friedlander by now has Applet of standard Billiards after Berthoumeau/Boyer. The general idea is different from Cylindrical Chess, which shows pieces exiting to reappear the other side of board, or Jeremy Good's derivatives of Cylinder, allowing multiple connectivity, in Game Courier Presets. One alternative in the Betza article is for a player to have Billiards' empowerment and the other player to play Cylindrically. Untold millions of possibilities, considering that the three different Mutators(Boyer, Castelli, Good) are universally applicable to generic Rules-sets, and the Mutators themselves extensively modifiable, as begun already by Betza and Good(Flying Kittens, Quadruple Beseige -- latter apparently by Gilman).