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A nice way to play 3-player game on rectangular board (a bit artifical, but still; I mean, artifical as invented to play on rectangular board, while as itself, topologically, it's quite natural). Also there is a good unusual choice of pieces, interesting special for this board. And a cube rule have special feeling at 3-player game. "This draught, with promotion, is taken from the version of draughts I grew up with; I am aware that it is different from the better known one played hereabouts" - interestingly, where are you from?
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Admittedly, though, I have not played draughts in a *very* long time, so I cannot remember exactly whether this is actually how it was. I do remember that:
(a) it is played on the diagonals of a 10x10 board, with each player's 1st 3 rows filled
(b) the unpromoted piece moves as in this variant, though I cannot remember whether it must be forwards. It could make multiple captures as described; I do not recall any obligation to capture
(c) the promoted piece, or 'queen' ('dama' in Spanish; the namesake of the game and isonymic with the chess queen) moves as the promoted draught in this variant. I think it must have had to stop on the square after its victim (otherwise it is quite overpowered), but I do not recall how multiple capture worked... What I have opted for in this variant seems a logical choice
With regard to the artificiality of the board, I agree, having playtested two prototype versions of this. However, it works, and, once you get used to it, it's not that difficult to deal with. :)