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H. G. Muller wrote on 2014-11-11 UTC
The problem with this is that when there are multiple color-bound pieces on each side you often know only in the end-game which color you need the piece to be on to not make it a hopeless draw (as with unlike Bishops). Depending on which of the pieces were traded and which survived to the end-game.

I was facing this problem when designing Team-Mate Chess, because I wanted to have several color-bound pieces there (because these can give interesting 4-men end-games, with checkmate not forcible in all corners). But if you start with 3 color-bound pieces they cannot all be on different color.

So I toyed with the idea to equip all color-bound pieces with the right to swap places with an orthogonally adjacent King, once in their lifetime. This to minimize the possibility that they can abuse this right for tactical purposes. An alternative, using a more conventional move would be to allow them a one-time Moa-hop over a friendly King. An idea from Superchess is to allow a permanent extra color-changing W-step in the board corners of the final rank (to which you presumably only have access in the end-game).

In the end I used none of that, but solved it by putting only two color-bound pieces in the initial setup, starting on different colors, and making the third piece (the strongest one) only available through promotion. (So that the player could under-promote to a non-color-bound piece if the promotion square happened to be of the wrong-color.)

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