[boothby]

Likewise, if you generalize to 3-face array, you'd need an octagonal unit painted in a 2^3 debruijn sequence...

[zahlman]

... But that's as far as you could take it, since 16-gons would show at least 7 faces while only having an encoding for 4.

I also thought of using hexagonal prisms, showing two faces at a time in paired colours but using three colours. These would also need much less clearance in order to rotate freely, compared to face-on cubes.