Polycube Snakes

The , and cubes can be filled with a polycube "snake" that turns by right angles at every step. The solutions for these three cubes are given here.
The display of the polycubes is optional; their positions are represented by black points connected by a polyline.
Such a snaking path can be found for all cuboids except for the following cases:
and ,
odd, odd, .
(Karl Scherer 1982; the proof is very easy.)

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