Patrick Hamlyn has provided the following list of numbers for the chequered polycubes.
Peter Esser has provided the following solution of a 2x5x29 box made with the chequered pentacubes.
As noted with the polyominoes above, the chequered polycubes of odd volume have double the number of pieces of the unchequered set and so we can form the above by using the block at the top below and then using another copy with colours alternated to form the 2x5x29. The figures at the bottom show 2x5x6 and 3x4x5 boxes made with half the chequered solid pentominoes.
Patrick Hamlyn has produced seventeen 3x6x6 and thirty four 3x3x6 blocks with the chequered hexacubes.