I have a pile of uniform cuboids, of side a,b,c. I would like to make a regular cube. The sides are not in harmonic ratio ( cf. de Bruijn) but are in fact white sugar lumps.

The minimum regular cube I can envisage is of side a+b+c. I have experimented making a 3x3x3 cube of 27 cuboids using a 3x3x3 magic cube to direct operations but find that

  1. I get a non tight pack on 2 out of 3 levels
  2. I cannot make a 3rd level that is compatible with the first two, one cuboid has to be omitted from the arrangement.

It seems more likely that I should be using a 6x6x6 magic cube formulation to match the 6 orientations of the cuboids a.b, b.a, a.c, c.a, b.c, c.b. But while I can visualise a 3x3x3 I would need to know how to encode the cells of a published 6x6x6 cube into these 6 possible cuboid orientations.

Any thoughts are appreciated


so it is obvious that I cannot get a tight pack, any one face of slice of the cube must have a 'hole' in it, so restricting to one 2D section laid out as a magic square ABC CAB BCA where e.g. A is a cuboid laid down with its b and c edges face down, and so occupying a footprint of b*c the length of a side is a+b+c, the overall area is (a+b+c)^2 but the component cuboids will present a total area 3ab+3ac+3bc and the difference not be equated to zero (a+b+c)^2 - 3ab+3ac+3bc = 0 except in the case a=b=c


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