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Hint to puzzle 151: Painted cubes

Consider separately the four types of cubes upon disassembly:

  1. 8 cubes with three faces painted black;
  2. 12 cubes with two black faces;
  3. 6 cubes with one black face;
  4. 1 completely white cube.

Then count the number of ways in which all cubes of each type may be correctly positioned and oriented.