5
$\begingroup$

Suppose I have 36 black blocks of dimensions 1x2x3. I can stack them 2 across, 3 deep and 6 high to make a nice looking cube of dimensions 6x6x6. I then proceed to paint the surface of this cube red. Now, when I look at the individual blocks, some faces are red and some have remained black.

In how many ways can I reassemble the large 6x6x6 cube so that its surface still appears all red?

You can choose to count rotations separately or as the same if you would like. I'm not so much looking for a final answer as a process. This is a very simple version of a more general problem I'm trying to solve. Even a nudge in the direction of some good literature would be of help. I've never encountered a counting problem like this before.

$\endgroup$
4
  • $\begingroup$ Am I to take it that you don't mind if the blocks are assembled in a different pattern as long as they form a correctly coloured $6\times 6\times 6$ cube? $\endgroup$
    – Ben Barber
    Oct 2, 2018 at 13:57
  • $\begingroup$ Yup. Same cube, but I'm wanting to count the different ways to assemble. So, I'm sort of interested in the patterns of the cracks that separate the pieces, so to speak. $\endgroup$ Oct 2, 2018 at 14:45
  • $\begingroup$ What symmetries are equivalent? If I have a brick with no paint and I rotate it, does that count as a different packing? I think this is equivalent to counting tilings or packings, which has been studied for some special cases like Aztec diamond. In particular, this problem can be considered as tiling an interval of 216 integers with special subsets of size three, with a coloring constraint added. Gerhard "Sorry, No References Right Now" Paseman, 2018.10.02. $\endgroup$ Oct 2, 2018 at 16:17
  • $\begingroup$ I am fine with all 24 rotations of an all black cube being counted separate if it makes any formulas easier. With this particular question and the initially specified stacking, each of the blocks will have at least some red on them. $\endgroup$ Oct 2, 2018 at 18:43

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.