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Ref: https://en.wikipedia.org/wiki/Squaring_the_square

A perfect cubing of a cube is a partition of the cube into some finite number of smaller cubes that are pair-wise non-congruent. The above page gives a nice proof that perfect cubing is impossible.

  • How to cut a cube into the least number of cubes so that only 2 of the smaller cubes are mutually congruent?

Note: Whether this construction is possible or not, we can replace 2 with 3, 4... to generate other questions (obviously, if 8 of the smaller cubes can be congruent, the problem becomes trivial). Alternatively, one can consider 2 pairs of mutually congruent cubes etc.

These questions could have analogs in higher dimensions.

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    $\begingroup$ It is not even clear whether there is an integer cube that can be decomposed into smaller cubes, none of which appear more than SEVEN times. i couldn't construct one. $\endgroup$
    – Erich Friedman
    Commented Nov 3 at 19:02
  • $\begingroup$ Thank you. Thus there appear to be at least 6 (in fact many more) questions - 2 of the cubes identical, 3 identical, 2 different pairs of mutually identical cubes and so forth. Just a hunch: with 4 of the smaller cubes identical, one might have a better chance than with 7 identical. I don't know how to even begin though. $\endgroup$
    – Nandakumar R
    Commented Nov 5 at 13:56

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