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

2. https://mathoverflow.net/questions/469655/cubing-the-cube-as-perfectly-as-possible

3. https://mathoverflow.net/questions/469868/perfect-squaring-of-rectangles

We try to add a bit to ref 2. In this post, by 'cuboid', we mean only rectangular cuboids - hexahedra with all faces rectangles and adjacent faces meeting only at right angles. A special case of such cuboids is when a pair of opposite faces are squares - the square rectangular cuboid. 

**Definition:** a perfect cuboiding of a cuboid C is a partition of C into a finite number of cuboids that are similar to one another (equal length: width: height ratios) but are mutually non-congruent. Requiring that the small cuboids be similar to C as well is a potentially interesting subcase. Obviously, the pieces needing to be cubes (perfect cubing) is a very restricted case of cuboiding. 

Note: As proved in ref 1, a cube has no perfect cubing. The same argument also implies that no perfect cubing can be done for any general cuboid.  

- Does a cube allow a perfect cuboiding - a partition into finitely many mutually similar and mutually non-congruent rectangular cuboids? Special case: Does a cube have a perfect cuboiding into square rectangular cuboids?

- And what can one say about perfect cuboidings of general rectangular cuboids/square rectangular cuboids?