At Tiling a square with rectangles the problem of dividing a square into distinct equal area rectangles. This is the same as the Blanche dissection with the added restraint that all sides are integers. The Mondrian Art Problem allows a defect which is the area difference between largest and smallest rectangle.

New problems:
Part 1: Divide an integer-sided rectangle into distinct integer-sided rectangles with the same area.
Part 2: Divide a cuboid into distinct cuboids of the same volume.
Part 3: Divide a cuboid into distinct integer-sided cuboids of the same volume.
Part 4: Divide a side N cube into distinct integer sided cuboids so that the defect (largest volume - smallest volume) is minimized.

Trivially, a dimension could be added to the 2D Blanche dissections. Multiple Blanche dissections could be stacked, or an extra layer could be added. There is no need to point out these trivial solutions.

A specific case: can a $24\times24\times20$ cuboid be divided into 16 distinct cuboids with volume 720?

Some extra Mondrian solutions have been found, earlier solutions are shown below. Mondrian Art

  • $\begingroup$ It is a bit confusing what is the status of each part of your Questions. Looking at the links it seems that all parts may be open. If so, I suggest limiting attention to settle part 1 (planar dissection) before proceeding to higher dimensions, unless you have motivation for a "specific case" in three dimensions. $\endgroup$ – hardmath Aug 27 '16 at 17:43

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