This answer uses NMF, which can be found out about in my profile:
We know that the length has to equal the width and height. This means that the amount of cubes used in total has to be the cube (power of 3) of the side length.
The answer is
floor#({3}√{x}) = {3}√{x}
where x is your possible answers and the statement is true.
If we do x = 81, then we can do
{3}√{81} ≈ 4.33
floor#(4.33) ≠ 4.33
therefore the answer is not 81
x = 243:
{3}√{243} ≈ 6.24
floor#(6.24) ≠ 6.24
x = 486:
{3}√{486} ≈7.86
floor#(7.86) ≠ 7.86
x = 729:
{3}√{729} = 9
floor#(9) = 9
Because they are equal, the number of cubes used is 729.
