Most perfect magic hypercube per order

 

[3x3x3x3]  Including mirrorring and/or rotating there are 22.272 possible (simple) 3x3x3x3

magic cubes. You need 2 out of 28 different grids. There are 348 combination

possibilities. For each grid you can use 1 of the 8 magic 3x3 squares (n.b.: 348

x 8 x 8 = 22.272).

[4x4x4x4]  You can construct a 4x4x4x4 pantriagonal and panquadragonal cube.

[5x5x5x5]  You can construct a 5x5x5x5, which is (two dimensional) panmagic in the levels or a 5x5x5x5 pantriagonal cube. 
 

[8x8x8x8]  You can construct a 8x8x8x8 pantriagonal and panquadragonal magic cube. 
 
[9x9x9x9]  You can construct a 9x9x9x9 pantriagonal magic cube or a 9x9x9x9 panquadra-gonal magic cube.

 

[12x12x12x12]  You can construct a 12x12x12x12 pantriagonal and panquadragonal magic cube.
  
[16x16x16x16] 
You can construct a 16x16x16x16 most perfect (= more than Nasik) tesseract.