Features of the magic cube

Look at the most magic 4x4x4 cube of Walter Trump.

It consists of 4x4 cells in each of the 4 levels and the numbers 1 up to (4x4x4=) 64 are in it. I have numbered the cells as follows:

 level I Level II Level III Level IV a1 a2 a3 a4 a1 a2 a3 a4 a1 a2 a3 a4 a1 a2 a3 a4 b1 b2 b3 b4 b1 b2 b3 b4 b1 b2 b3 b4 b1 b2 b3 b4 c1 c2 c3 c4 c1 c2 c3 c4 c1 c2 c3 c4 c1 c2 c3 c4 d1 d2 d3 d4 d1 d2 d3 d4 d1 d2 d3 d4 d1 d2 d3 d4

The possible magic features of a magic cube are:

[All rows in each level]
give the magic sum (in each of the levels I, II, III and IV: a1+a2+a3+a4 = b1+b2+b3+b4 = c1+c2+c3+c4 = d1+d2+d3+d4 = magic sum)

[All columns in each level]
give the magic sum (in each of the levels I, II, III and IV: a1+b1+c1+d1 = a2+b2+c2+d2 = a3+b3+c3+d3 = a4+b4+c4+d4 = magic sum)

[All main diagonals in each level] give the magic sum (in each of the levels I, II, III and IV: a1+b2+c3+d4 = a4+b3+c2+d1 = magic sum)

[All pandiagonals in each level]  give the magic sum (in each of the levels I, II, III and IV:  a2+b3+c4+d1 = a3+b4+c1+d2 = a4+b1+c2+d3 = b4+c3+d2+a1 = c4+d3+a2+b1 = d4+a3+b2+c1 = magic sum)

[All 2x2 subsquares in each level]
give the magic sum (in each of the levels I, II, III and IV: a1+a2+b1+b2 = a2+a3+b2+b3 = a3+a4+b3+b4 = b1+b2+c1+c2 = b2+b3+c2+c3 = b3+b4+c3+c4 = c1+c2+d1+d2 = c2+c3+d2+d3 = c3+c4+d3+d4 = magic sum)

[All pillars through the levels]
give the magic sum (I a1 + II a1 + III a1 + IV a1 = I a2 + II a2 + III a2 + IV a2 = ... = I d4 + II d4 + III d4 + IV d4 = magic sum {for all 16 pillars})

[All diagonals from left to right through the levels] give the magic sum (I a1 + II a2 + III a3 + IV a4 = I b1 + II b2 + III b3 + IV b4 = I c1 + II c2 + III c3 + IV c4 = I d1 + II d2 + III d3 + IV d4 = magic sum)

[All diagonals from right to left through the levels] give the magic sum (I a4 + II a3 + III a2 + IV a1 = I b4 + II b3 + III b2 + IV b1 = I c4 + II c3 + III c2 + IV c1 = I d4 + II d3 + III d2 + IV d1 = magic sum)

[All diagonals from up to down through the levels] give the magic sum (I a1 + II b1 + III c1 + IV d1  = I a2 + II b2 + III c2 + IV d2  =  I a3 + II b3 + III c3 + IV d3  =  I a4 + II b4 + III c4 + IV d4  =  magic sum)

[All diagonals from down to up through the levels] give the magic sum (I d1 + II c1 + III b1 + IV a1 = I d2 + II c2 + III b2 + IV a2 = I d3 + II c3 + III b3 + IV a3 = I d4 + II c4 + III b4 + IV a4 = magic sum)

[All space diagonals through the levels]
give the magic sum (I a1 + II b2 + III c3 + IV d4 = I a4 + II b3 + III c2 + IV d1 = I d1 + II c2 + III b3 + IV a4 = I d4 + II c3 + III b2 + IV a1 = magic sum)

[All pandiagonals from left to right through the levels]
give the magic sum (I a2 + II a3 + III a4 + IV a1 = I a3 + II a4 + III a1 + IV a2 = I b2 + II b3 + III b4 + IV b1 = I b3 + II b4 + III b1 + IV b2 = I c2 + II c3 + III c4 + IV c1 = I c3 + II c4 + III c1 + IV c2 = I d2 + II d3 + III d4 + IV d1 = I d3 + II d4 + III d1 + IV d2 = magic sum)

[All pandiagonals from right to left through the levels]
give the magic sum (I a3 + II a2 + III a1 + IV a4 = I a2 + II a1 + III a4 + IV a3 = I b3 + II b2 + III b1 + IV b4 = I b2 + II b1 + III b4 + IV b3 = I c3 + II c2 + III c1 + IV c4 = I c2 + II c1 + III c4 + IV c3 = I d3 + II d2 + III d1 + IV d4 = I d2 + II d1 + III d4 + IV d3 = magic sum)

[All pandiagonals from up to down through the levels]  give the magic sum (I b1 + II c1 + III d1 + IVa1 = I c1 + II d1 + III a1 + IV b1 = I b2 + II c2 + III d2 + IV a2 = I c2 + II d2 + III a2 + IV b2 = I b3 + II c3 + III d3 + IV a3 = I c3 + II d3 + III a3 + IV b3 = I b4 + II c4 + III d4 + IV a4 = I c4 + II d4 + III a4 + IV b4 = magic sum)

[All pandiagonals from down to up through the levels]
give the magic sum (I c1 + II b1 + III a1 + IV d1 = I b1 + II a1 + III d1 + IV c1 = I c2 + II b2 + III a2 + IV d2 = I b2 + II a2 + III d2 + IV c2 = I c3 + II b3 + III a3 + IV d3 = I b3 + II a3 + III d3 + IV c3 = I c4 + II b4 + III a4 + IV d4 = I b4 + II a4 + III d4 + IV c4 = magic sum)

[All pantriagonals, first direction, through the levels]
give the magic sum (I a2 + II b3 + III c4 + IV d1 = I a3 + II b4 + III c1 + IV d2 = I a4 + II b1 + III c2 + IV d3 = I b1 + II c2 + III d3 + IV a4 = I b2 + II c3 + III d4 + I a1 = I b3 + II c4 + III d1 + IV a2 = I b4 + II c1 + III d2 + IV a3 = I c1 + II d2 + III a3 + IV b4 = I c2 + II d3 + III a4 + IV b1 = I c3 + II d4 + III a1 + IV b2 = I c4 + II d1 + III a2 + IV b3 = I d1 + II a2 + III b3 + IV c4 = I d2 + II a3 + III b4 + IV c1 = I d3 + II a4 + III b1 + IV c2 = I d4 + II a1 + III b2 + IV c3 = magic sum)

[All pantriagonals, second direction, through the levels]
give the magic sum (I a1 + II b4 + III c3 + IV d2 = I a2 + II b1 + III c4 + IV d3 = I a3 + II b2 + II c1 + IV d4 = I b1 + II c4 + III d3 + IV a2 = I b2 + II c1 + III d4 + IV a3 = I b3 + II c2 + III d1 + IV a4 = I b4 + II c3 + III d2 + IV a1 = I c1 + II d4 + III a3 + IV b2 = I c2 + II d3 + III a4 + IV b1 = I c3 + II d2 + III a1 + IV b2 = I c4 + II d1 + III a2 + IV b3 = I d1 + II a4 + III b3 + IV c2 = I d2 + II a1 + III b4 + IV c3 = I d3 + II a2 + III b1 + IV c4 = I d4 + II a3 + III b2 + IV c1 = magic sum)

[All pantriagonals, third direction, through the levels]
give the magic sum (I a1 + II d2 + III c3 + IV b4 = I a2 + II d3 + III c4 + IV b1 = I a3 + II d4 + III c1 + IV b2 = I a4 + II d1 + III c2 + IV b3 = I b1 + II a2 + III d3 + IV c4 = I b2 + II a3 + III d4 + IV c1 = I b3 + II a4 + III d1 + IV c2 = I b4 + II a1 + III d2 + IV c3 = I c1 + II b2 + III a3 + IV d4 = I c2 + II b3 + III a4 + IV d1 = I c3 + II b4 + III a1 + IV d2 = I c4 + II b1 + III a2 + IV d3 = I d2 + II c3 + III b4 + IV a1 = I d3 + II c4 + III b1 + IV a2 = I d4 + II c1 + III b2 + IV a3 = magic sum)

[All pantriagonals, fourth direction, through the levels]  give the magic sum (I a1 + II d4 + III c3 + IV b2  = I a2 + II d1 + III c4 + IV b3  =  I a3 + II d2 + III c1 + IV b4  =  I a4 + II d3 + III c2 + IV b1  = I b1 + II a4 + III d3 + IV c2  =  I b2 + II a1 + III d4 + IV c3  =  I b3 + II a2 + III d1 + IV c4  = I b4 + II a3 + III d2 + IV c1  =  I c1 + II b4 + III a3 + IV d2  =  I c2 + II b1 + III a4 + IV d3  = I c3 + II b2 + III a1 + IV d4  =  I c4 + II b3 + III a2 + IV d1  =  I d1 + II c4 + III b3 + IV a2  = I d2 + II c1 + III b4 + IV a3  =  I d3 + II c2 + III b1 + IV a4  =  magic sum)

[Symmetric]
In a symmetric magic cube each time addition of two digits, which can be connected with a straight line through the centre of the magic cube and which are at the same distance to the centre, gives the same sum. The centre of a(n even, for example the) 4x4x4 magic is the virtual crosspoint of the middle 2x2 cells of the second and third level. For example II b2 + III c3 = II b3 + III c2 = II c2 + III b3 = II c3 + III b2 = proportional part (= 1/2) of the magic sum ór I a1 + IV d4 = I a2 + IV d3 = I a3 + IV d2 = I a4 + IV d1 = proportional part (= 1/2) of the magic sum.

[half of]
the rows/columns/diagonals in each level or pillars/space diagonals through the levels give half of the magic sum.

• Simple magic cubes have the red marked (see above) magic features.

• Diagonal magic cubes have the red & orange marked (see above) magic features

• Pantriagonal cubes have the red & yellow marked (see above) magic futures

• Pandiagonal magic cubes have the red & orange & pink marked (see above) magic features.

• Perfect (Nasik) magic cubes have the red & orange & pink & yellow marked (see above) magic features.

• More than perfect magic cubes have red & orange & yellow & [part of the] green marked (see above) magic features.

Order 3 is simple magic. Diagonal magic cubes exist from order 5 and up. Pantriagonal magic cubes exist from order 4 and up. Pandiagonal magic cubes exist from order 7 and up. Nasik perfect (= pandiagonal & pantriagonal) magic cubes exist for odd orders from order 9 and up and for order 8, 16, 24, 32, …