bimagic cubes
These magic cubes are bimagic, that is, they are still magic when each cell of them is squared.
order-(2x) bordered diagonal magic cubes
The following magic cubes are probably the first bordered (namely, consecutively concentric) magic cubes of even order in the world, and probably the first bordered diagonal magic cubes in the world.
Each of these cubes is a diagonal magic cube and contains (non-normal) subcubes of all even orders lower than the order of the cube.
Every subcube consists of consecutive integers, and every subcube (except the order-4 one) is a diagonal magic cube.
(An order-4 magic cube cannot be diagonal unless it contains duplicate numbers.)
Note William H. Benson & Oswald Jacobi published the first order-14 diagonal magic cube in 1981. The cube is shown in the following book:
Benson & Jacoby, Magic Cubes New Recreations, Dover,1981, ISBN 0-486-24140-8, pp.116-126.
Note An order-(4x+2) associated diagonal magic cube cannot exist. If such a cube were possible, its oblique squares would be order-(4x+2) (non-normal) associated magic squares with the odd constant. However, such a magic square cannot exist.
Note An order-(4x+2) associated magic hypercube of even dimension cannot exist, but this does not hold for odd dimensions.
Note An order-(4x+2) panmagic hypercube of even dimension cannot exist, but this does not hold for odd dimensions.
In fact, Abhinav Soni made some order-(4x+2) panmagic hypercubes of odd dimension higher than 3 in July 2004.
order-(8x) Nasik magic cubes (also associated)
The following magic cubes are probably the first order-(8x) associated Nasik magic cubes (that is, pan-2,3-agonal magic cubes) in the world.
Note A Nasik magic cube of even order can exist only if the order is divisible by 8. On the other hand, a Nasik magic cube of odd order can exist only if the order is higher than 8.
Note An order-8 associated Nasik magic cube cannot exist. Look at this.
Note Order-4 diagonal magic cubes cannot exist. Richard Schroeppel proved this in 1972. Walter Trump says that an order-4 s-magic cube cannot exist, either.
order-(2x+1) pseudo-pantriagonal diagonal magic cubes (also associated)
The folowing magic cubes have a special feature, that is, though they are not pantriagonal magic cubes, all pantriagonals on their oblique squares are magic. I call the feature pseudo-pantriagonal.
These cubes are both pseudo-pantriagonal and diagonal. I made these cube by exchanging some cells of pan-2,3-agonal magic cubes of the same order.
Open problem Can there exist an order-7 pseudo-pantriagonal diagonal magic cube?
(An order-7 pan-2,3-agonal magic cube cannot exist, so we cannot make order-7 pseudo-pantriagonal diagonal magic cubes by using pan-2,3-agonal magic cubes.)
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