2016 Update:

The cube can now be evaluated for water retention. Note the 1st crossover is for the 1,2,3,4 and 1,2,3,4,5 random levels as shown in italics below.

 

Three lakes are shown in blue by Trump in the order 6 cube below.

        Lake dimensions of nxnxn

 

Retention in the planes of a cube. Each interior cell has 6 nearest neighbors.

Pc for the 3d lattice is 0.3116.

 

 

Most-perfect Space

The most-perfect magic square introduces the idea of all 2x2 subsets in the square having the same sum. This idea is extended to the 4x4x4 cube below. There are 108 of these 2x2 subsets in the order 4 cube. The F1 compiler enumerates all 285,504 solutions in about 6 weeks run time. If magic criteria are added to the 108 criteria there are 960 solutions.

There are 8 central numbers in a 4x4x4 cube. The cube below with the values 1 and 2 in the center provide my best guess as to which of these 285,504 solns has the maximum incarceration value.

https://oeis.org/A275359

 

Most perfect space.png

 

 

 

 

 

Magic in 3 dimensions

https://en.wikipedia.org/wiki/File:Magic_square_Hilbert_space_filling_curve.png

The Hilbert space filling curve can be bent and folded into a 2 or 3 dimensional structure. I searched for a structure that had magic properties in all 3 dimensions. I could only find a semi-magic solution for the 8x8 magic square. Note that the total number of cells for the 4x4x4 cube (64) = the number cells in the 8x8 square(64).