I had known about the base cubes derived from base lines since the 70's and it is easy to derive base cubes from the magic cubes made by Barnard, et al. The question arose as to whether there were other base cubes that could be used to make nasik magic cubes. To test all possible base cubes containing 256 zeros and 256 ones is a daunting task. Keith was able generate all possible 8x8 pan magic squares containing 32 zeros and 32 ones. There are over a million. Combining these to make all possible base cubes was still a daunting task.
A postulate was made that in one direction in the cube the squares had to alternate between a square and its inverse. This postulate was tested on the million plus squares and 1024 were found to generate unique base cubes. Among them were the 16 base line category, the 48 Barnard category, and the 64 new alternating line category discussed in Barnard's Cube. The remaining 896 are discussed below. Each of these can also be rotated into six different dimensional orientations to make six different base cubes. The remainder of the 48 possible aspects of each unique base cube are redundant.
Of course after finding all possible base cubes that undergo inversion of the squares in one direction a question remained. Are there base cubes that disprove this postulate? The answer is yes.
The base cubes discussed in Barnard's Cube used one of the lines described as base lines in Magic Cube or its inverse for every line in the figure. There are base cubes that do not have these special lines for every line of the figure. Every line in a base cube must contain four zeros and four ones. There are 70 eight bit lines that contain four zeros and four ones. Half start with zero and the other half with one. The base lines account for 22 of these leaving 48 other line possibilities. Sixteen of these possibilities are used in the base cubes of this section.
The line 00011011 and all of its translations make half of the sixteen and the inverses of the first eight are the other eight. Notice that these new lines are asymmetrical in that they have neither repetitive nature nor point of inversion. If the base cubes made using only the base line types are split in half on any axis, the two halves are either the same or inverses. This is not true of base cubes made using the asymmetric lines. The new lines can be described using the method described for Barnard's Cube. The lines are all shown on the Unique Square worksheet of the BaseCubes Excel spreadsheet available on the Downloads page. The decimal equivalent of the new lines are 27, 54, 99, 108, 141, 177, 198, and 216 for the lines with three consecutive zeros and 39, 57, 78, 114, 147, 156, 201, and 228 for those with three consecutive ones.
Base cubes in this category do not have B or C type lines for any of the lines in the cube. An A type line is used for every line in one direction, the tertiary direction, as described in Barnard's Cube. Determination of primary and secondary directions is arbitrary as the same line types are used in both directions. The same set of decimal codes will be generated if the squares are rotated 90°.
An example square of this category is shown at right. There are just four unique square patterns as shown in BaseCubes. Every zero in each of the unique squares can be translated to the upper left corner resulting in a different square configuration. There are thus 128 x 6 (for dimensional orientation) different base cubes of this category. The lines are of two types, one type has three 0's in a row and the other three 1's. Call the group of lines with three 0's, X and the group with three 1's, Y regardless of translation. The rows then have either an XYXYYXYX pattern or an XXXXYYYY pattern and so do the columns. The patterns may be translated. The four unique squares each represent one of the four ways that the patterns can be placed in the rows and columns. For every other line in both the rows and columns another configuration is line, line shifted 4, inverse line shifted 4, inverse line.
The squares are converted to cubes by alternating with the squares inverse in the z direction. There are 128x6 = 768 base cubes after considering all dimensional directions. This gives 7689 = 9.30E25 possible magic cubes. Only about 90 out of a billion of the randomly generated cubes using only base cubes of this category contained all 512 numbers and are thus nasik magic cubes. This means that there are ~ 1.7E17 magic cubes of this category after correcting for aspects.
This category of base cube has C and B type lines alternating in one direction. In the other direction are the asymmetric lines. The third direction is represented by the A type lines and is the tertiary direction. The primary direction of the cube has been designated as the direction of the B and C type lines, and the secondary is thus the direction of the asymmetric lines.
An example square of this category is shown at right. The primary direction is always highly structured. The C type lines are in every other row and alternate between the line and its inverse. The B type lines are in the remaining rows. They come in pairs of two the same shifted by one or every successive line shifted by one in the same direction. Using the convention described above for the asymmetric lines, the secondary direction has either an XYXYYXYX pattern or an XXXXYYYY pattern of asymmetric lines. For every other line in the columns another configuration for the asymmetric lines is line, line shifted 4, inverse line shifted 4, inverse line. There are 256 different squares of this category with a zero in the upper left corner.
There are 256x6 = 1536 different base cubes in this category with a zero in the upper left corner of the first square. This can make 15369 = 4.76E28 different magic squares. Random generation of magic cubes using nine base cubes form this category yielded about 8.3 magic cubes for every billion combinations tested. After adjusting for the 48 aspects there are thus about 8.2E18 magic cubes using base cubes of just this category.
This category of base cube has the asymmetric lines alternating with C type lines in one direction and the asymmetric lines alternating with B type lines in a second. The primary direction for the resulting cubes has been designated as the direction of the C line types and the secondary as the direction of the B line types. As with all the cubes discussed to this point the tertiary direction is an A type line and is obtained in practice by inverting the zeros and ones of the previous square when going in the z direction.
The C type row lines in the example square shown at right alternate between a C type line and its inverse. This is true of all examples of the category. The B type lines in the columns are either B type line, repeat, inverse, inverse or the B type line is shifted 1 in the same direction with each successive line. For the asymmetric lines in both the rows and columns the pattern is line, line shifted 4, inverse line shifted 4, inverse line. On the Unique Square page of the BaseCubes Excel spreadsheet available on the Downloads page are decimal codes for 512 squares of this category. This is the largest category of squares that can be converted to base cubes by inversion of the square in the z direction.
There are 512x6 = 3072 different base cubes in this category with a zero in the upper left corner of the first square. This can make 30729 = 2.44E31 different magic cubes. Random generation of magic cubes using nine base cubes from this category yielded about 1.5 magic cubes for every billion combinations tested. After adjusting for the 48 aspects there are thus about 7.5E20 magic cubes using base cubes of just this category. There are examples of nasik magic cubes made using the three categories of base cubes in this section on the BaseCubes Excel sheet on the Downloads page. Cubes made from these base cubes have not been shown previously but there is nothing remarkable about them.