All of the previous categories of base cubes can be looked at as being derived from 8x8 pan-magic base squares. They are completed in the third direction by alternating between the square and its inverse. This inversion is equivalent to an exclusive OR of all numbers in the square with A0. Out of about a million possible 8x8 pan-magic base squares, only 1024 can be used to make base cubes with this property. They all can be rotated into 6 different dimensional orientations to make 6144 different base cubes.
I was convinced that having the A0 inversion in one direction of the cube was a necessity as I could not think of another way to ensure that the 2x2x2 cubes would always add to the appropriate sum and I felt that this would be a requirement for a nasik magic cube. Keith had been trying various approaches to determine if my postulate was correct and had almost decided it was. Then Keith found a few examples of non A0 base cubes. The "other way" is obvious when you see some examples. The 2x2x2 sub-cubes in the non A0 base cubes all add to 4.
Finding a few examples was the impetus needed to push for an approach that would exhaustively determine all possible order-8 pan-2,3-agonal base cubes. This was not an easy task. It could be done by testing all possible stacks of valid 8x8 base squares, but there are about a million of them. A million base squares can be stacked 1,000,0008 ways to make a cube. This is well beyond our current computers abilities. Stacking the second and third squares perpendicular to the previous squares trims the tree faster than stacking them parallel, but it still requires too many iterations to test all possibilities. Nakamura had shown that all numbers a (4,4,4) vector apart in order-8 pan-2,3-agonal magic cubes are complements. For order-8 pan-2,3-agonal base cubes this means that numbers located a (4,4,4) vector apart are inverses. This means that if a square in a base cube is known, the square located a vector of 4 away is also known. This fact made determining all possible base cubes by an exhaustive search possible by greatly reducing the number of combinations that had to be tested.
The base cubes in this section are best described completely rather than being built from smaller units of the cube. Some can be described as smaller units, i.e. the second half is the inverse of the first half or in other ways, but in general, such a description will not describe an entire category. An inclusive code that can define these cubes is given in Barnard's Cube. The 8x8 matrix required for this description is probably more cumbersome than the 8x8x8 matrix of zeros and ones when considering that a conversion between the two is needed. It is more compact for inclusion in a spreadsheet and I find it more useful when programming. The base cubes of all nine categories are listed this way in BaseCubes2 available by request from the author. This spreadsheet lists all the possible order-8 pan-2,3-agonal base cubes including the different orientations when they are unique. It also shows graphically the last three categories.
| 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
| 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
| 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
| 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
One of Keith's first examples is shown at right. It is a simple pattern using only A, B, and C line types throughout the cube. The first two xy planes are shown. The third and fourth xy planes are the inverses of the first two. The last four xy planes are repetitions of the first four. In the z direction are just B type lines.
As shown at right, the squares of the cube in this cross section look like the first squares of a C0 A0 B0 base line cube. Unlike the base line cube, the second square in the cube at right is rotated 90° and translated a vector of 1 to the right. The third square is obtained by just rotating the second square 90°, the fourth by rotation and translation, and the last four are the same as the first four.
There are nineteen pattern types within this category. The different pattern types are usually not as easily described as the above example except by one orientation. The number of different base cubes in each pattern type varies from 96 to 768 (In multiples of 2). There are 6144 different base cubes in this category. Coincidentally, there are 6144 different base cubes in the first six categories combined.
Most patterns within this group have one orientation in which every square can be described as an 8x8 A0or1B0,1,2,or3 or B0,1,2,or3A0or1 square. In this orientation some are the AB type and others the BA type. The three categories in Barnard's Cube are all either AB or BA in this orientation, they could thus be considered subsets of this category. When viewed in this orientation, the first four squares of all of these cubes are the same as their last four squares. All possible cubes that can be built from a repetition of a combination of four AB and/or BA type squares will make one of the base cubes in this category or in one of the three categories described in Barnard's Cube.
Three patterns are exceptions to the above description. One is illustrated above. For these three patterns, there is an A0or1 line in either all the rows or all the columns of each square. The A0or1 direction alternates between rows and columns for each successive square. In the other direction is one of the C type lines. In the example illustrated above the C pattern is 00001111 or one of its shifted analogs. It is present in all the squares going perpendicular to the A0or1 type line. In a second example the C type line is 00101101 or one of its shifted analogs in all squares. For the third exception, the successive square alternate between the two C type lines.
There are 6144 base cubes in this category. This gives 61449 = 1.25E34 possible magic cubes.
As indicated earlier there are a total of 70 ways to arrange four zeros and four ones in an eight bit line. The base lines account for 22 of these and the asymmetric lines 16 leaving 32. All 32 of these are used in this category of base cubes. The lines are 00010111, 00101011, 11010100 , and 11101000 and all possible translations. The second two are inverses of the first two. The two halves of the lines as written are mirror inverses of each other, thus the category name.
There are thirteen different pattern types within this category. Each pattern type contains from 384 to 3072 (In multiples of 2) different base cubes. There are a total of 12,288 different base cubes in this category. Coincidentally, there are a total of 12,288 different base cubes in the first seven categories combined.
Unlike all the other categories, in this category there are no 8x8 squares perpendicular to an axis that contain an A type line in either all its rows or all its columns. There is always one cross section in which all of the squares have only mirror type lines in both directions. Squares perpendicular to the two axes always have only mirror type lines in one direction. In the other direction, half of the lines are A type lines and the other half are B type lines. There are two patterns of these lines in the square, AAAABBBB or AABABBAB and all possible shifts. For a set of eight squares in one direction, half of the squares have the A and B type lines in the rows and half in the columns.
There are 12,288 base cubes in this category. This gives 122889 = 6.39E36 possible magic cubes.
This is the largest category of base cubes that I have defined. There are seventeen different pattern types within this category. Each pattern type contains from 384 to 3072 (In multiples of 2) different base cubes. There are 19,968 different base cubes in this category. Except for the first two defined categories, each successive category has had twice as many base cubes as the preceding category. That would lead to the expectation of 24,576 base cubes. This category is either an exception to the trend or we missed some. I lean towards the former because the search was thorough.
The defining feature of this grouping is that all members have some asymmetric lines as defined in Other A0 Cubes. In one dimensional orientation every base square of these cubes has an A type line in every row or every column. The A type line will be in rows for some squares and columns for others. There is no consistent pattern for the direction of the A type line other than the direction will be the same in the second four squares as it was in the first.
In the direction perpendicular to the A type lines are either B type lines, C type lines, or asymmetric type lines. A square can only have one type. Squares 1, 3, 5, and 7 will all have the same type and squares 2, 4, 6, and 8 will all have the same type. If the odd squares and the even squares have different types or they both have asymmetric type lines then the cube produced will belong to this category. If both have B type lines or both have C type lines then the cube will be of "the non A0 base cubes containing only ABC type lines" type. If one has a B type line and the other a C type line, then of course there will be no asymmetric lines in this projection, but there will be asymmetric lines in the z direction.
As stated above the direction of the A type line appears to be independent of the alternating squares. The alternating squares are subject to other restrictions. Successive squares that have B type lines always have their inverse located at the square four removed. The intervening squares with B type lines may be the same as one of the other two, it may be shifted, or it may be rotated with or without shifting. Successive squares that have C type lines always are the inverses of those on either side so that squares located four apart are the same. Squares with asymmetric type lines that are spaced four apart have their lines shifted by four and inverted. In successive squares, the lines are either shifted by four or inverted relative to the next square.
There are 19,968 base cubes in this category. This gives 199689 = 5.05E38 possible magic cubes.