The figures discussed in the other sections of this site can all be constructed using base lines as defined in Basic Construction. From the base lines, base figures are constructed and from the base figures, magic figures are constructed. Most magic figures cannot be made using master base lines and base lines as discussed on the other pages. There are nasik magic cubes that cannot be made using the base line concept. The first order-8 pan-2,3-agonal magic cube made by Barnard1 was of this category as were others that followed.2,3 As the base line method was being fleshed out it became clear that these other cubes could be built using base cubes but not base lines. It would be somewhat more difficult because the basic unit of construction, the base cube, contains 512 values rather than the three groups of 8 of the base lines and there are many more base cubes than base lines to deal with, but it would still be possible.
The major differences between the cubes generated by base lines and all the other possible order-8 nasik magic cubes is the presence or lack of master base lines. The 24 numbers of the cubes master base lines can be used to generate the remainder of the cube, but they are only present in cubes that can be made using base lines and most magic figures including other nasik order-8 magic cubes cannot be made from master base lines.
All but one of the other literature cubes were made by an algorithm like that on Nakamura's site and not by using base cubes. The magic cubes can be broken into base cubes by the process described for the pan-magic square on the History page and more fully by Grogono. These base cubes all have certain features in common and comprise a second category of base cubes. There are other combinations of 256 zero's and 256 one's that add to four in all the required ways and thus are also base cubes. All possible base cubes can be classified into nine categories.
Three of the base cube categories are discussed in Barnard's Cube because of their similarities. One category includes all of the previously known literature cubes that are similar to Barnard's. The second and smallest of the base cube groups is the base line cube category. The third category as far as I can determine has not appeared in any known magic cube even though it has many similarities to the known literature cubes. All of the agonals for these three categories can be described as A, B, or C type lines as defined in Magic Cube. These three categories also have in common that in one dimensional direction all of the agonals can be described as A0 or A1 type lines. For two of the categories the base cubes themselves are not made from base lines although all of their agonals are A, B, or C type lines. The third is the base line cube category.
Note that in this and the following section I use the term A, B, or C type lines rather than A, B, or C base lines. This is to designate that the individual row, column and pillar base agonals have their bits arranged in the same pattern as the A, B, or C lines but they are not generated from base lines. Their pattern is independent of the lines in the cube that should be the base lines.
In the section, Other A0 Cubes are three additional categories of base cubes. Unlike those discussed above these base cubes have some 8-bit lines with patterns that are different than the A, B, and C type lines discussed in Magic Cube. These three categories also have in common that in one dimensional direction all of the lines can be described as A0 or A1 type lines.
By far the greatest number of base cubes is discussed is the Non A0 Cubes section. The three categories of base cubes in this section are more difficult to describe than those in the other sections because they must be described as a three dimensional structure.
The number of base cubes in each defined category varies from 96 to 19968 as shown in the table below. The focus of most of this site, the base line cubes, constitute the smallest category. Not all of the base cubes are compatible, i.e. the combination of 2x one base cube plus a second base cube does not give an intermediate cube with even integral distribution and thus together will be unable to be part of the same magic cube. There is a more extensive discussion of base cube compatibility in the Magic Cube main page and in Base Cube Rules.
If two base line base cubes picked at random are combined, 95.79% of the time the resulting pair will be compatible. If two random base squares are picked from any of the other categories, they will be less likely to form a compatible pair as shown in the table below. The non A0 base cubes are significantly less likely to form compatible pairs.
Nine base cubes must be combined to form a valid magic cube. The table also shows the percentage of all possible combinations of three base cubes of each category that are compatible. As discussed in Base Cube Rules incompatibilities also show up in larger groups of base squares that are not due to the incompatibility of just pairs. For the base line cubes it was shown that these additional incompatibilities show up in groups of four, six, and eight but not in groups of three, five, and seven. By testing randomly generated combinations of nine base cubes from the various categories it is possible to get an estimate of how many magic cubes of each type exist. This is discussed in greater detail in the subsections of this section.
|category||# base cubes||possible pairs||compatible pairs||% of pairs||% of triplets||determined for 9|
|Alternating Lines||384||73728||65088||88.28||68.06||3.87 ppm|
|Asymmetric in 2 Dir.||768||294912||259200||87.89||67.06||85 ppb|
|Asymmetric in 1 Dir.||1536||1179648||985344||83.53||57.29||8.3 ppb|
|Asymmetric Alternate||3072||4718592||3838464||81.35||52.50||1.5 ppb|
|Non A0 ABC||6144||18874368||13919232||73.75||40.04||?|
|Non A0 Mirror||12288||75497472||34842624||46.15||?||?|
|Non A0 Asymmetric||19968||199360512||100988160||50.66||?||?|
Only the compatibilities of base squares in each individual category are considered in the table. Base squares are also compatible across groups. But I reached my limit before going there.