# ORDER-2p CUBES

The smallest nasik magic cube that can be made is the order-8 cube. For the order-8 nasik cubes only the A, B, and C type base lines can be used to make the base cubes and the resulting pan-magic cubes all have the same features. For the larger order-2p nasik cubes, the D, E, etc. type base lines become available as long as their base line units are equal to or shorter than the cube's order. Three base lines are needed for each base cube. Therefore, some of the available base line letter types are left out the larger base cubes and possibly left out of all of the base cubes in the final nasik cube.

With the cubes, features can be looked at in three dimensions or in just the two dimensions of the individual squares. The pan-2-agonal feature is concerned with the diagonals of the individual squares while the pan-3-agonal feature looks at diagonals through the three dimensions of the cube. Although this site focuses on nasik figures, it is possible to make pan-3-agonal cubes using master base lines. These figures will also be discussed as they do have pan features.

The three base lines needed to construct base cubes introduce more variety in possible feature combinations. It also increases the complexity of their interactions. Much of this is lost because the most interesting figures are also large and thus difficult to display.

## ORDER-8 MAGIC CUBES

The order-8 nasik magic line cubes are covered extensively in Magic Cube. Additional research is discussed in Base Cube Rules and Magic Constant Groups. The Cube Generator Guide provides guidance for those who would like to make and manipulate a 3-D cube. The program to do this is available on the Downloads page.

The 8 cube worksheet in the CubeLines Excel Spreadsheet available on the Downloads page contains the seven order-8 cubes shown below. That download also displays many sums for features. For all of these cubes, any two planes that are separated by 4 can be exchanged making a new cube with the same properties as the original.

All the order-8 cubes below except cube 4 are complete and {3compact2,3,5}. They all, except cube 4, also have all the addition properties described in Magic Constant Groups.

To view the seven magic cubes generated below, enter a master base line value from 1-6 to get the corresponding cube. The order of the base cubes can be shuffled to make 9! different cubes with the same features. Enter the integers 1-9 in the base cube order boxes in any order. There can be no duplicates in the base cube order as that will lead to duplicate numbers in the cube. The downloads permit even more manipulations.

All the order-8 cubes except cube 4 are {3compact2,3,5}. They also have all the addition properties described in Magic Constant Groups.

The first two are the literature cubes that can be made using master base lines. Cube 1 is the cube that appeared in the Old Farmers Almanac6. All of the two-dimensional 8x8 squares contain W shaped Franklin bent diagonals.

Cube 2 is Gakuho Abe's4 literature cube. Abe's cube has Franklin V and W bent diagonals in the two-dimensional 8x8 squares in the xy planes and is {2compact2} in the xz planes. Half rows add to S/2 in all xy and yz squares.

Cube 3 is a structured cube built using the approach described in n-Dimensions. The structure is in the base lines but the cube also shows a more structured pattern than most nasik cubes. All of the two-dimensional 8x8 squares contain Franklin W shaped bent diagonals. The xy squares are also {2compact2} and the yz squares are {2compact3}.

Cube 4 illustrates a new feature that can be expressed in figures of three dimensions or more. This feature is expressed by breaking one of the rules for making magic figures with binary base lines. By using three C type base lines in the same base cube, a base cube is made that has all triagonals intact but many of the diagonals do not add to S. This base cube will cause the resulting magic cube to become a pan-3-agonal magic cube. The cube will also no longer be {3compact2} or {3compact3}. Cube 4 was made by modifying just the last base cube added to the Old Farmers Almanac cube. The master base lines (which can be seen in the CubeLines Excel Spreadsheet available on the Downloads page) and the numbers in the cube are nearly the same with some numbers only being ±1. This is because only the last base cube was changed. If the first base cube had been changed then the differences would be ±256.

Although the cubes made by Barnard1 and others2,3 cannot be made using master base lines, they do have rows, columns, and pillars that contain the zero (after subtracting 1 from all numbers to make the analytic cube). Cube 5 was made from these lines extracted from Barnard's1 cube. The resulting cube is a nasik magic cube but it is not the same as Barnard's.

For a long time I felt that there was little to distinguish among the many possible nasik order-8 cubes other than the ability to make some using master base lines. Then I realized that it should be possible to have Franklin diagonals in some of the squares. Analysis of possible base line combinations revealed that it would not be possible to have Franklin V diagonals in all the squares but Franklin W diagonals should be possible in them all or just some of them. Half rows can also be present in all or some of the squares. Compact features could also be present in some of the squares. Cube 6 contains Franklin V bent diagonals and is {2compact3} in all the xy squares, i.e. the eight squares numbered I to VIII. It is also {2compact2} in all xz squares. This was planned for cube 6, however, as can be seen above the square features were present in some earlier cubes that Abe and I had made. Barnard's1, Plank's3, Hendricks'3, and Nakamura's cubes have no Franklin bent diagonals and no compact features in any of their squares.

All of the half rows, half columns, and half pillars in cube 7 add to S/2. This cube also has Franklin W diagonals in all squares. All yz squares are {2compact2} and all xz squares are {2compact3}.

## ORDER-16 MAGIC CUBES

The order-16 magic cubes have much more variety than the order-8 cubes since A, B, C, and D base lines can all be used in their construction and only three of the four are needed. In all of the cubes in this section planes spaced eight apart may be exchanged to give new magic cubes. Inlaid cubes and association will be lost in such an exchange but other features will be retained.

The 16 cube worksheet in the CubeLines Excel Spreadsheet available on the Downloads page contains the seven order-16 cubes shown below. That download also displays many sums for features. For all of these cubes, any two planes that are separated by 8 can be exchanged making a new cube with the same properties as the original.

Cube 1 is complete and {3compact2,3,9}. The two-dimensional 16x16 squares contain all V, W, and WW shaped Franklin bent diagonals.

Cube 2 is complete and {3compact2,5,9}. The two-dimensional 16x16 squares contain all V, W, and WW shaped Franklin bent diagonals.

Cube 3 is complete and {3compact3,5,9}. The two-dimensional 16x16 squares contain all V, W, and WW shaped Franklin bent diagonals and all {zigzag2} lines. The xy squares are {2compact3} and the yz squares are {2compact5}.

If the cube is {3compact2} then the corners of every rectangular prism with even numbered sides also adds to S/2. This is shown by the proof shown elsewhere for cubes. Likewise, a cube that is {3compact3} will have the corners of all rectangular prisms with sides of (3+4r) equal to S/2 where r is independent for each dimension. And a cube that is {3compact5} will have the corners of all rectangular prisms with sides of (5+8r) equal to S/2 where r is independent for each dimension.

Cube 4 is associated and {3compact2,3}. As with the squares the association feature is lost when planes are exchanged. It would be possible to also make cubes incorporating with the association feature either {3compact2,5} or {3compact3,5}, but this was not illustrated here.

Cube 5 illustrates that it is possible to make nasik cubes with nasik inlaid cubes. There is an order-8 pan-2,3-agonal magic cube in each octant of this order-16 pan-2,3-agonal magic cube. The larger cube is {3compact2,3}. The smaller cubes of course have all the features of all order-8 pan-2,3-agonal magic cubes. The 16x16 squares all have Franklin V diagonals.

Cube 6 is similar to Cube 5 except that larger cube is an order-16 pan-3-agonal magic cube. It is thus not {3compact2 or 3}. The larger cube is pan-3-agonal because all three base lines in one of the base cubes are D type. The order-8 cubes are all pan-2,3-agonal and {3compact2,3,5}.

Cube 7 is {3compact5}. The squares in the xy plane have Franklin V, W, and WW bent diagonals. These squares also all have order-8 pan-magic squares in their quadrants. The yz squares have {zigzag2} lines.

## ORDER-32 MAGIC CUBES

Even more variety can be expressed in the order-32 cubes but only five examples are shown in the 32 cube worksheet in the CubeLines Excel Spreadsheet available on the Downloads page. Only the master base lines and a converter to the base lines are given in the download as the cube and checkers are large. The Excel converter to a cube and associated checker are available by request to the author. The order-32 cubes are also not shown on this page due to their size. A description of the five cubes is given below.

Cube 1 is {3compact2} and contains an order-16 pan-2,3-agonal magic cube in each of its octants. In each of the octants of each of the order-16 inlaid cubes lies an order-8 pan-2,3-agonal magic cube. Even more inlaid cubes are possible as shown in Cube 2 which is {3compact2,3}. This cube has 64 inlaid order-16 and 64 inlaid order-8 pan-2,3-agonal magic cubes. These have their upper, front, left corner at the 8n+1 positions within the big cube where n takes on the values 0-3 independently for the three dimensions. There are probably 64 order-24 cubes as well but this was not confirmed.

Cubes 3 and 4 are again examples where there is apparently little difference in the master base lines but the resulting cubes properties are quite different. Some of the numbers in the master base lines of Cube 3 are just ±1 of the comparable numbers in Cube 4. Both cubes are {3compact2,3} but Cube 4 is complete and Cube 3 is not. Each cube has an order-16 pan-2,3-agonal magic cube in each of its octants but those in Cube 3 are associated while those in Cube 4 are not.

Cube 5 is {3compact2} and associated. It contains eight associated order-16 pan-2,3-agonal magic cubes. These associated cubes are located at the 8n+1 positions where n is independently 1 or 3 for the three dimensions, i.e. one of the associated order-16 cubes is centered and the others are aligned with its edges. In addition, it contains 56 inlaid order-16 pan-3-agonal magic cubes and 64 inlaid order-8 pan-3-agonal magic cubes.