FEATURE BUILDER

Scattered through this site are discussions of how to build magic squares of various dimensions and orders. For the order-2p figures, these focused on the alphanumeric base line types. There is some discussion of the relationship between features and base lines in Order-2p Squares and also a description of how to construct them. Below is an interactive demonstration of the construction of order-8 pan-magic squares with selected features.

FEATURE/BASE LINE RELATIONSHIP

Order-8 Base Lines
binarysymbol
01010101A0
00110011B0
01100110B1
00001111C0
00011110C1
00101101C2
00111100C3
01001011C4
01011010C5
01101001C6
01111000C7

Only the 11 base lines in the Order-8 Base Lines table at right will be used for the construction. In the discussion, I will use the shorthand symbolic representation, but the binary base line will be entered into the figures.

Features are expressed in order-8 base squares made with the following base lines.

  1. Base squares with an A0 base line are {2compact2}. Base squares with a B type base line are {2compact3}. Base squares with a C type base line are {2compact5}.
  2. Base squares with a C type base line are complete.
  3. The combinations of A0 with B1, C3, or C5 yield an associated base square. The combinations of B0 with C3 or C5 do also. And so do the combinations of B1 with C0, or C6.
  4. The combinations of A0 with B0, B1, C0, C3, or C6 yield a base square with Franklin V bent diagonals. The combinations of B0 with C0, C5, or C6 do also.
  5. All combinations of different alpha types of base lines, except the B1 base line, yield base squares with Franklin W bent diagonals. No base line combination that includes the B1 base line has Franklin W bent diagonals.
  6. All combinations of different B type base lines with C type base lines yield base squares with {zigzag2} lines. No base line combination that includes the A0 base line has {zigzag2} lines.
  7. The combinations of A0 with B0, B1, C3, or C6 yield a base square with inlaid order-4 squares in its quadrants. The combinations of B0 or B1 with C5 do also.

Six sets of base lines are needed to construct the six base squares required to build an order-8 magic square. A pair of base lines can be used twice because the A and B pair can be used as the row and column base lines and also as the column and row base lines. Therefore, even if there are only three pairs of base squares that have all the desired features, a pan-magic square with those features can often be built. An example construction is shown below. You can follow the process using the features shown in the example or be adventurous and pick your own set to build a magic square. No guarantees.

MAGIC ORDER-8 SQUARE BUILDER

The above process can be repeated with a new set of features based on the feature/base line relationship. The process can also be applied to larger squares, cubes or hypercubes.