Computer Algorithms for Siamese and Uniform Step Squares (Part V)

15^th Order Squares and its Multiples

This section is a continuation of Part IV which covered methods for constructing Siamese and Uniform Step squares via new computer algorithms. This section will be used to predict which 45th order squares are magic and which are not. Using JavaScript (JS) code it will be shown that these large order squares can be constructed from nine 15th order squares. The tables posted will be presented in Soduko fashion with four symbols represented by:

in which one of these symbols is placed into the same cell of a square where the initial number 1 is to be placed. A diabolic square, labeled here as an O with horns (el Diablo), is a magic square in which every row, column, two major diagonals and its negative diagonals (all the diagonals going from the left to the right) sum to the magic number. In addition, we also have the non diabolic magic square, and the two latter non magic squares, one which can be constructed and the other not. These symbols are used to fill a squaretype table after determining the type of square using the new JS computer programs.

Since the squares are of large order, JS computer methods were required in order to reduce the amount of construction time. For example, both Siamese, i.e., Staircase, and Uniform Step methods each require 225 runs to construct the 15th order squares and 2025 runs to verify the 45th order squares. However, no 45th order squares or squaretype table will be depicted. The three 15th order squaretype tables, on the other hand, will be shown and these tables, as said previously, are used to construct the large 45th order just like fractals where the large structure is similar to its component parts.

I have found that a 15th order squaretype table must be constructed from scratch, i.e, it cannot be be contructed from three 5 order squares or five 3 order squares, since five order squaretype tables contain diabolic squares which are not allowed in the composite 15th or 45th order squares. In fact, all primes, except for 3, produce diabolic squares. The sequence below shows that only third order squares may be used to construct 9, 27, 81,... order squares, while the 15th order can be use to construct 45, 135, 405,.. order squares. Thus, for the three different orders the number of 15th order squares is:

which means that it is easier to string together a certain number of 15th order squares instead of constructing the large square from scratch. All the other non bold, non color type must be constructed from scratch since they are equal to 3×p where p is prime, while the color ones are composite multiples of the original color:

The next section section Part VI will show that this sequence has some special properties.

As I mentioned previously, a 45th order squaretype table requires nine 15th order squaretype subtables which can be strung together into a 3x3 configuration as in Table A. For example, if we pick the coordinates in the first subtable to be (3,7), the coordinates of the eight other subtables are multiples of (3,7) just by adding 15 or 30 to either the first or last number. These values according to (row,column) are:

Since each of these positions correspond to the same coordinate in a 15th order, squaretype table, each of these sites are considered equivalent. Indeed, placing an initial 1 into any of these coordinate positions produces, for the Staircase 45, all magic squares. On the other hand, only non magic squares are produced for the Uniform Step 45 squares (as verified by the JS programs). We must remember that these are only nine positions out of a total of 2025. However, as mentioned above, every position on the 45th order squaretype tables has been verified, as predicted.

15th Order Squaretype Tables

In the three squaretype tables below, the first row and last column denote the column and row (the coordinate of each cell), respectively. For example, (1,2) signifies row 1, column 2 of a 15th order square where the initial number 1 is placed. Row 2 and the next to the last column signify the break move employed for each square after a diagonal is filled. From the second row and next to last column (the j/k numbers) it can be seen that the (1,2) cell position uses a (5,3) break move. i.e., 5 column moves to the right and 3 row moves down. The O in the cell shows that this particular square is magic as verified using the JS program Staircase 15 via the method discussed in Part IV. The picture shows that the square is magic but not diabolic (the diagonals don't sum to 1695) and posts the break move in the caption. For the Uniform Step 15, using the coordinates (5,4), this square is also magic, as shown in Uniform 15.

As can be seen an axis of symmetry runs along the right diagonals of squaretype table Staircase 15 and Staircase/Uniform Step 15 but not in the Uniform Step 15 table. This means that the cells above and below the axis of symmetry of the two former squaretype tables are mirror images of each other, making the squares on both sides of the left diagonal equivalent. This, however, is not possible for the Uniform Step squaretype table which has no similar axis of symmetry.

All 45th order squares of the Staircase/Uniform Step have been constructed and every square type symbol found to be identical to those of the 15th order squaretype table shown below. An example of one of these magic squares, (coordinate (10,12), is shown in program: Stair/Uniform 15,

In summary, this method also applies to other composite squares. For example, a 225th order squaretype may be composed of 15²=225 squaretype subsquares, i.e., fifteen 15th order squaretypes per row. In addition, as long as we start with squaretypes that don't include diabolic entries we're good to go.

This completes this section (Part V). To go Part VI which analizes the sequence in this section. To go back to Part IV. To return to homepage.