NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

GENERATION OF MAGIC SQUARES WITH SEVEN SQUARES (Part IG)

Magic Squares of Seven Squares with Imaginary Numbers-Section I

Previously using non imaginary integers, the magic square below containing seven squares was discovered by Andrew Bremner and independently by Lee Sallows:

This section deals with novel magic squares of squares containing at least one imaginary number having the form (ni)², which when squared becomes the square of a negative number, -n². It has been found that there exists an infinite number of tuples (ai,b,c) containing at least one imaginary square which can be used as the right diagonal in magic squares. Methods for construction of these tuples are listed in Part IA, Part IA2, Part IB, Part IC, Part IIC, Part IIIC, Part ID, Part IE and Part IF.

To date eight magic squares of squares containing imaginanary numbers have been found. These are listed below with their sums and deltas (Δ) where:

Magic Squares with Imaginary Numbers-Section II

A second type of Magic squares with Imaginary numbers consists of the type (-ni, 0, n) whuch is constructed as follows:

whereby d² = a²+b² and c² = a²-b². In addition, both squares have Δ = b² and Sum=0.

If we use for example n = 8, the following two magic squares containing ((8i)², 0, 8²) as the right diagonal can be generated. Furthermore, increasing n produces larger amounts of magic squares of seven squares as shown for the first four squares for n = 60. In addition, Squares D, E and F are divisible by 5², 4² and 15², respectively.

We can determine if it is possible to construct magic squares having more than seven squares of the type shown here by analyzing the equations in Square P₂:

However, this is only possible when a = any number and b = 0, or when b = any number and a = 0 or when a = b = 0. Thus, we have either a square where all cells are 0 or one where either the right diagonal contains all zeros or the left diagonal contains all zeros and the rest, each row, column and the other diagonal consists of non zero integers as shown below in Squares G and H:

Alternatively, we can use the Pythagorean theorem to generate the table below using Square P_{1 and 2} as template:

We can generate the following series of squares based on the Pythagorean theorem: a² + b² = d² and force all the numbers in row 1 to be squares. Listed below are three examples. Example I uses the lowest set of squares 3² + 4² = 5², example II uses 6² + 8² = 10² and example III uses 9² + 12² = 15²:

Examining these squares, c² is actually equal to 7n² where n is a multiple of a square and, therefore, c = √7n. Thus, c can never be an integer and only seven squares are possible with this types of magic square.

The Square Variables for Each Cell of a 3x3 Square with a Negative a²

A 3x3 magic square of squares, having a -a² in the left lower corner cell, is composed of nine cells having the structure as shown in Square I when four variables a², b², c² and d² are used to specify the square. Since the sum of the four corner cells equals the sum of the four outside central cells, e.g.,:

then substituting c² - 2a² for d² affords Square II in terms of only a², b² and c²:

Furthermore Δ, the variable added to those cells having a b², c² and d² as shown in the picture at the beginning of the page, is equal to a² + b² and is part of cells 1, 3, 4, 5, and 6 moving in a horizontal direction (→). Thus, all we need to know is a, b and c to generate a magic square of squares having an imaginary a in its major diagonal. In addition, the Magic Sum for each line or diagonal is 3b². And finally we can conclude that we can generate magic squares having at least 7 squares, but finding one with 8 or 9 squares is a major effort.

This concludes Part IG.

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