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A magic square M over an integral domain D is a 3 x 3 matrix with entries from D such that the elements from each row, column, and diagonal add to the same sum. If all the entries in M are perfect squares in D, we call M a magic square of squares over D. Martin LaBar raised an open question in 1984, which states, “Is there a magic square of squares over the ring Z of the integers which has all the nine entries distinct?” We approach to answering a similar question in case D is a finite field. Our main result confirms that a magic square of squares over a finite field F of characteristic greater than 3 can only hold 3, 5, 7, or 9 distinct entries. Corresponding to LaBar’s question, we claim that there are infinitely many prime numbers p such that, over a finite field of characteristic p, magic squares of squares with nine distinct elements exist. Constructively, we build magic squares of squares using consecutive quadratic residue triples derived from twin primes. We classify all the magic squares of squares over any finite fields of characteristic 2. Description of magic squares over a finite field of characteristic 3 is provided.
The science of magic squares witnessed an important development in the Islamic world during the Middle Ages, with a great variety of construction methods being created and ameliorated. The initial step was the translation, in the ninth century, of an anonymous Greek text containing the description of certain highly developed arrangements, no doubt the culmination of ancient research on magic squares.
This innovative work replaces magic square numbers with two-dimensional forms. The result is a revelation that traditional magic squares are now better seen as the one-dimensional instance of this self-same geometrical activity.
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 edition. Excerpt: ...the result, but it is not probable that he derived his square according to the scheme employed here. Our 16X16 square is not exactly the same as the square of Franklin, but it belongs to the same class. Our method gives the key to the construction, and it is understood that the system here represented will allow us to construct many more squares by simply pushing the square beyond its limits into the opposite row which by this move has to be transferred. There is the same relation between Franklin's 16X16 square and our square constructed by alternation with quaternate transposition, that exists between the corresponding 8X8 squares. REFLECTIONS ON MAGIC SQUARES. MATHEMATICS, especially in the field where it touches philosophy, has always been my foible, and so Mr. W. S. Andrews's article on "Magic Squares" tempted me to seek a graphic key to the interrelation among their figures which should reveal at a glance the mystery of their construction. THE ORDER OF FIGURES. In odd magic squares, 3X3, 5X5, 7X7, etc., there is no difficulty whatever, as Mr. Andrews's diagrams show at a glance (Fig. 213). The consecutive figures run up slantingly in the form of a staircase, so as to let the next higher figure pass over into the next higher or lower cell of the next row, and those figures that according to this method would fall outside of the square, revert into it as if the magic square were for the time (at the moment of crossing its boundary) connected with its opposite side into the shape of a cylinder. This cannot be clone at once with both its two opposite vertical and its two opposite horizontal sides, but the process is easily represented in the plane by having the magic square extended on all its sides, and on passing its limits...
Comment concerning a recent paper (1) on the confounded factorial approach to the construction of incomplete block designs indicates the desirability of considering other, less complex, construction methods. This paper is limited to the finite field approach to the generation of orthogonalized squares. Three general cases are discussed: (a) where the number of elements, m, in a row or column of a square is a prime; (b) where m is the power of a prime; and (c) where m is a product of several primes or powers of primes. These three cases, of course, cover all positive integers. The construction methods are limited, however in that while the minimum number of orthogonalized squares is determined by the least prime power law (i.e., s = min(p superscript i - 1), where s is the number of squares and p superscript i is the smallest number in the product m = (p superscript i)(p superscript j)(p superscript k), the maximum number of squares is not directly determinable. For instance, in cases where the least prime power is equal to 2 (i.e., p superscript i = 2, and, in general, numbers of the form 4k + 2), it was long thought that not even orthogonal pairs of squares existed. However, although at least pairs of squares have been found for such cases using other methods, these methods are not covered here. (Author).