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Magic Squares With Perfect Square Sum of Entries: Orders 3 to 47

Authors: Taneja, Inder J.;

Magic Squares With Perfect Square Sum of Entries: Orders 3 to 47

Abstract

This paper shows how to create magic squares with perfect square sum of entries. There are three types of entries, consecutive odd numbers, natural numbers and fraction numbers. The consecutive odd numbers entries are for all order magic squares. The consecutive natural numbers entries are for odd order magic squares or the consecutive fraction numbers entries are for even order magic squares resulting in equal sum magic squares. This process is applied in three types of situations. One is uniformity, i.e., k, k^2, k^3, k^4, where k is the order of a magic square. This property means, we have magic square of order k, with k^2 entries. The magic sum is k^3 and the total sum of entries is k^4. The second process is generating magic squares from Pythagorean triples with least possible sum of entries. The third process is having directly magic squares with minimum perfect square sum of entries. For each order, there are five types of magic squares with perfect square sum of entries having total three different types of magic sums. The sum is minimum possible only in the third case. The work is for the magic square orders 3 to 47. It include the revised version of of author's previous works for orders 3 to 31. The idea of this work is applied to area-representations of magic squares.

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Keywords

Magic squares, Uniformity property, Pythagorean triples

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