This work shows how to create magic squares with a perfect square number for the total sum entries. This has been done in five ways. Initially, the two ways are using entries as consecutive odd numbers and consecutive natural numbers for odd order magic squares and consecutive fraction numbers for even order magic squares. This process sastify an interesting property known by uniformity property. The second way is also give two kind of magic squares with entries sum as perfect square. It is based on magic squares generated by Pythagorean triples. This procedure give many magic squares based on interval and order of magic squares (for more details click here) . In this work, only the first value is considered to have least possible perfect square sum of entries. In this way also there are two possibilities, one using entries as consecutive odd numbers and second using entries as consecutive natural numbers for odd order magic squares and consecutive fraction numbers for even order magic squares. In all the four possibilities given above not even a single one give us minimum perfect square sum of entries. By using the entries as nonnegative numbers, a third procedure is considered to get minimum perfect square sum of entries. In this case the entries are either consecutive natural numbers for odd order magic squares and consecutive fraction numbers for even order magic squares. For each order, the work is divided in three parts resulting in five magic squares with perfect square sum of entries. Recently, author applied this idea of perfect square sum of entries to write area representations magic squares. This work is for the magic squares of orders 3 to 31. Further orders from 32 to 47 are given in another work.

Below is a link for download the work:

Below are Examples of Magic Squres studied in the work:

Order 3

Order 4

Order 5

Order 6

Order 7

Order 8

Order 9

Order 10

Order 11

Order 12

Order 13

Order 14

Order 15

Order 16

Order 17

Order 18

Order 19

Order 20

Order 21

Order 22

Order 23

Order 24

Order 25

Order 26

Order 27

Order 28

Order 29

Order 30

Order 31

Perfect square sum of entries for higher order magic square shall be given in another work. Below is again the link of above work for download:

  • Inder J. Taneja , Magic Squares With Perfect Square Sum of Entries: Orders 3 to 31, July 19, 2021, pp. 1-181, http://doi.org/10.5281/zenodo.5115214.
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