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:

- 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.

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:

## Leave a Reply