Magic Numbers

Minimum Perfect Square Sum Bordered and Block-Wise Bordered Magic Squares: Orders 3 to 47

There are many ways of writing magic or bordered magic squares, where the sum of entries is always a perfect square. In one of the possibility, the magic sums are such that they satisfy uniformity property. Another way is to write magic squares generated by Pythagorean triples. Based on these idea, we can always write magic squares or bordered magic squares where the sum of entries is always a perfect square. Still, these sums are not minimum. There is a third way to write magic squares or bordered squares such that we get minimum perfect square sum of entries. This what we have done in this work with bordered and block-wise bordered magic squares. In case of even order magic squares, we can always write \textbf{block-wise bordered} magic squares with equal sum sub-blocks. In case of odd order magic squares, we can write them as block-wise bordered magic squares, but with different sub-blocks sums. This work is for the orders 3 to 47. Recently, author applied this idea of perfect square sum of entries to write area representations of magic squares. The whole work is written in two parts. See the links below for download:

Below are examples of bordred and block-wise bordred magic squares of orders 3 to 47 with minimum perfect square sum of the entries.

Magic Squares of Orders 3 and 4

Bordered Magic Squares of Orders 5 to 9

Bordered and Block-Wised Bordered Magic Squares of Order 10

Bordered Magic Square of Order 11

Bordered and Block-Wised Bordered Magic Squares of Order 12

Bordered Magic Square of Order 13

Bordered and Block-Wised Bordered Magic Squares of Order 14

Bordered Magic Square of Order 15

Bordered and Block-Wised Bordered Magic Squares of Order 16

Bordered Magic Square of Order 17

Bordered and Block-Wised Bordered Magic Squares of Order 18

Bordered Magic Square of Order 19

Bordered and Block-Wised Bordered Magic Squares of Order 20

Bordered Magic Square of Order 21

Bordered and Block-Wised Bordered Magic Squares of Order 22

Bordered Magic Square of Order 23

Bordered and Block-Wised Bordered Magic Squares of Order 24

Bordered Magic Square of Order 25

Bordered and Block-Wised Bordered Magic Squares of Order 26

Bordered Magic Square of Order 27

Bordered and Block-Wised Bordered Magic Squares of Order 28

Bordered Magic Square of Order 29

Bordered and Block-Wised Bordered Magic Squares of Order 30

Bordered Magic Square of Order 31

Bordered and Block-Wised Bordered Magic Squares of Order 32

Bordered Magic Square of Order 33

Bordered and Block-Wised Bordered Magic Squares of Order 34

Bordered Magic Square of Order 35

Bordered and Block-Wised Bordered Magic Squares of Order 36

Bordered Magic Square of Order 37

Bordered and Block-Wised Bordered Magic Squares of Order 38

Bordered Magic Square of Order 39

Bordered and Block-Wised Bordered Magic Squares of Order 40

Bordered Magic Square of Order 41

Bordered and Block-Wised Bordered Magic Squares of Order 42

Bordered Magic Square of Order 43

Bordered and Block-Wised Bordered Magic Squares of Order 44

Bordered Magic Square of Order 45

Bordered and Block-Wised Bordered Magic Squares of Order 46

Bordered Magic Square of Order 47

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