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:
- Inder J. Taneja, Minimum Perfect Square Sum Bordered and Block-Wise Bordered Magic Squares: Orders 3 to 31. July 20, 2021, pp. 1-82, Zenodo, http://doi.org/10.5281/zenodo.5116408
- Inder J. Taneja, Minimum Perfect Square Sum Bordered and Block-Wise Bordered Magic Squares: Orders 32 to 47. July 20, 2021, pp. 1-64, Zenodo, http://doi.org/10.5281/zenodo.5116410
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
Numbers Magic 