Magic Squares with Perfect Square Number Sums

This short paper shows how to create magic squares in such a way that total sum of their numbers becomes a perfect square. This has been done in two ways: Firstly, Take the sum of odd numbers, and secondly, take the numbers in a sequential way. In the first case, for all orders of magic squares, one can always have a perfect square sum. In the second case, magic squares with perfect square magic sums exist, but only for odd order magic squares. For the even order magic squares, such as 4, 6, 8, etc. it is not possible to write sequential number magic squares with perfect square sums. For the odd order magic squares, at least three examples of sequential numbers are given in each case. This is done from the 3rd to the 25th orders of magic squares. For the prime number orders, examples are considered up to order 11. Further results for order 13, 17, etc. are presented along similar lines. Finally, we reach a conclusion that we can always create a magic square, so that, if the order of magic square is k, then the number of elements are k2, the magic sum is k^3, and the sum of all numbers on the square is k^4, for all k=3,4,5,…

trump-1Download: Magic Squares with Perfect Square Number Sums

Excel file

 

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