Different Digits Magic Squares and Magic Sums Number Patterns

Magic squares are generally constructed using sequential or consecutive numbers such as 1, 2, …, n^2. Here in this work, we shall write magic squares using nonconsecutive different digits numbers. This work, we have done for the magic squares of orders 5, 7, 8, 9 and 10. As commonly, the entries are not in a sequential way, but each cell is with different digits. Also, all the entries in a magic square are with different numbers. The aim is to write in such a way that in each case as the number of digits in cells increases, we get number patterns with magic sums, i.e., these sums are in symmetrical way. In case of orders 8 and 9, the magic squares considered are pan diagonal. In case of order 9, there are examples of bimagic squares, while in case of order 8 there are examples where, we have bimagic only in rows sums. In both these cases, there are interesting number patterns with bimagic sums. See below the table of number patterns for the different orders and digits that we worked in this paper:

From the table, we observe that still, we don’t have magic squares with 3 digits in each cell for the magic squares of orders 7 and 10. Magic squares of orders 4 and 6 are not dealt here, as in case of order 4, we have only two possibilities, i.e., for the digits 3 and 4, and for the magic squares of order 6, we have results for the digits 5 and 6, but for the case of 4 digits, it becomes semi-magic square. For details see author’s previous work. The bimagic sums patterns are for the orders 8 and 9. In case of order 8, it is only in rows.

Below are links of author’s works for download:
1. Multi-Digits Magic Squares (old work)
2. Different Digits Magic Squares and Number Patterns (new work)

Below are examples written separately in each case:

• Pattern with Magic Sums of Different Digits Magic Squares of Order 5: The numbers used are from 1 to 5:

• Pattern With Magic Sums of Different Digits Magic Squares of Order 7: The numbers used are from 1 to 7:

• Patterns With Magic Sums of Different Digits Magic Squares of Order 8: The numbers used are from 1 to 8. In this case the magic squares are bimagic only in rows:

• Patterns With Magic Sums of Different Digits Magic Squares of Order 8: The numbers used are from 1 to 8. In this case the magic squares are pan diagonal:

• Patterns With Magic Sums of Different Digits Magic Squares of Order 9: The numbers used are from 1 to 9. In this case the magic squares are pan diagonal:

• Patterns With Magic Sums of Different Digits Magic Squares of Order 9: The numbers used are from 1 to 9. In this case the magic squares are bimagic:

• Patterns With Magic Sums of Different Digits Magic Squares of Order 10: The numbers used are from 0 to 9.

Below are again links of work to download:
1. Multi-Digits Magic Squares (2015) – previous work.
2. Different Digits Magic Squares and Number Patterns (2019) – revised work.