This work brings more concepts in magic squares. In past we studied a lot of upside-down and mirror looking magic squares. These are based some kind of digital/special fonts. To understand better let’s consider the following 10 digits from 0 to 9:

Let’s make 180 degrees rotation over the above 10 digits, we get

We observe that the numbers 0, 1, 2, 5, 6, 8 and 9 are still there. The difference is that 6 becomes 9.

Let’s see how these numbers can seen in mirror:

We observe that the numbers 0, 1, 2, 5 and 8 are still there. In this case the numbers 2 and 5 interchanges, i.e., 2 becomes 5 and 5 as 2.

In general there are two kinds of flips, i.e., horizontal flip and vertical filp. See the image below:

Source: https://www.mathsisfun.com/definitions/vertical-flip.html

Making vertical flip over the digits 0 to 9, we get

We observe the numbers 0, 1, 2, 3, 5 and 8 remains the same. The numebrs 0, 1, 2, 5 and 8 are already in mirror looking except the number. That’s why we call these numbers as universal numbers. Here also 2 becomes 5 and 5 as 2. Thus using vertical flip, we get an extra number as 3. Let’s call the operation as water reflection. This means that the numbers 0, 1, 2, 3, 5 and 8 are water reflexive.

The aim this work is write magic squares based on the numbers 0, 1, 2, 3, 5 and 8, where 3 is always there, i.e., making combinations of 3 with the numbers 0, 1, 2, 5 and 8. This work is for orders 14 to 16. Full work can be downloaded from the following link:

Below is the index of the work given below for the order 24.

  • Magic Squares of Order 24: Blocks of Semi-Magic Squares of Order 3
    • a) 5-Digits Cell Entries:
      • Example 1. The Digits (1,2,3,5,8)
      • Example 2. The Digits (0,2,3,5,8)
      • Example 3. The Digits (0,1,2,3,5)
    • b) 6-Digits Cell Entries:
      • Example 4. The Digits (2,3,5)
      • Example 5. The Digits (1,3,8)
      • Example 6. The Digits (0,3,8)
      • Example 7. The Digits (0,1,3)
    • c) 10-Digits Cell Entries:
      • Example 8. The Digits (3,8)
      • Example 9. The Digits (1,3)
      • Example 10. The Digits (0,3)
  • Magic Squares of Order 24: Blocks of Magic Squares of Order 4
    • a) 4-Digits Cell Entries:
      • Example 11. The Digits (1,2,3,5,8)
      • Example 12. The Digits (0,2,3,5,8)
      • Example 13. The Digits (0,1,2,3,5)
    • b) 6-Digits Cell Entries:
      • Example 14. The Digits (2,3,5)
      • Example 15. The Digits (1,3,8)
      • Example 16. The Digits (0,3,8)
      • Example 17. The Digits (0,1,3)
    • c) 10-Digits Cell Entries:
      • Example 18. The Digits (3,8)
      • Example 19. The Digits (1,3)
      • Example 20. The Digits (0,3)
  • Magic Squares of Order 24: Blocks of Order 8
    • a) 4-Digits Cell Entries:
      • Example 21. The Digits (1,2,3,5,8)
      • Example 22. The Digits (0,2,3,5,8)
      • Example 23. The Digits (0,1,2,3,5)
    • b) 6-Digits Cell Entries:
      • Example 24. The Digits (2,3,5)
      • Example 25. The Digits (1,3,8)
      • Example 26. The Digits (0,3,8)
      • Example 27. The Digits (0,1,3)
    • c) 10-Digits Cell Entries:
      • Semi-Bimagic Squares
      • Example 28. The Digits (3,8)
      • Example 29. The Digits (1,3)
      • Example 30. The Digits (0,3)
  • References

For the previous work on orders 3 to 13 refer the followig links:

Magic Squares of Order 24:
Blocks of Semi-Magic Squares of Order 3

Thus, we have three examples of semi-magic squares of order 24 having 4-digits cells entries with five numbers combinations (1, 2, 3, 5, 8), (0, 2, 3, 5, 8) and (0, 1, 2, 3, 5). The internal blocks of order 3 are semi-magic squares with different semi-magic sums. The blocks of order 12 are semi-magic squares with equal semi-magic sums. These three examples are not water reflexive as these doesn’t give semi-magic squares after applying vertical flip.

Thus, we have four examples of water reflection semi-magic squares of order 24 having 6-digits cells entries with 3 numbers combinations (2, 3, 5), (1, 3, 8), (0, 3, 8) and (0,1, 3). The internal blocks of order 3 are semi-magic square with different semi-magic sums. The blocks of order 12 are semi-magic squares with equal semi-magic sums.

Above we have three examples of water reflection semi-magic squares of order 24 having 10-digits cells entries with 2 numbers combinations (3, 8), (1, 3) and (0, 3). In each case, the internal blocks of order 3 are semi-magic squares with different semi-magic sums. The blocks of order 12 are semi-magic squares with equal semi-magic sums.

Magic Squares of Order 24
Blocks of Magic Squares of Order 4

Thus, we have three examples of semi-magic squares of order 24 having 4-digits cells entries with five numbers combinations (1, 2, 3, 5, 8), (0, 2, 3, 5, 8) and (0, 1, 2, 3, 5). The internal blocks of order 4 are magic squares with different magic sums. The blocks of order 12 are semi-magic squares with equal semi-magic sums. These three examples are not water reflexive as these doesn’t give semi-magic squares after applying vertical flip.

Thus, we have four examples of water reflection semi-magic squares of order 24 having 6-digits cells entries with 3 numbers combinations (2, 3, 5), (1, 3, 8), (0, 3, 8) and (0,1, 3). The internal blocks of order 4 are magic square with different magic sums. The blocks of order 12 are semi-magic squares with equal semi-magic sums.

Above we have three examples of water reflection pandiagonal magic squares of order 24 having 10-digits cells entries with 2 numbers combinations (3, 8), (1, 3) and (0, 3). In each case, the internal blocks of order 4 are also pandiagonal magic squares with equal magic sums.

Magic Squares of Order 24:
Blocks of Order 8

Thus, we have three examples of water reflection semi-magic squares of order 24 having 4-digits cells entries with five numbers combinations (1, 2, 3, 5, 8), (0, 2, 3, 5, 8) and (0, 1, 2, 3, 5). The internal blocks of order 8 are magic squares with different magic sums.

Thus, we have four examples of water reflection semi-magic squares of order 24 having 6-digits cells entries with 3 numbers combinations (2, 3, 5), (1, 3, 8), (0, 3, 8) and (0,1, 3). The internal blocks of order 8 are magic square with different magic sums.

Above we have three examples of water reflection semi-bimagic squares of order 24 having 10-digits cells entries with 2 numbers combinations (3, 8), (1, 3) and (0, 3). In each case, the internal blocks of order 8 are also pandiagonal bimagic squares with equal magic square sums.

References

Below are references of complete project in 8 parts for the orders 3 to 25. Mainly, it includes three things: Upside-down, Mirror Looking and Water Reflection properties of magic squares.

  1. Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 3 to 6, Zenodo, January 07, 2025, pp. 1-93, https://doi.org/10.5281/zenodo.14607070.
  2. Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 7 to 10, Zenodo, January 07, 2025, pp. 1-171, https://doi.org/10.5281/zenodo.14607071.
  3. Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 11 to 13, Zenodo, January 15, 2025, pp. 1-146, https://doi.org/10.5281/zenodo.14649320.
  4. Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 14 to 16, Zenodo, January 15, 2024, pp. 1-140, https://doi.org/10.5281/zenodo.14649519.
  5. Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 17 to 20, Zenodo, January 17, 2025, pp. 1-86, https://doi.org/10.5281/zenodo.14676293.
  6. Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 21 to 23, Zenodo, January 20, 2025, pp. 1-83, https://doi.org/10.5281/zenodo.14688709.
  7. Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Order 24, Zenodo, Janeiro 20, 2025, pp. 1-150, https://doi.org/10.5281/zenodo.14700325
  8. Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Order 25, Zenodo, January 21, 2025, pp. 1-79, https://doi.org/10.5281/zenodo.14715162.
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