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:

Upside-down (180 degrees rotation)
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.
Mirror Looking (Horizontal Flip)
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
Water Reflection (Vertical Flip)
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:
- 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.
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)
- a) 5-Digits Cell Entries:
- 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)
- a) 4-Digits Cell Entries:
- 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)
- a) 4-Digits Cell Entries:
- References
For the previous work on orders 3 to 13 refer the followig links:
- Inder J. Taneja, Water Reflection Magic Squares: Order 3 to 6.
- Inder J. Taneja, Water Reflection Magic Squares: Order 7 to 10.
- Inder J. Taneja, Water Reflection Magic Squares: Order 11 to 13.
- Inder J. Taneja, Water Reflection Magic Squares: Orders 14 to 16.
- Inder J. Taneja, Water Reflection Magic Squares: Order 17 to 20.
- Inder J. Taneja, Water Reflection Magic Squares: Orders 21 to 23.
- Inder J. Taneja, Water Reflection Magic Squares: Order 24. (This work).
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)

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.
b) 6-Digits Cell Entries:
Example 4. The Digits (2, 3, 5)

Water Reflection Image

Example 5. The Digits (1, 3, 8)

Water Reflection Image

Example 6. The Digits (0, 3, 8)

Water Reflection Image

Example 7. The Digits (0, 1, 3)

Water Reflection Image

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.
c) 10-Digits Cell Entries:
Example 8. The Digits (3, 8)

Water Reflection Image

Example 9. The Digits (1, 3)

Water Reflection Image

Example 10. The Digits (0, 3)

Water Reflection Image

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
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)

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.
b) 6-Digits Cell Entries:
Example 14. The Digits (2, 3, 5)

Water Reflection Image

Example 15. The Digits (1, 3, 8)

Water Reflection Image

Example 16. The Digits (0, 3, 8)

Water Reflection Image

Example 17. The Digits (0, 1, 3)

Water Reflection Image

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.
c) 10-Digits Cell Entries:
Example 18. The Digits (3, 8)

Water Reflection Image

Example 19. The Digits (1, 3)

Water Reflection Image

Example 20. The Digits (0, 3)

Water Reflection Image

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
a) 4-Digits Cell Entries:
Example 21. The Digits (1, 2, 3, 5, 8)

Water Reflection Image

Example 22. The Digits (0, 2, 3, 5, 8)

Water Reflection Image

Example 23. The Digits (0, 1, 2, 3, 5)

Water Reflection Image

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.
b) 6-Digits Cell Entries:
Example 24. The Digits (2, 3, 5)

Water Reflection Image

Example 25. The Digits (1, 3, 8)

Water Reflection Image

Example 26. The Digits (0, 3, 8)

Water Reflection Image

Example 27. The Digits (0, 1, 3)

Water Reflection Image

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.
c) 10-Digits Cell Entries:
Semi-Bimagic Squares
Example 28. The Digits (3, 8)

Water Reflection Image

Example 29. The Digits (1, 3)

Water Reflection Image

Example 30. The Digits (0, 3)

Water Reflection Image

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.
- 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.
- 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.
- 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.
- Site link: Universal and Upside-Down Magic Squares of Orders 11 to 15 (new site)
- Site link: Water Reflection Magic Squares: Order 11 to 13 (new site).
- Site link: Universal and Upside-Down Magic Squares of Orders 11 to 15 (old site)
- Site link: Water Reflection Magic Squares: Orders 11 to 13 (old site)
- 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.
- Site link: Universal and Upside-Down Magic and Bimagic Squares of Order 16 (new site)
- Site link: Water Reflection Magic Squares: Order 14 to 16 (new site).
- Site link: Universal and Upside-Down Magic and Bimagic Squares of Order 16 (old site)
- Site link: Water Reflection Magic Squares: Orders 14 to 16 (old site)
- 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.
- 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.
- 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
- 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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