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: Orders 14 to 16, Zenodo, January 14, 2025, pp. 1-140, https://doi.org/10.5281/zenodo.14649519.
Below is the index of the work given below for the orders 14 to 16.
- Magic Squares of Order 14
- a) 4-Digits Cell Entries:
- Example 1. The Digits (2,3,5,8)
- Example 2. The Digits (1,2,3,5)
- Example 3. The Digits (0,2,3,5)
- Example 4. The Digits (0,1,3,8)
- b) 6-Digits Cell Entries:
- Example 5. The Digits (2,3,5)
- Example 6. The Digits (1,3,8)
- Example 7. The Digits (0,3,8)
- Example 8. The Digits (0,1,3)
- c) 8-Digits Cell Entries:
- Example 9. The Digits (3,8)
- Example 10. The Digits (1,3)
- Example 11. The Digits (0,3)
- a) 4-Digits Cell Entries:
- Magic Squares of Order 15: Blocks of Magic Squares of Order 5
- a) 4-Digits Cell Entries:
- Example 12. The Digits (2,3,5,8)
- Example 13. The Digits (1,2,3,5)
- Example 14. The Digits (0,2,3,5)
- Example 15. The Digits (0,1,3,8)
- b) 6-Digits Cell Entries:
- Example 16. The Digits (2,3,5)
- Example 17. The Digits (1,3,8)
- Example 18. The Digits (0,3,8)
- Example 19. The Digits (0,1,3)
- c) 8-Digits Cell Entries:
- Example 20. The Digits (3,8)
- Example 21. The Digits (1,3)
- Example 22. The Digits (0,3)
- a) 4-Digits Cell Entries:
- Magic Squares of Order 15: Blocks of Semi-Magic Squares of Order 3
- d) 8-Digits Cell Entries:
- Example 23. The Digits (3,8)
- Example 24. The Digits (1,3)
- Example 25. The Digits (0,3)
- d) 8-Digits Cell Entries:
- Magic and Bimagic Squares of Order 16: Blocks of Magic Squares of Order 4
- a) 4-Digits Cell Entries:
- Example 26. The Digits (2,3,5,8)
- Example 27. The Digits (1,2,3,5)
- Example 28. The Digits (0,2,3,5)
- Example 29. The Digits (0,1,3,8)
- b) 6-Digits Cell Entries:
- Example 30. The Digits (2,3,5)
- Example 31. The Digits (1,3,8)
- Example 32. The Digits (0,3,8)
- Example 33. The Digits (0,1,3)
- c) 8-Digits Cell Entries: Bimagic Squares
- Example 34. The Digits (3,8)
- Example 35. The Digits (1,3)
- Example 36. The Digits (0,3)
- d) 8-Digits Cell Entries: Pandiagonal Magic Squares
- Example 37. The Digits (3,8)
- Example 39. The Digits (1,3)
- Example 39. The Digits (0,3)
- a) 4-Digits Cell Entries:
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.
Magic Squares of Order 14
a) 4-Digits Cell Entries:
Example 1. The Digits (2, 3, 5, 8)

Water Reflection Image

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

Water Reflection Image

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

Water Reflection Image

Example 4. The Digits (0, 1, 3, 8)

Water Reflection Image

Thus, we have four examples of water reflection magic squares of order 14 having 4-digits cells entries with four numbers combinations (2, 3, 5, 8), (1, 2, 3, 5), (0, 2, 3, 5) and (0,1, 3, 8). The internal block of order 4 is a magic square.
b) 6-Digits Cell Entries:
Example 5. The Digits (2, 3, 5)

Water Reflection Image

Example 6. The Digits (1, 3, 8)

Water Reflection Image

Example 7. The Digits (0, 3, 8)

Water Reflection Image

Example 8. The Digits (0, 1, 3)

Water Reflection Image

Thus, we have four examples of water reflection magic squares of order 14 having 6-digits cells entries with 3 numbers combinations (2, 3, 5), (1, 3, 8), (0, 3, 8) and (0,1, 3). The internal block of order 4 is a magic square.
c) 8-Digits Cell Entries:
Example 9. The Digits (3, 8)

Water Reflection Image

Example 10. The Digits (1, 3)

Water Reflection Image

Example 11. The Digits (0, 3)

Water Reflection Image

Thus, we have three examples of water reflection magic squares of order 14 having 8-digits cells entries with 2 numbers combinations (3, 8), (1, 3) and (0, 3). The internal block of order 4 is a magic square.
Magic Squares of Order 15:
Blocks of Magic Squares of Order 5
a) 4-Digits Cell Entries:
Example 12. The Digits (2, 3, 5, 8)

Water Reflection Image

Example 13. The Digits (1, 2, 3, 5)

Water Reflection Image

Example 14. The Digits (0, 2, 3, 5)

Water Reflection Image

Example 15. The Digits (0, 1, 3, 8)

Water Reflection Image

Thus, we have four examples of water reflection semi-magic squares of order 15 having 4-digits cells entries with four numbers combinations (2, 3, 5, 8), (1, 2, 3, 5), (0, 2, 3, 5) and (0,1, 3, 8). The blocks of order 5 are pandiagonal magic squares with different magic sums
b) 6-Digits Cell Entries:
Example 16. The Digits (2, 3, 5)

Water Reflection Image

Example 17. The Digits (1, 3, 8)

Water Reflection Image

Example 18. The Digits (0, 3, 8)

Water Reflection Image

Example 19. The Digits (0, 1, 3)

Water Reflection Image

Thus, we have four examples of water reflection semi-magic squares of order 15 having 6-digits cells entries with three numbers combinations (2, 3, 5), (1, 3, 8), (0, 3, 8) and (0,1, 3). The blocks of order 5 are pandiagonal magic squares with different magic sums.
c) 8-Digits Cell Entries:
Example 20. The Digits (3, 8)

Water Reflection Image

Example 21. The Digits (1, 3)

Water Reflection Image

Example 22. The Digits (0, 3)

Water Reflection Image

Thus, we have three examples of water reflection semi-magic squares of order 15 having 8-digits cells entries with 2 numbers combinations (3, 8), (1, 3) and (0, 3). The blocks of order 5 are pandiagonal magic squares of order 5 with different magic sums.
Magic Squares of Order 15:
Blocks of Semi-Magic Squares of Order 3
8-Digits Cell Entries:
In this case we have only the results for 2-digits combinations where each cell have 8-digits. See below three examples
Example 23. The Digits (3, 8)

Water Reflection Image

Example 24. The Digits (1, 3)

Water Reflection Image

Example 25. The Digits (0, 3)

Water Reflection Image

Thus, we have three examples of water reflection semi-magic squares of order 15 having 8-digits cells entries with 2 numbers combinations (3, 8), (1, 3) and (0, 3). The blocks of order 3 are semi-magic squares of order 3 with different semi-magic sums.
Magic and Bimagic Squares of Order 16:
Blocks of Magic Squares of Order 4
a) 4-Digits Cell Entries:
Example 26. The Digits (2, 3, 5, 8)

Water Reflection Image

Example 27. The Digits (1, 2, 3, 5)

Water Reflection Image

Example 28. The Digits (0, 2, 3, 5)

Water Reflection Image

Example 29. The Digits (0, 1, 3, 8)

Water Reflection Image

Thus, we have four examples of water reflection bimagic squares of order 16 having 4-digits cells entries with four numbers combinations (2, 3, 5, 8), (1, 2, 3, 5), (0, 2, 3, 5) and (0,1, 3, 8). The blocks of order 4 are magic squares of order 4 with equal magic sums.
b) 6-Digits Cell Entries:
Example 30. The Digits (2, 3, 5)

Water Reflection Image

Example 31. The Digits (1, 3, 8)

Water Reflection Image

Example 32. The Digits (0, 3, 8)

Water Reflection Image

Example 33. The Digits (0, 1, 3)

Water Reflection Image

Thus, we have four examples of water reflection magic squares of order 16 having 6-digits cells entries with three numbers combinations (2, 3, 5), (1, 3, 8), (0, 3, 8) and (0,1, 3). The blocks of order 4 are magic squares of order 4 with different magic sums.
c) 8-Digits Cell Entries: Bimagic Squares
Example 34. The Digits (3, 8)

Water Reflection Image

Example 35. The Digits (1, 3)

Water Reflection Image

Example 36. The Digits (0, 3)

Water Reflection Image

Thus, we have three examples of water reflection bimagic squares of order 16 having 8-digits cells entries with 2 numbers combinations (3, 8), (1, 3) and (0, 3). The blocks of order 4 are magic squares of order 4 with equal magic sums.
d) 8-Digits Cell Entries: Pandiagonal Magic Squares
Example 37. The Digits (3, 8)

Water Reflection Image

Example 39. The Digits (1, 3)

Water Reflection Image

Example 39. The Digits (0, 3)

Water Reflection Image

Thus, we have three examples of water reflection pandiagonal squares of order 16 having 8-digits cells entries with 2 numbers combinations (3, 8), (1, 3) and (0, 3). The blocks of order 4 are also pandiagonal magic squares of order 4 with equal magic 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.