Water Reflection Magic Squares: Orders 21 to 23

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 20, Zenodo, January 17, 2025, pp. 1-140,

Below is the index of the work given below for the order 21.

• Magic Squares of Order 21: 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 (0,1,2,3,5)
    • Example 3. The Digits (0,2,3,5,8)
  • 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 21: Blocks of Magic Squares of Order 7
  • a) 4-Digits Cell Entries:
    • Example 11. The Digits (1,2,3,5,8)
    • Example 12. The Digits (0,1,2,3,5)
    • Example 13. The Digits (0,2,3,5,8)
  • 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) 8-Digits Cell Entries:
    • Example 18. The Digits (3,8)
    • Example 19. The Digits (1,3)
    • Example 20. The Digits (0,3)
• Magic Squares of Orders 22 and 23
• 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 20. (This work)

Magic Squares of Order 21:
Blocks of Semi-Magic Squares of Order 3

a) 5-Digits Cell Entries:

Example 1. The Digits (1, 2, 3, 5, 8)

Water Reflection Image

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

Water Reflection Image

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

Water Reflection Image

Thus, we have three examples of water reflection magic squares of order 21 having 4-digits cells entries with five numbers combinations (1, 2, 3, 5, 8), (0, 1, 2, 3, 5) and (0, 2, 3, 5, 8). The internal blocks of order 3 are semi-magic squares with different semi-magic sums. The magic squares and their water reflection images are also semi-magic squares.

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 magic squares of order 21 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 magic squares and their water reflection images are also semi-magic squares.

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 21 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 ual semi-magic sums.

Magic Squares of Order 21
Blocks of Magic Squares of Order 7

a) 4-Digits Cell Entries:

Example 11. The Digits (1, 2, 3, 5, 8)

Water Reflection Image

Example 12. The Digits (0, 1, 2, 3, 5)

Water Reflection Image

Example 13. The Digits (0, 2, 3, 5, 8)

Water Reflection Image

Thus, we have three examples of water reflection semi-magic squares of order 20 having 4-digits cells entries with five numbers combinations (1, 2, 3, 5, 8), (0, 1, 2, 3, 5) and (0, 2, 3, 5, 8). The internal blocks of order 7 are pandiagonal magic squares with different magic sums.

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 20 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 7 are pandiagonal magic square with different magic sums.

c) 8-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 semi-magic squares of order 21 having 10-digits cells entries with 2 numbers combinations (3, 8), (1, 3) and (0, 3). In each case, the internal blocks of order 7 are pandiagonal magic squares with different magic sums.

Magic Squares of Orders 22 and 23

Results on magic squares of order 22 are similar to magic squares of order 14. Since 23 is prime numbers. The results for the order 23 are similar to magic squares of orders 11 or 13. See the links below:

• Inder J. Taneja, Water Reflection Magic Squares: Order 11 to 13.
• Inder J. Taneja, Water Reflection Magic Squares: Orders 14 to 16.

References

1. Inder J. Taneja, Universal and Upside-Down Magic Squares of Orders 3 to 6, Zenodo, November 05, 2024, pp. 1-61, https://doi.org/10.5281/zenodo.14041149
   • Site link: Universal and Upside-Down Magic Squares of Order 3 to 6 (new site)
   • Site link: Water Reflection Magic Squares: Order 3 to 6 (new site)
   • Site link: Universal and Upside-Down Magic Squares of Order 3 to 6 (old site)
2. Inder J. Taneja, Universal and Upside-Down Magic Squares of Orders 7 to 10, Zenodo, November 05, 2024, pp. 1-120, https://doi.org/10.5281/zenodo.14041164
   • Site link: Universal and Upside-Down Magic Squares of Orders 7 to 10 (new site)
   • Site link: Water Reflection Magic Squares: Order 7 to 10 (new site)
   • Site link: Universal and Upside-Down Magic Squares of Orders 7 to 10 (old site)
3. Inder J. Taneja, Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 11 to 15, Zenodo, January 14, 2024, pp. 1-233, https://doi.org/10.5281/zenodo.14642199.
   • Site link: Universal and Upside-Down Magic Squares of Orders 11 to 15 (new site)
   • Site link: Water Reflection Magic Squares: Order 11 to 15 (new site)
   • Site link: Universal and Upside-Down Magic Squares of Orders 11 to 15 (old site)
4. Inder J. Taneja, Universal and Upside-Down Magic Squares of Order 16, Zenodo, October 16, 2024, pp. 1-28, https://doi.org/10.5281/zenodo.13942620
   • Site link: Universal and Upside-Down Magic and Bimagic Squares of Order 16 (new site)
   • Site link: Universal and Upside-Down Magic and Bimagic Squares of Order 16 (old site)
5. Inder J. Taneja, Universal and Upside-Down Magic Squares of Order 20, Zenodo, October 20, 2024, pp. 1-56, https://doi.org/10.5281/zenodo.13958700.
   • Site link: Universal and Upside-Down Magic Squares of Order 20 (new site)
   • Site link: Universal and Upside-Down Magic Squares of Order 20 (old site)
6. Inder J. Taneja, Universal and Upside-Down Magic Squares of Order 21, Zenodo, October 23, 2024, pp. 1-49, https://doi.org/10.5281/zenodo.13982859
   • Site link: Universal and Upside-Down Magic Squares of Order 21 (new site)
   • Site link: Universal and Upside-Down Magic Squares of Order 21 (old site)
7. Inder J. Taneja, Universal and Upside-Down Magic Squares of Order 24, Zenodo, October 29, 2024, pp. 1-82, https://doi.org/10.5281/zenodo.14004788
   • Site link: Universal and Upside-Down Magic Squares of Order 24 (new site).
   • Site link: Universal and Upside-Down Magic Squares of Order 24 (old site)
8. Inder J. Taneja, Universal and Upside-Down Magic and Bimagic Squares of Order 25, Zenodo, October 30, 2024, pp. 1-53, https://doi.org/10.5281/zenodo.14014851.
   • Site link: Universal and Upside-Down Magic and Bimagic Squares of Order 25 (new site)
   • Site link: Universal and Upside-Down Magic and Bimagic Squares of Order 25 (old site)