Water Reflection Magic Squares: Order 20 – Recreating Numbers and Magic Squares

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. 1-86, https://doi.org/10.5281/zenodo.14676293.

Below is the index of the work given below for the order 20.

Magic Squares of Order 20: Blocks of Magic Squares of Order 4
- a) 4-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) 8-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 20: Blocks of Magic Squares of Order 5
- 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)
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 20
Blocks of Magic Squares of Order 4

a) 4-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 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 4 are magic squares with different magic sums.

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 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 4 are magic square with different magic sums.

c) 8-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 pandiagonal magic squares of order 20 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 pandiagonal magic squares with equal magic sums.