Water Reflection Magic Squares: Order 11 to 15

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 11 to 15. Full work can be downloaded from the following link:

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.

For the previous work on orders 3 to 6 and 7 to 10 refer the links below:

Inder J. Taneja, Water Reflection Magic Squares: Order 3 to 6.
Inder J. Taneja, Water Reflection Magic Squares: Order 7 to 10.

Magic Squares of Order 11

a) 4-Digits Cell Entries:

Example 1. Magic Square with 4-Digits (2, 3, 5, 8)

Water Reflection Image

Example 2. Magic Square with 4-Digits (1, 2, 3, 5)

Water Reflection Image

Example 3. Magic Squares with 4-Digits (0, 2, 3, 5)

Water Reflection Image

Example 4. Magic Squares with 4-Digits (0,1, 3, 8)

Water Reflection Image

Thus, we have four examples of water reflection pandiagonal magic squares of order 11 having 4-digits cells entries with four number combinations (2,3,5,8), (1,2,3,5), (0,2, 5, 8) and, (0,1,3,8).

b) 6-Digits Cell Entries:

Example 5. Magic Square with 3-Digits (2, 3, 5)

Water Reflection Image

Example 6. Magic Square with 3-Digits (1, 3, 8)

Water Reflection Image

Example 7. Magic Square with 3-Digits (0, 3, 8)

Water Reflection Image

Example 8. Magic Square with 3-Digits (0, 1, 3)

Water Reflection Image

Thus, we have four examples of water reflection pandiagonal magic squares of order 11 having 6-digits cells entries with three numbers combinations (2, 3, 5), (1, 3, 8), (0,3,8) and, (0, 1, 3).

c) 8-Digits Cell Entries:

Example 9. Magic Square with 2-Digits (3, 8)

Water Reflection Image

Example 10. Magic Square with 3-Digits (1, 3)

Water Reflection Image

Example 11. Magic Square with 3-Digits (0, 3)

Water Reflection Image

Thus, we have three examples of water reflection pandiagonal magic squares of order 11 having 8-digits cells entries with two numbers combinations (3, 8), (1, 3) and, (0, 3).