Universal and Upside-Down Magic Squares of Order 21

There are many ways of representing magic squares with palindromic type entries. Also, we can write magic squares in the composite forms based on pair of Latin squares. This paper works with magic and bimagic squares of order 16. By upside-down, we understand than by making 180^o it remains same. When the magic square is of both type, i.e., upside-down and mirror looking, we call it as universal magic square. By mirror looking, we understand that putting in front of mirror, still we see the image as a magic square. In case of mirror looking, writing as digitais fonts, 2 becoms 5 and 5 as 2. In case of upside-down, 6 becomes 9 and 9 as 6.

This work brings examples in different categories of entries. These are divided in three parts. In each part we worked with blocks of magic squares of orders 7 and 3. The order 7 magic squares are always pandigonal and the order 3 are always semi-magic squares.

1. 4-Digits Cell Entries
   • In this case, the work is in 5-digits, such as, (0,1,6,8,9) and (0,1,2,5,8). 5-Digits combinations lead us to 4-digits cell entries.
2. 6-Digits Cell Entries
   • In this case, the work is in 3-digits, such as, (1,6,9) and (2,5,8). 3-Digits combinations lead us to 6-digits cell entries.
3. 10-Digits Cell Entries
   • In this case, the work is in 2-digits, such as, (1,8), (2,5) and (6,9). 2-Digits combinations lead us to 10-digits cell entries.

For this work refer the following link. For complete work the see the reference list at the end of this work.

• 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

Universal and Upside-Down Magic Squares of Order 21

a) 4-Digits Cell Entries:

Upside-Down Magic Squares with 5-Digits (0,1,6,8,9)

Example 1: Blocks of Magic Squares of Order 7

It is upside-down semi-magic square with semi-magic sum:

Sm_21×21(0,1,6,8,9):=131098.

Blocks of order 7 are pandiagonal magic squares with different magic sums.

Example 2: Blocks of Semi-Magic Squares of Order 3

It is upside-down semi-magic square with semi-magic sum:

Sm_21×21(0,1,6,8,9):=131098.

Blocks of order 3 are semi-magic squares with different magic sums.

Universal Magic Squares with 5-Digits (0,1,2,5,8)

Example 3: Blocks of Magic Squares of Order 7

The above semi-magic square is universal with semi-magic sum:

S_21×21(0,1,2,5,8):=86658.

Blocks of order 7 are pandiagonal magic squares with different magic sums.

Example 4: Blocks of Magic Squares of Order 3

The above semi-magic square is universal with semi-magic sum:

S_21×21(0,1,2,5,8):=86658.

Blocks of order 3 are semi-magic squares with different magic sums.

b) 6-Digits Cell Entries:

Upside-Down Magic Squares with 3-Digits (1,6,9)

Example 5: Blocks of Magic Squares of Order 7

It is upside-down semi-magic square with semi-magic sum:

Sm_21×21(1,6,9):=12314302.

Blocks of order 7 are pandiagonal magic squares with different magic sums.

Example 6: Blocks of Semi-Magic Squares of Order 3

It is upside-down semi-magic square with semi-magic sum:

Sm_21×21(1,6,9):=12314302.

Blocks of order 3 are semi-magic squares with different magic sums.

Universal Magic Squares with 3-Digits (2,5,8)

Example 7: Blocks of Magic Squares of Order 7

The above semi-magic square is universal with semi-magic sum:

S_21×21(2,5,8):=11756745.

Blocks of order 7 are pandiagonal magic squares with different magic sums.

Example 8: Blocks of Magic Squares of Order 3

The above semi-magic square is universal with semi-magic sum:

S_21×21(2,5,8):=11756745.

Blocks of order 3 are semi-magic squares with different magic sums.

c) 10-Digits Cell Entries:

Universal Magic Squares with 2-Digits (1,8)

Example 9: Blocks of Magic Squares of Order 7

It is universal semi-magic square with semi-magic sum:

Sm_21×21(1,8):=108888888878.

Blocks of order 7 are pandiagonal magic squares with different magic sums.

Example 10: Blocks of Semi-Magic Squares of Order 3

It is universal semi-magic square with semi-magic sum:

Sm_21×21(1,8):=108888888878

Blocks of order 3 are semi-magic squares with different magic sums.

Universal Magic Squares with 2-Digits (2,5)

Example 11: Blocks of Magic Squares of Order 7

It is universal semi-magic square with semi-magic sums:

SO_21×21(2,5):=SU_21×21(2,5):=83333333325 and SM_21×21(2,5):=79999999992.

This means that semi-magic sums of original and upside-down versions are the same, while in case of mirror looing, this sums is different.

Blocks of order 7 are pandiagonal magic squares with different magic sums.

Example 12: Blocks of Semi-Magic Squares of Order 3

It is universal semi-magic square with semi-magic sums:

SO_21×21(2,5):=SU_21×21(2,5):=83333333325 and SM_21×21(2,5):=79999999992.

This means that semi-magic sums of original and upside-down versions are the same, while in case of mirror looing, this sums is different.

Blocks of order 3 are semi-magic squares with different magic sums.

Upside-Down Magic Squares with 2-Digits (6,9)

Example 13: Blocks of Magic Squares of Order 7

It is upside-down semi-magic square with semi-magic sum:

Sm_21×21(6,9):=176666666649.

Blocks of order 7 are pandiagonal magic squares with different magic sums.

Example 14: Blocks of Semi-Magic Squares of Order 3

It is upside-down semi-magic square with semi-magic sum:

Sm_21×21(6,9):=176666666649.

Blocks of order 3 are semi-magic squares with different magic sums.

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: 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: Universal and Upside-Down Magic Squares of Orders 7 to 10 (old site)
3. Inder J. Taneja, Universal and Upside-Down Magic Squares of Orders 11 to 15, Zenodo, November 05, 2024, pp. 1-141, https://doi.org/10.5281/zenodo.14041168
   • Site link: Universal and Upside-Down Magic Squares of Orders 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)