Universal and Upside-Down Magic Squares of Order 24

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 based on blocks of magic squares of orders 3, 4, 6 and 8. In some cases the these blocks are pandiagonal magic squares. The blocks of order 3 are always semi-magic squares.

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. This part is done only in case of blocks of order 8. Still, we don’t have results for the blocks of orders 3, 4 and 6.
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. This part is done for the blocks of orders 3, 4 and 8.
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. This part is done for the blocks of orders 3, 4, 6 and 8.

For complete work on universal and upside-down magic, bimagic and pandiagonal squares of orders 3 to 24 excep the orders 17, 18, 19, 22 and 23 see the reference list give at the end. More details of this work are given in following online work:

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.

a) 4-Digits Cell Entries:

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

Example 1: Blocks of Magic Squares of Order 8

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

Sm_24×24(0,1,6,8,9):=133320.

Blocks of order 8 are magic squares with different magic sums.

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

Example 2: Blocks of Magic Squares of Order 8

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

S_24×24(0,1,2,5,8):=88880.

Blocks of order 8 are magic squares with different magic sums.

b) 6-Digits Cell Entries:

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

Example 3: Blocks of Magic Squares of Order 3

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

Sm_24×24(1,6,9):=14222208

Blocks of order 3 are semi-magic squares with different magic sums. Blocks of order 12 are of equal semi-magic sums:

Sm_12×12(1,6,9):=7111104..

Example 4: Blocks of Magic Squares of Order 4

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

Sm_24×24(1,6,9):=12314302.

Blocks of order 4 are magic squares with different magic sums.

Example 5: Blocks of Magic Squares of Order 8

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

Sm_24×24(1,6,9):=12314302.

Blocks of order 8 are magic squares with different magic sums.

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

Example 6: Blocks of Magic Squares of Order 3

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

Sm_24×24(2,5,8):=13333320

Blocks of order 3 are semi-magic squares with different magic sums. Blocks of order 12 are of equal semi-magic sums:

Sm_12×12(2,5,8):=6666660.

Example 7: Blocks of Magic Squares of Order 4

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

Sm_24×24(2,5,8):=13333320

Blocks of order 4 are magic squares with different magic sums. Blocks of order 12 are of equal semi-magic sums:

Sm_12×12(2,5,8):=6666660.

Example 8: Blocks of Magic Squares of Order 8

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

Sm_24×24(2,5,8):=13333320

Blocks of order 8 are 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 3

It is universal pandiagonal magic square with magic sum:

S_24×24(1,8):=119999999988.

Blocks of order 3 are semi-magic squares with different magic sums. Blocks of order 12 are magic squares with equal magic sums

S_12×12(1,8):=59999999994.

Example 10: Blocks of Magic Squares of Order 4

It is universal pandiagonal magic square with magic sum:

Sm_24×24(1,8):=119999999988.

Blocks of order 4 are pandiagonal magic squares with equal magic sums:

S_4×4(1,8):=19999999998.

Example 11: Blocks of Magic Squares of Order 6

It is universal magic square with magic sum:

Sm_24×24(1,8):=119999999988.

Blocks of order 6 are magic squares with equal magic sums:

S_6×6(1,8):=29999999997.

Example 12: Blocks of Magic Squares of Order 8

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

S_24×24(1,8):=119999999988 and Sb_24×24(1,8):=9581393369685581393369700.

Blocks of order 8 are magic squares with equal magic sums:

S_8×8(1,8):=39999999996.

These are either bimagic or semi-bimagic squares.

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

Example 13: Blocks of Magic Squares of Order 3

It is universal pandiagonal magic square with magic sum:

S_24×24(2,5):=93333333324

Blocks of order 3 are semi-magic squares with different magic sums. Blocks of order 12 are magic squares with equal magic sums

S_12×12(2,5):=46666666662.

Example 14: Blocks of Magic Squares of Order 4

It is universal pandiagonal magic square with magic sum:

S_24×24(2,5):=93333333324

Blocks of order 4 are pandiagonal magic squares with equal magic sums.

S_4×4(2,5):=15555555554

Example 15: Blocks of Magic Squares of Order 6

It is universal magic square with magic sum:

S_24×24(2,5):=93333333324

Blocks of order 6 are magic squares with equal magic sums.

S_6×6(2,5):=23333333331

Example 16: Blocks of Magic Squares of Order 8

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

S_24×24(2,5):=93333333324 and Sb_24×24(2,5):=4287436575077028177315828

Blocks of order 8 are magic squares with equal magic sums.

S_8×8(2,5):=3111111111108

These are either bimagic or semi-bimagic squares.

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

Example 17: Blocks of Magic Squares of Order 3

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

S_24×24(6,9):=19999999999980

Blocks of order 3 are semi-magic squares with different magic sums. Blocks of order 12 are magic squares with equal magic sums

S_12×12(6,9):=9999999999990.

Example 18: Blocks of Magic Squares of Order 4

It is universal pandiagonal magic square with magic sum:

S_24×24(6,9):=19999999999980

Blocks of order 4 are pandiagonal magic squares with equal magic sums.

S_4×4(6,9):=3333333333330.

Example 19: Blocks of Magic Squares of Order 6

It is universal magic square with magic sum:

S_24×24(6,9):=19999999999980

Blocks of order 6 are magic squares with equal magic sums.

S_6×6(6,9):=4999999999995.

Example 20: Blocks of Magic Squares of Order 8

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

S_24×24(6,9):=19999999999980 and Sb_24×24(6,9):=1719415777078837889604.

Blocks of order 8 are magic squares with equal magic sums.

S_8×8(2,5):=6666666666660

These are either bimagic or semi-bimagic squares.

References

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)
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)
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)
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)
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)
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)
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)
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)

Note;
The result for the magic squares of orders 17, 18, 19, 22 and 23 are missing. The reason is that the numbers 17, 19 and 23 are prime numbers. It can be done similar to orders 11 and 13. The other numbers missing are of orders 18 and 22. It can be done similar to orders 10 or 14.

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ByInder J. Taneja