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 180o 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 4 and 5.
- 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.
- 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.
- 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 complete work on universal and upside-down magic, bimagic and pandiagonal squares see the reference list at the end. For this work online refer the following link:
- 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.
Universal and Upside-Down Magic of Order 20
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 4
The above magic square is upside-down with magic sum:
S20×20(0,1,6,8,9):=11120109.
Blocks of order 4 are magic squares with different magic sums
Example 2: Blocks of Magic Squares of Order 5
The above magic square is upside-down with magic sum:
S20×20(0,1,6,8,9):=11120109.
Blocks of order 5 are pandiagonal 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 4
The above magic square is universal with magic sum:
S20×20(0,1,2,5,8):=81103.
Blocks of order 4 are magic squares with different magic sums
Example 4: Blocks of Magic Squares of Order 5
The above magic square is universal with magic sum:
S20×20(0,1,2,5,8):=81103.
Blocks of order 5 are pandiagonal 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 4
The above magic square is upside-down with magic sum:
S20×20(1,6,9):=12203191.
Blocks of order 4 are magic squares with different magic sums.
Example 6: Blocks of Magic Squares of Order 5
The above magic square is upside-down with magic sum:
S20×20(1,6,9):=12203191.
Blocks of order 5 are pandiagonal magic squares with different magic sums.
Universal Magic Squares with 2-Digits (1,8)
Example 7: Blocks of Magic Squares of Order 4
The above magic square is universal with magic sum:
S20×20(2,5,8):=10867857.
Blocks of order 4 are magic squares with different magic sums.
Example 8: Blocks of Magic Squares of Order 5
The above magic square is universal with magic sum:
S20×20(2,5,8):=10867857.
Blocks of order 5 are pandiagonal 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 4
The above magic square is universal with magic sum:
S20×20(1,8):=99999999990.
Blocks of order 4 are pandiagonal magic squares with equal magic sums:
S4×4(1,8):=19999999998.
Example 11: Blocks of Magic Squares of Order 5
The above magic square is universal with magic sum:
S20×20(1,8):=99999999990.
Blocks of order 5 are pandiagonal magic squares with different magic sums.
Universal Magic Squares with 2-Digits (2,5)
Example 12: Blocks of Magic Squares of Order 4
The above magic square is universal with magic sum:
S20×20(2,5):=77777777770.
Blocks of order 4 are pandiagonal magic squares with equal magic sums:
S4×4(2,5):=15555555554.
Example 10: Blocks of Magic Squares of Order 5
The above magic square is universal with magic sum:
S20×20(2,5):=77777777770.
Blocks of order 5 are pandiagonal magic squares with different magic sums.
Upside-Down Magic Squares with 2-Digits (6,9)
Example 13: Blocks of Magic Squares of Order 4
The above magic square is upside-down with magic sum:
S20×20(6,9):=166666666650.
Blocks of order 4 are pandiagonal magic squares with equal magic sums:
S4×4(6,9):=33333333330.
Example 14: Blocks of Magic Squares of Order 5
The above magic square is upside-down with magic sum:
S20×20(6,9):=166666666650.
Blocks of order 5 are pandiagonal magic squares with different magic sums.
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
- 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
- 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
- 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
- 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.
- 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
- 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
- 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.