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 squares of order 11 to 15. 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 an 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.
For complete work for the orders 3 to 15, the readers can access the following links:
- Inder J. Taneja, Universal and Upside-Down Magic Squares of Orders 3 to 6, Zenodo, September 30, 2024, pp. 1-59, https://doi.org/10.5281/zenodo.13858891.
- Inder J. Taneja, Universal and Upside-Down Magic Squares of Orders 7 to 10, Zenodo, September 30, 2024, pp. 1-87, https://doi.org/10.5281/zenodo.13858893.
- Inder J. Taneja, Universal and Upside-Down Magic Squares of Orders 11 to 15, Zenodo, October 06, 2024, pp. 1-91, https://doi.org/10.5281/zenodo.13896515
Magic Squares of Order 11
a) 4-Digits Cell Entries:
Example 1
It is pandiagonal upside-down magic square with magic sum S11×11(1,6,8,9):=63327.
Example 2.
It is pandiagonal universal magic square of order 11. Its magic sum is S7×7(0,2,5,8):=58883.
b) 6-Digits Cell Entries:
Example 3.
It is pandiagonal upside-down magic square with magic sum S7×7(1,6,9):=6361355.
Example 4.
It is pandiagonal universal magic square. In this case the magic sums are different, i.e., the magic sums of original and mirror looking are the same, but in upside-down situation the sum is different. The sums are as follows:
SO11×11(2,5,8):=SM11×11(2,5,8):=6471465 and SO11×11(2,5,8):=6174168.
Magic Squares of Order 12
a) 4-Digits Cell Entries: Blocks of Order 4
Example 5.
It is upside-down semi-magic square of order 12 in 4-digits (1,6,8,9) with magic sum S12×12(1,6,8,9) := 79992. Blocks of order 4 are magic squares with different magic sums.
Example 6.
It is universal semi-magic square of order 12 in 4-digits (1,2,5,8) with magic sum S12×12(1,6,8,9):=53328. Blocks of order 4 are magic squares with different magic sums.
b) 6-Digits Cell Entries: Blocks of Order 4
Example 7.
It is upside-down semi-magic square of order 12 in 3-digits (1,6,9) with magic sum S12×12(1,6,9) := 7916909. Blocks of order 4 are magic squares with different magic sums.
Example 8.
It is universal semi-magic square of order 12 in 3-digits (2,5,8) with magic sum S12×12(2,5,8):=6666660. Blocks of order 4 are magic squares with different magic sums.
c) 4-Digits Cell Entries: Blocks of Order 3
Example 9.
It is upside-down semi-magic square of order 12 in 4-digits (1,6,8,9) with magic sum S12×12(1,6,8,9) := 87769. Blocks of order 3 are semi-magic squares with different semi-magic sums..
Example 10.
It is universal semi-magic square of order 12 in 4-digits (0,2,5,8) with magic sum S12×12(0,2,5,8):=49995. Blocks of order 3 are semi-magic squares with different semi-magic sums
d) 6-Digits Cell Entries: Blocks of Order 3
Example 11.
It is upside-down semi-magic square of order 12 in 3-digits (1,6,9) with magic sum S12×12(1,6,9) := 7916909. Blocks of order 3 are semi-magic squares with different semi-magic sums.
Example 12.
It is universal semi-magic square of order 12 in 3-digits (2,5,8) with magic sum S12×12(0,2,5,8):=6666660. Blocks of order 3 are semi-magic squares with different semi-magic sums
Bimagic Squares of Order 13
a) 4-Digits Cell Entries:
Example 13.
It is upside-down pandiagonal magic square of order 13 in 4-digits (1,6,8,9) with magic sum S13×13(1,6,8,9) := 87769.
Example 14.
It is universal pandiagonal magic square of order 13 in 4-digits (1,2,5,8) with magic sum S13×13(1,2,5,8) := 62216.
b) 6-Digits Cell Entries:
Example 15.
It is upside-down pandiagonal magic square of order 13 in 3-digits (1,6,9) with magic sum S13×13(1,6,9) := 7897890.
Example 16.
It is universal pandiagonal magic square. In this case the magic sums are different, i.e., the magic sums of original and mirror looking are the same, but in upside-down situation the sum is different. The sums are as follows:
SO13×13(2,5,8):=SM13×13(2,5,8):=7249242 and SR13×13(2,5,8):=6951945.
Pandiagonal magic Squares of Order 14
a) 4-Digits Cell Entries:
Example 17.
It is upside-down magic square of order 14 in 4-digits (1,6,8,9) with magic sum S14×14(1,6,8,9) := 96657.
Example 18.
It is universal magic square of order 14 in 4-digits (0,2,5,8) with magic sum S14×14(0,2,5,8) := 58883.
b) 6-Digits Cell Entries:
Example 19.
It is upside-down magic square of order 14 in 3-digits (1,6,9) with magic sum S14×14(1,6,9) := 9453444.
Example 20.
It is universal magic square of order 14 in 3-digits (2,5,8) with magic sum S14×14(2,5,8) := 8048040.
Magic Squares of Order 15
a) 4-Digits Cell Entries:
Example 21.
It is upside-down magic square of order 15 in 4-digits (1,6,8,9) with magic sum S15×15(1,6,8,9) := 105545. The blocks of order 5 are pandiagonal magic squares with different magic sums.
Example 22.
It is universal magic square of order 15 in 4-digits (0,2,5,8) with magic sum S15×15(0,2,5,8) := 66660. The blocks of order 5 are pandiagonal magic squares with different magic sums
a) 6-Digits Cell Entries:
Example 23.
It is upside-down magic square of order 15 in 3-digits (1,6,9) with magic sum S15×15(1,6,9) := 9564555. The blocks of order 5 are pandiagonal magic squares with different magic sums
Example 24.
It is universal magic square. In this case the magic sums are different, i.e., the magic sums of original and mirror looking are the same, but in upside-down situation the sum is different. The sums are as follows:
SO15×15(2,5,8):=SM15×15(2,5,8):=8936928 and SR15×15(2,5,8):=8639631.
The blocks of order 5 are pandiagonal magic squares with different magic sums