Universal and Upside-Down Magic Squares of Orders 7 to 10

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 3 to 6. 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 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.

The magic squares of order 9, we have considered in two different situations. One as bimagic and another as pandiagonal. In case of order 8, we considered the situations with pandiagonal and bimagic. For complete work for the orders 3 to 10, 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.
  • 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, September 30, 2024, pp. 1-87, https://doi.org/10.5281/zenodo.13858893.
  • 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)

Magic Squares of Order 7

a) 2-Digits Cell Entries:

• Example 1.

It is pandiagonal upside-down magic square with magic sum S_7×7(0,1,2,5,6,8, 9):=341.

b) 3-Digits Cell Entries:

• Example 2.

It is pandiagonal upside-down magic square with magic sum S_7×7(0,1,2,5,6,8, 9):=3441. In this case, the entries are palindromes.

c) 4-Digits Cell Entries:

• Example 3.

It is pandiagonal upside-down magic square with magic sum S_7×7(1,6,9):=36663.

• Example 4.

It is pandiagonal universal magic square with magic sum S_7×7(2,5,8):=42218.

Magic Squares of Order 8

a) 4-Digits Cell Entries:

• Example 5.

It is upside-down magic square with magic sum S_8×8(1,6,9):=52217.

• Example 6.

It is universal magic square with magic sum S_8×8(2,5,8):=41107.

a) 5-Digits Cell Entries:

• Example 7.

It is pandiagonal universal semi-bimagic square with magic sums S_8×8(2,5,6,9):=488884 and Sb_8×8(2,5,6,9):=34928450732.

• Example 8.

It is universal semi-magic square with semi-magic sum S_8×8(0,2,5,8):=333330.

b) 11-Digits Cell Entries:

• Example 9.

It is pandiagonal upside-down bimagic magic square with magic sums S_8×8(6,9):=666666666660 and Sb_8×8(6,9):=57373737374426262626268.

• Example 10.

It is pandiagonal universal bimagic magic square with magic sums S_8×8(1,8):=399999999996 and Sb_8×8(6,9):=29898989908389898989900.

• Example 11.

It is pandiagonal universal bimagic magic square with magic sums S_8×8(2,5):=311111111108 and Sb_8×8(6,9):=13916947251838608305276.

Bimagic Squares of Order 9

a) 4-Digits Cell Entries:

• Example 12.

It is upside-down bimagic magic square with magic sums S_9×9(1,6,9):=53328 and Sb_9×9(1,6,9):=414976074.

• Example 13.

It is universal bimagic magic square with magic sums S_9×9(2,5,8):=49995 and Sb_9×9(2,5,8):=332267679.

a) 7-Digits Cell Entries:

• Example 14.

It is upside-down bimagic square of order 9 with 3-digits (1,6,9). The magic and bimagic sums are given by S_9×9(1,6,9):=53333328 and Sb_9×9(1,6,9):=415039806496074 respectively.

• Example 15.

It is universal bimagic square of order 9 with 3-digits (2,5,8). The magic and bimagic sums are given by S_9×9(2,5,8):=49999995 and Sb_9×9(1,6,9):=332323500767679 respectively.

Pandiagonal magic Squares of Order 9

a) 4-Digits Cell Entries:

• Example 16.

It is upside-down pandiagonal magic square of order 9 with 3-digits (6,8,9). The magic sum is given by S_9×9(6,8,9):=76659.

• Example 17.

It is pandiagonal universal magic square of order 9 with 3-digits (0,2,5). The magic sum is given by S_9×9(0,2,5):=23331.

a) 7-Digits Cell Entries:

• Example 18.

It is upside-down pandiagonal magic square of order 9 with 3-digits (6,8,9). The magic sum is given by S_9×9(6,8,9):=76666659.

• Example 19.

It is pandiagonal universal magic square of order 9 with 3-digits (0,2,5). The magic sum is given by S_9×9(2,5,8):=49999995.

Magic Squares of Order 10

a) 4-Digits Cell Entries:

• Example 20.

It is upside-down magic square of order 10 with 4-digits (1,6,8,9). The magic sum is given by S_10×10(1,6,8,9):=62216.

• Example 21.

It is universal magic square of order 10 with 4-digits (0,2,5,8). The magic sum is given by S_10×10(0,2,5,8):=32219.

a) 6-Digits Cell Entries:

• Example 22.

It is upside-down magic square of order 10 with 3-digits (1,6,9). The magic sum is given by S_10×10(1,6,9):=7019012.

• Example 23.

It is universal magic square of order 10 with 3-digits (2,5,8). The magic sum is given by S_10×10(1,6,9):=5708703.