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 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.

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:

Magic Squares of Order 7

  • Example 1.

It is pandiagonal upside-down magic square with magic sum S7×7(0,1,2,5,6,8, 9):=341.

  • Example 2.

It is pandiagonal upside-down magic square with magic sum S7×7(0,1,2,5,6,8, 9):=3441. In this case, the entries are palindromes.

  • Example 3.

It is pandiagonal upside-down magic square with magic sum S7×7(1,6,9):=36663.

  • Example 4.

It is pandiagonal universal magic square with magic sum S7×7(2,5,8):=42218.

Magic Squares of Order 8

  • Example 5.

It is upside-down magic square with magic sum S8×8(1,6,9):=52217.

  • Example 6.

It is universal magic square with magic sum S8×8(2,5,8):=41107.

  • Example 7.

It is pandiagonal universal semi-bimagic square with magic sums S8×8(2,5,6,9):=488884 and Sb8×8(2,5,6,9):=34928450732.

  • Example 8.

It is universal semi-magic square with semi-magic sum S8×8(0,2,5,8):=333330.

  • Example 9.

It is pandiagonal upside-down bimagic magic square with magic sums S8×8(6,9):=666666666660 and Sb8×8(6,9):=57373737374426262626268.

  • Example 10.

It is pandiagonal universal bimagic magic square with magic sums S8×8(1,8):=399999999996 and Sb8×8(6,9):=29898989908389898989900.

  • Example 11.

It is pandiagonal universal bimagic magic square with magic sums S8×8(2,5):=311111111108 and Sb8×8(6,9):=13916947251838608305276.

Bimagic Squares of Order 9

  • Example 12.

It is upside-down bimagic magic square with magic sums S9×9(1,6,9):=53328 and Sb9×9(1,6,9):=414976074.

  • Example 13.

It is universal bimagic magic square with magic sums S9×9(2,5,8):=49995 and Sb9×9(2,5,8):=332267679.

  • 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 S9×9(1,6,9):=53333328 and Sb9×9(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 S9×9(2,5,8):=49999995 and Sb9×9(1,6,9):=332323500767679 respectively.

Pandiagonal magic Squares of Order 9

  • Example 16.

It is upside-down pandiagonal magic square of order 9 with 3-digits (6,8,9). The magic sum is given by S9×9(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 S9×9(0,2,5):=23331.

  • Example 18.

It is upside-down pandiagonal magic square of order 9 with 3-digits (6,8,9). The magic sum is given by S9×9(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 S9×9(2,5,8):=49999995.

Magic Squares of Order 10

  • Example 20.

It is upside-down magic square of order 10 with 4-digits (1,6,8,9). The magic sum is given by S10×10(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 S10×10(0,2,5,8):=32219.

  • Example 22.

It is upside-down magic square of order 10 with 3-digits (1,6,9). The magic sum is given by S10×10(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 S10×10(1,6,9):=5708703.

Leave a Reply

Your email address will not be published. Required fields are marked *

WP Twitter Auto Publish Powered By : XYZScripts.com