Universal and Upside-Down Magic Squares of Order 3 to 6

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 becomes 5 and 5 as 2. In case of upside-down, 6 becomes 9 and 9 as 6.

The magic squares of 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. The orders 7 to 10 are considered in another work. See the links below:

• 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 3

a) 2-Digits Cell Entries:

• Example 1.

It is upside-down semi-magic square of order 3 with 3-digits (1,6,9). The magic sum is S_3×3(1,6,9):=176. This magic square becomes becomes semi-magic in case of mirror looking.

• Example 2.

It is universal magic square of order 3 with 3-digits (2,5,8). The magic sum is S_3×3(2,5,8):=165. This magic square becomes becomes semi-magic in case of mirror looking.

b) 3-Digits Cell Entries:

• Example 3.

It is upside-down semi-magic square of order 3 with 3-digits (1,6,9). In this case, the entries are palindromes. The magic sum is S_3×3(1,6,9):=1776.

• Example 4.

It is universal magic square of order 3 with 3-digits (2,5,8). The magic sum is S_3×3(2,5,8):=1665. In this case, the entries are palindromes. This magic square becomes becomes semi-magic in case of mirror looking.

Magic Squares of Order 4

a) 2-Digits Cell Entries:

• Example 5.

It is upside-down magic square of order 4 with 4-digits (1,6,8,9). The magic sum is S_4×4(1,6,8,9):=264.

Example 6.

It is universal magic square of order 4 with 4-digits (0,1,2,5). The magic sum is S_4×4(0,1,2,5):=88.

It is well-known author’s stamp published at Macau – China – 2015. See below:

b) 3-Digits Cell Entries:

• Example 7.

It is upside-down magic square of order 4 with 4-digits (1,6,8,9). In this case, the entries are palindromes. The magic sum is S_4×4(1,6,8,9):=2664.

• Example 8.

It is universal magic square of order 4 with 4-digits (1,2,5,8). In this case, the entries are palindromes. The magic sum is S_4×4(1,2,5,8):=1776.

c) 4-Digits Cell Entries:

• Example 9.

It is pandiagonal upside-down magic square of order 4 with 2-digits (6,9). The magic sum is S_4×4(6,9):=33330.

• Example 10.

It is pandiagonal universal magic square of order 4 with 2-digits (1,8). The magic sum is S_4×4(1,8):=19998.

• Example 11.

It is pandiagonal universal magic square of order 4 with 2-digits (2,5). The magic sum is S_4×4(2,5):=15554.

d) 7-Digits Cell Entries:

• Example 12.

It is pandiagonal upside-down magic square of order 4 with 2-digits (6,9). The entries are palindromes. The magic sum is S_4×4(6,9):=33333330.

• Example 13.

It is pandiagonal universal magic square of order 4 with 2-digits (1,8). The entries are palindromes. The magic sum is S_4×4(1,8):=19999998.

• Example 14.

It is pandiagonal universal magic square of order 4 with 2-digits (2,5). The entries are palindromes. The magic sum is S_4×4(2,5):=15555554.

Magic Squares of Order 5

a) 5-Digits Cell Entries:

• Example 15.

It is pandiagonal upside-down magic square of order 5 with 5-digits (0,1,6,8,9). The magic sum is S_5×5(0,1,6,8,9):=264.

• Example 16.

It is pandiagonal upside-down magic square of order 5 with 5-digits (2,5,6,8,9). The magic sum is S_5×5(2,5,6,8,9):=330.

• Example 17.

It is pandiagonal universal magic square of order 5 with 5-digits (0,1,2,5,8). The magic sum is S_5×5(0,1,2,5,8):=176.

b) 3-Digits Cell Entries:

• Example 18.

It is pandiagonal upside-down magic square of order 5 with 5-digits (1,2,5,6,9). In this case the entries are palindromes. The magic sum is S_5×5(1,2,5,6,9):=2553.

• Example 19.

It is pandiagonal universal magic square of order 5 with 5-digits (0,1,2,5,8). In this case the entries are palindromes. The magic sum is S_5×5(0,1,2,5,8):=1776.

c) 4-Digits Cell Entries:

• Example 20.

It is pandiagonal upside-down magic square of order 5 with 3-digits (1,6,9). The magic sum is S_5×5(1,6,9):=34441.

• Example 21.

It is pandiagonal upside-down magic square of order 5 with 3-digits (6,8,9). The magic sum is S_5×5(6,8,9):=42218.

• Example 22.

It is pandiagonal universal magic square of order 5 with 3-digits (2,5,8). The magic sum is S_5×5(2,5,8):=24442.

Magic Squares of Order 6

a) 4-Digits Cell Entries:

• Example 23.

It is upside-down magic square of order 5 with 3-digits (6,8,9). The magic sum is S_6×6(2,5,8):=33330. This magic square is only upside-down because its mirror looking image is no more a magic square.