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 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.

For complete work for the orders 3 to 25, see the reference list at end of this work. For this work, see the online link given below:

**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

### Magic Squares of Order 11

### a) 4-Digits Cell Entries

### Example 1: The Digits (1,6,9)

It is **pandiagonal** **upside-down** magic square with magic sum **S _{11×11}(1,6,8,9):=63327**.

### Example 2. The Digits (0,2,5,8)

It is **pandiagonal** **universal** magic square of order 11. Its magic sum is **S _{7×7}(0,2,5,8):=58883**.

### b) 6-Digits Cell Entries

### Example 3. The Digits (1,6,9)

It is **pandiagonal** **upside-down** magic square with magic sum **S _{7×7}(1,6,9):=6361355**.

### Example 4. The Digits (2,5,8)

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:

**SO _{11×11}(2,5,8):=SM_{11×11}(2,5,8):=6471465** and

**SO**.

_{11×11}(2,5,8):=6174168### c) 8-Digits Cell Entries

### Example 5. The Digits (1,8)

It is **universal pandiagonal** magic square of order 11 for the digits (1,8). Its magic sum is **S _{11×11}(1,8):=511111106**.

### Example 6. The Digits (2,5)

It is **universal pandiagonal** magic square of order 11 for the digits (2,5). Its magic sums are **S _{11×11}(2,5):=411111107** and

**S**. The

_{11×11}(2,5):=444444440**S**refers to

_{11×11}(2,5):=444444440**mirror-looking**verion of magic square.

### Example 7. The Digits (6,9)

It is **upside-down pandiagonal** magic square of order 11 for the digits (1,8). Its magic sum is **S _{11×11}(6,9):=905730564**.

### Magic Squares of Order 12

### a) 4-Digits Cell Entries: Blocks of Order 4

### Example 8. The Digits (1,6,8,9)

It is **upside-down semi-magic** square of order 12 in 4-digits (1,6,8,9) with magic sum **S _{12×12}(1,6,8,9) := 79992**. Blocks of order 4 are magic squares with different magic sums.

### Example 9. The Digits (1,2,5,8)

It is **universal semi-magic** square of order 12 in 4-digits (1,2,5,8) with magic sum **S _{12×12}(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 10. The Digits (1,6,9)

It is **upside-down semi-magic** square of order 12 in 3-digits (1,6,9) with magic sum **S _{12×12}(1,6,9) := 7916909**. Blocks of order 4 are magic squares with different magic sums.

### Example 11. The Digits (2,5,8)

It is **universal semi-magic** square of order 12 in 3-digits (2,5,8) with magic sum **S _{12×12}(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 12. The Digits (1,6,8,9)

It is **upside-down semi-magic** square of order 12 in 4-digits (1,6,8,9) with magic sum **S _{12×12}(1,6,8,9) := 87769**. Blocks of order 3 are

**semi-magic**squares with different

**semi-magic**sums..

### Example 13. The Digits (0,2,5,8)

It is **universal semi-magic** square of order 12 in 4-digits (0,2,5,8) with magic sum **S _{12×12}(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 14. The Digits (1,6,9)

It is **upside-down semi-magic** square of order 12 in 3-digits (1,6,9) with magic sum **S _{12×12}(1,6,9) := 7916909**. Blocks of order 3 are

**semi-magic**squares with different

**semi-magic**sums.

### Example 15. The Digits (2,5,8)

It is **universal semi-magic** square of order 12 in 3-digits (2,5,8) with magic sum **S _{12×12}(0,2,5,8):=6666660**. Blocks of order 3 are

**semi-magic**squares with different

**semi-magic**sums

### d) 8-Digits Cell Entries: Blocks of Order 3

### Example 16. The Digits (1,8): The Blocks of Order 4

It is **universal pandiagonal** magic square with magic sum: **S _{12×12}(1,8):=599999994**. Block of order 4 are

**pandiagonal**magic square with equal magic sums:

**S**.

_{4×4}(1,8):=199999998### Example 17. The Digits (1,8): The Blocks of Order 6

It is **universal **magic square with magic sum: **S _{12×12}(1,8):=599999994**. Blocks of order 6 are magic square with equal magic sums:

**S**.

_{6×6}(1,8):=299999997### Example 18. The Digits (1,8): The Blocks of Order 3

It is **universal pandiagonal** magic square with magic sum: **S _{12×12}(1,8):=599999994**. Blocks of order 3 are

**semi-magic**squares with different

**semi-magic**sums.

### Example 19. The Digits (2,5): The Blocks of Order 4

It is **universal pandiagonal** magic square with magic sum: **S _{12×12}(2,5):=466666662**. The blocks of order 4 are

**pandiagonal**magic square with equal magic sums, i.e.,

**S**.

_{4×4}(2,5):=155555554### Example 20. The Digits (2,5): The Blocks of Order 6

It is **universal **magic square with magic sum: **S _{12×12}(2,5):=466666662**. Blocks of order 6 are magic square with equal magic sums:

**S**.

_{6×6}(2,5):=233333331### Example 21. The Digits (2,5): The Blocks of Order 3

It is **universal pandiagonal** magic square with magic sum: **S _{12×12}(2,5):=466666662**. Blocks of order 3 are

**semi-magic**squares with different

**semi-magic**sums.

### Example 22. The Digits (6,9): The Blocks of Order 4

It is **universal pandiagonal** magic square with magic sum: **S _{12×12}(6,9):=999999990**. The blocks of order 4 are

**pandiagonal**magic square with equal magic sums, i.e.,

**S**.

_{4×4}(6,9):=333333330### Example 23. The Digits (6,9): The Blocks of Order 6

It is **universal **magic square with magic sum: **S _{12×12}(6,9):=999999990**. Blocks of order 6 are magic square with equal magic sums:

**S**.

_{6×6}(6,9):=499999995### Example 24. The Digits (6,9): The Blocks of Order 3

It is **universal pandiagonal** magic square with magic sum: **S _{12×12}(6,9):=999999990**. Blocks of order 3 are

**semi-magic**squares with different

**semi-magic**sums.

### Bimagic Squares of Order 13

### a) 4-Digits Cell Entries

### Example 25. The Digits (1,6,8,9)

It is **upside-down pandiagonal **magic square of order 13 in 4-digits (1,6,8,9) with magic sum **S _{13×13}(1,6,8,9) := 87769**.

### Example 26. The Digits (1,2,5,8)

It is **universal pandiagonal **magic square of order 13 in 4-digits (1,2,5,8) with magic sum **S _{13×13}(1,2,5,8) := 62216**.

### b) 6-Digits Cell Entries

### Example 27. The Digits (1,6,9)

It is **upside-down pandiagonal **magic square of order 13 in 3-digits (1,6,9) with magic sum **S _{13×13}(1,6,9) := 7897890**.

### Example 28. The Digits (2,5,8)

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:

**SO _{13×13}(2,5,8):=SM_{13×13}(2,5,8):=7249242** and

**SR**.

_{13×13}(2,5,8):=6951945### b) 8-Digits Cell Entries

### Example 29. The Digits (1,8)

It is **pandiagonal universal** magic square of order 13 using only two digits (1,8) with magic sum **S _{13×13}(1,8):=688888882**.

### Example 30. The Digits (2,5)

It is **pandiagonal universal** magic square of order 13 using only two digits (2,5) with with different magic sums, i.e., **S _{13×13}(2,5):=522222217** and

**S**. The sum

_{13×13}(2,5):=488888884**S**refers to mirror looking version.

_{13×13}(2,5):=488888884### Example 31. The Digits (6,9)

It is **upside-down pandiagonal **magic square of order 13 using only two digits (6,9) with magic sum **S _{13×13}(6,9):=1066996689**.

### Pandiagonal magic Squares of Order 14

### a) 4-Digits Cell Entries:

### Example 32. The Digits (1,6,8,9)

It is **upside-down **magic square of order 14 in 4-digits (1,6,8,9) with magic sum **S _{14×14}(1,6,8,9):= 96657**.

### Example 33. The Digits (0,2,5,8)

It is **universal **magic square of order 14 in 4-digits (0,2,5,8) with magic sum **S _{14×14}(0,2,5,8) := 58883**.

### b) 6-Digits Cell Entries

### Example 34. The Digits (1,6,9)

It is **upside-down **magic square of order 14 in 3-digits (1,6,9) with magic sum **S _{14×14}(1,6,9) := 9453444**.

### Example 35. The Digits (2,5,8)

It is **universal **magic square of order 14 in 3-digits (2,5,8) with magic sum **S _{14×14}(2,5,8) := 8048040**.

### b) 8-Digits Cell Entries

### Example 36. The Digits (1,8)

It is **universal **magic square of order 14 in 2-digits (1,8) with magic sum **S _{14×14}(1,8):=699999993**.

### Example 37. The Digits (2,5)

It is **universal **magic square of order 14 in 2-digits (1,8) with magic sum **S _{14×14}(2,5):=544444439**.

### Example 38. The Digits (6,9)

It is **universal **magic square of order 14 in 2-digits (1,8) with magic sum **S _{14×14}(1,8):=1166666655**.

### Magic Squares of Order 15

### a) 4-Digits Cell Entries

### Example 39. The Digits (1,6,8,9)

It is **upside-down **magic square of order 15 in 4-digits (1,6,8,9) with magic sum** S _{15×15}(1,6,8,9) := 105545**. The blocks of order 5 are

**pandiagonal**magic squares with different magic sums.

### Example 40. The Digits (0,2,5,8)

It is **universal **magic square of order 15 in 4-digits (0,2,5,8) with magic sum **S _{15×15}(0,2,5,8) := 66660**. The blocks of order 5 are

**pandiagonal**magic squares with different magic sums.

### b) 6-Digits Cell Entries

### Example 41. The Digits (1,6,9)

It is **upside-down **magic square of order 15 in 3-digits (1,6,9) with magic sum** S _{15×15}(1,6,9) := 9564555**. The blocks of order 5 are

**pandiagonal**magic squares with different magic sums

### Example 42. The Digits (2,5,8)

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:

**SO _{15×15}(2,5,8):=SM_{15×15}(2,5,8):=8936928** and

**SR**.

_{15×15}(2,5,8):=8639631The blocks of order 5 are **pandiagonal** magic squares with different magic sums.

### c) 8-Digits Cell Entries

### Example 43. The Digits (1,8): Blocks of Order 5

It is **universal semi-magic** square of order 15 for 2-digits (1,8) with **semi-magic** sum:** S _{15×15}(1,8):=711111104**. Blocks of order 5 are

**pandiagonal**magic squares with different magic sums.

### Example 44. The Digits (1,8): Blocks of Order 3

It is **universal semi-magic** square of order 15 for 2-digits (1,8) with **semi-magic** sum:** S _{15×15}(1,8):=711111104**. Blocks of order 3 are

**semi-magic**squares with different

**semi-magic**sums.

### Example 45. The Digits (2,5): Blocks of Order 5

It is **universal semi-magic** square of order 15 for 2-digits (2,5) with **semi-magic** sums:** S _{15×15}(2,5):=566666661**. and

**S**. The magic sum

_{15×15}(2,5):=599999994**S**refers to mirror-looking version. Blocks of order 5 are

_{15×15}(2,5):=599999994**pandiagonal**magic squares with different magic sums.

### Example 46. The Digits (2,5): Blocks of Order 3

It is **universal semi-magic** square of order 15 for 2-digits (2,5) with **semi-magic** sums:** S _{15×15}(2,5):=566666661**. and

**S**. The magic sum

_{15×15}(2,5):=599999994**S**refers to mirror-looking version. Blocks of order 3 are

_{15×15}(2,5):=599999994**semi-magic**squares with different

**semi-magic**sums.

### Example 47. The Digits (6,9): Blocks of Order 5

It is **upside-down semi-magic** square of order 15 for 2-digits (6,9) with **semi-magic** sums:** S _{15×15}(6,9):=1233336654**. Blocks of order 5 are

**pandiagonal**magic squares with different magic sums.

### Example 48. The Digits (6,9): Blocks of Order 3

It is **upside-down semi-magic** square of order 15 for 2-digits (6,9) with **semi-magic** sum:** S _{15×15}(6,9):=1233336654**. Blocks of order 3 are

**semi-magic**squares with different

**semi-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.