Self-Made Algebraic Magic, Semi-Magic and Pandiagonal Magic Squares of Order 9

This work brings self-made algebraic magic, semi-magic and pandiagonal magic squares . By self-made or reduced or less entries, we understand that instead of normal n^2 entries of a magic square order n, we are using less numbers, where the magic square is complete in itself. This is just put any integer values for the less entries, one will get always a magic square. Moreover, in these situations the entries are no more sequential numbers. These entries are non-sequential positive  and negative  numbers. In some cases, these may be decimal or fractional values depending on the way of chosing the entries. Sometime to avoid decimal or fractional entries we apply certain conditions. These conditions depends on the types of magic squares. The name self-made is not known in the literature of magic squares. It is being introduced for first time here in this work. The work is based on different types of magic squares, i.e., pandiagonal, block-wise, cornered, single-digit bordered, double-digit bordered, etc. It is not necessary, but we worked with magic rectangles with equal width and length for the same category within a magic square. If we relax this condition, i.e., by considering only equality of width, still we have good results. For more details refer author’s previous works. Previously, the author brought similar kind of work for the orders 3 to 12, specially for the for the dates and days of the year 2025, where the dates are few entries and days are the sums of magic squares. Total there are 46 magic squares, out of them 10 are just magic squares, 9 are semi-magic later these 9 are converted to magic squares,  and 18 are pandiagonal magic squares. This work is online available at the following links: 

Once again, self-made means that they are complete in themselves: once you choose the entries and the magic sum, a magic square will always result. These squares can contain integer, decimal, or fractional values.

For more details see the link given below:

Inder J. Taneja. Self-Made Algebraic Magic, Semi-Magic and Pandiagonal Magic Squares of Order 9, Zenodo, August 27, 2025, pp. 1-92, https://doi.org/10.5281/zenodo.16955571.

See below the details of the work

Self-Made Algebraic Magic and Sem-magic Squares of Order 9

This part is already discussed before. See the following links:

• Inder J. Taneja, Reduced Entries Magic and Semi-Magic Squares of Orders 3, 5, 7 and 9, Zenodo, July 01, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.15783321.
  • Site Link: Reduced Entries Algebraic Magic Squares of Orders 3, 5, 7 and 9 (new site)
  • Site Link: Reduced Entries Algebraic Magic Squares of Orders 3, 5, 7 and 9 (old site)

Self-Made Algebraic Pandiagonal Magic Squares of Order 9

Below are 18 pandiagonal magic squares of order 9. These are based on the results given above

Result 1. Pandiagonal Magic Square of Order 9: 9 Equal Sum Magic Squares of Order 3.

It is a pandiagonal magic square of order 9 with nine equal sums magic squares of order 9. Interestingly, the middle value of all these nine magics squares of order 3 are the same and is M/3, where M is the sums of the magic square of order 3. Obviously, in this case, the magic sum of order nine is given as T_9×9 = 3*M. As we observe that, this magic sum of order 3, i.e., M should be divisible by three, otherwise, we shall have fractional entries. See below two examples.

Result 2. Double-Digits Bordered Pandiagonal Magic Sqyare of Order 9

It is a double-digit bordered pandiagonal magic square of order 9 with 4 equal sums magic rectangles of order 2×5, with inner part as pandiagonal magic square of order 5. Moreover, whole the magic square is depending on the sum of the magic square of order 5. In this case T_9×9 =5*S/9, where S is the magic sum of order 5. Moreover, both the magic squares of orders 5 and 9 are pandiagonal. See below two examples.

Result 3. Double-Digits Bordered Pandiagonal Magic Sqyare of Order 9

It is a double-digit bordered pandiagonal magic square of order 9 with 4 equal sums magic rectangles of order 2×5, with inner part as cornered pandiagonal magic square of order 5, with a magic square of order 3 at the upper-left corner. The two small magic rectangles of order 2×5 are also of equal sums. Moreover, whole the magic square is depending on the sum of the magic square of order 3. In this case T_9×9 = 3*M, where M is the magic sum of order 3. Moreover, both the magic squares of orders 5 and 9 are pandiagonal. See below two examples.

Result 4. Double-Digits Bordered Pandiagonal Magic Sqyare of Order 9

It is a double-digit bordered pandiagonal magic square of order 9 with 4 equal sums magic rectangles of order 2×5, with inner part as single-digit bordered pandiagonal magic square of order 5, embedded with a magic square of order 3. Moreover, whole the magic square is depending on the sum of the magic square of order 3. In this case, T_5×5 = 5*M/3 and T_9×9=3*M, where M is the magic sum of order 3. Moreover, both the magic squares of orders 5 and 9 are pandiagonal, while the order 3 is just a magic square. See below two examples.

Result 5. Cornered Pandiagonal Magic Square of Order 9

It is a cornered pandiagonal magic square of order 9 with double-digit borered pandiagonal magic square of order 7 at the upper-left corner. It is embedded with a magic square of order 3. We observe that all the magic squares of order 3, 7 and 9 are based on the magic sum of order 3. In this case, T_7×7 = 7*M/3 and T_9×9 = 3*M, where M is the magic sum of order 3. The magic rectangles of orders 2×3 and 2×7 are of equal sums in each case. Moreover, the magic squares of orders 7 and 9 both are pandiagonal. See below two examples.

Result 6. Cornered Pandiagonal Magic Square of Order 9

It is a cornered pandiagonal magic square of order 9, where another cornered pandiagonal magic squares of orders 7 and are at the upper left corners having magic square of order 3 at the upper-left corner. The magic rectangles of orders 2×5 and 2×7 are of equal sums for each category. The magic sums of orders 5, 7 and 9 are given as T_5×5=5*M/3, T_7×7=7*M/3 and L_9×9=3*M where M is the magic sum of order 3. Moreover, the magic squares of orders 5, 7 and 9 are all pandiagonal. See below two examples:

Result 7. Cornered Pandiagonal Magic Square of Order 9

It is a cornered pandiagonal magic square of order 9, where another cornered pandiagonal magic squares of orders 7 and 5 are at the upper left corners having magic square of order 5 as single-digit bordered pandigonal magic square embedded with magic square of order 3. The magic rectangles of orders 2×5 and 2×7 are of equal sums for each category. The magic sums of orders 5, 7 and 9 are given as T_5×5 = 5* M/3, T_7×7 =7*M/3 and L_9×9 = 3*M, where M is the magic sum of order 3. Moreover, the magic squares of orders 5, 7 and 9 are all pandiagonal. See below two examples:

Result 8. Cornered Pandiagonal Magic Square of Order 9

It is a cornered pandiagonal magic square of order 9, where another cornered pandiagonal magic squares of orders 7 and 5 are at the upper-left corners. The magic rectangles of orders 2×5 and 2×7 are of equal sums for each category. The magic sums of orders 7 and 9 are given as T_7×7=7*S/5 and L_9×9 =9*S/5, where S is the magic sum of order 5. Moreover, the magic squares of orders 5, 7 and 9 are all pandiagonal. See below two examples:

Result 9. Cornered Pandiagonal Magic Square of Order 9

It is a cornered pandiagonal magic square of order 9, where pandiagonal magic squares of orders 7 is at the upper-left corner. In this case, the magic rectangles of orders 2 × 7 are of equal sums. The magic sum of order is given as T_9×9=9*T/7, where T is the magic sum of order 7. Both the magic squares of orders 7 and 9 are pandiagonal. See below two examples:

Result 10. Cornered Pandiagonal Magic Square of Order 9

It is a cornered pandiagonal magic square of order 9, where single-digit pandiagonal magic square of order 7 is at the upper-left corner embedded with a pandiagonal magic square of order 5. The magic rectangles of order 2×7 are of equal sums. The magic sum of orders 7 and 9 are given as T_7×7=7*S/5 and L_9×9=9*S/5, where S is the magic sum of order 5. The magic squares of orders 5, 7 and 9 are all pandiagonal. See below two examples:

Result 11. Cornered Pandiagonal Magic Square of Order 9

It is a cornered pandiagonal magic square of order 9, where single-digit bordered pandiagonal magic square of order 7 is at the upper-left corner embedded with a cornered pandiagonal magic square of order 5 having magic square of order 3 at upper-left corner. The magic rectangles of order 2×3 and 2×7 are of equal sums for each category. The magic sum of orders 5, 7 and 9 are given as S_5×5=5*M/3, T_7×7=7*M/3 and L_9×9=3*M, where M is the magic sum of order 3. The magic squares of orders 5, 7 and 9 are pandiagonal. See below two examples:

Result 12. Cornered Pandiagonal Magic Square of Order 9

It is a cornered pandiagonal magic square of order 9, where a single-digit pandiagonal magic square of order 7 is at the upper-left corner embedded with pandiagonal magic square of order 5 and a magic square of order 3 as an inner magic square. The magic rectangles of order 2×7 are of equal sums. The magic sum of orders 5, 7 and 9 are given as S_5×5=5*M/3, T_7×7=7*M/3, and L_9×9=3* M, where M is the magic sum of order 3. The magic squares of orders 5, 7 and 9 are pandiagonal. See below two examples:

Result 13. Single-Digit Bordered Pandiagonal Magic Square of Order 9

It is a single-digit bordered pandiagonal magic square of order 9 embedded with a double-digit pandiagonal magic square of order 7 having magic square of order 3 in the inner part. The magic rectangles of order 2×3 are of equal sums. The magic sum of orders 7 and 9 are given as T_7×7=7* M/3, and T_9×9=3* M, where M is the magic sum of order 3. The magic squares of orders 7 and 9 are pandiagonal. See below two examples:

Result 14. Single-Digit Bordered Pandiagonal Magic Square of Order 9

It is a single-digit bordered pandiagonal magic square of order 9 embedded with a cornered pandiagonal magic square of order 7 having magic square of order 5 and at the upper left-corner. In each case, the magic rectangles of order 2×3 and 2×5 are of equal sums. The magic sum of orders 5, 7 and 9 are given as S_5×5=5*M/3, T_7×7=7*M/3 and L_9×9=3*M, where M is the magic sum of order 3. The magic squares of orders 5, 7 and 9 are pandiagonal. See below two examples:

Result 15. Single-Digit Bordered Pandiagonal Magic Square of Order 9

It is a single-digit bordered pandiagonal magic square of order 9 embedded with a cornered pandiagonal magic square of order 7 having pandiagonal magic square of orders 5 at the upper left-corner. The magic rectangles of order 2×5 are of equal sums. The magic sum of orders 7 and 9 are given as T_7×7 = 7 *S/5 and T_9×9 = 9*S/5, where S is the magic sum of order 5. The magic squares of orders 5, 7 and 9 are pandiagonal. See below two examples:

Result 16. Single-Digit Bordered Pandiagonal Magic Square of Order 9

It is a single-digit bordered pandiagonal magic square of order 9 embedded with a pandiagonal magic squares of orders 7 and 5. The magic sum of orders 7 and 9 are given as T_7×7=7*S/5 and L_9×9=9*S/5, where S is the magic sum of order 5. The magic squares of orders 5, 7 and 9 are pandiagonal. See below two examples:

Result 17. Single-Digit Bordered Pandiagonal Magic Square of Order 9

It is a single-digit bordered pandiagonal magic square of order 9 embedded with a pandiagonal magic square of orders 7. It contains cornered pandiagonal magic square of order 5 having magic square of order 5 at the upper-left corner. The magic sum of orders 5, 7 and 9 are given as T_5×5=5*M/3, T_7×7=7*S/3, and L_9×9=3*M, where M is the magic sum of order 3. The magic squares of orders 5, 7 and 9 are pandiagonal. See below two examples:

Result 18. Single-Digit Bordered Pandiagonal Magic Square of Order 9

It is a single-digit bordered pandiagonal magic square of order 9 embedded with pandiagonal magic squares of orders 5 and 7. The magic square of order 3 is in the inner part. The magic sum of orders 5, 7 and 9 are given as T_5×5=5* M/3, T_7×7=7*M/3 and T_9×9=3*M, where M is the magic sum of order 3. The magic squares of orders 5, 7 and 9 are pandiagonal. See below two examples:

References

Part 1: Day and Dates of the Year – 2025 in Terms of Magic Squares

• Inder J. Taneja, Magic Squares of Orders 3 to 7 in Representing Dates and Days of the Year 2025, Zenodo, May 04, 2025, pp. 1-474, https://doi.org/10.5281/zenodo.15338142.
  • Site Link: Magic Squares of Orders 3 to 7 Representing Dates and Days of the Year 2025 (new site)
  • Site Link: Magic Squares of Orders 3 to 7 Representing Dates and Days of the Year 2025 (old site)
• Inder J. Taneja, Magic Squares of Order 8 Representing Days and Dates of the Year 2025, Zenodo, May 04, 2025, pp. 1-134, https://doi.org/10.5281/zenodo.15338246.
  • Site Link: Magic Squares of Order 8 Representing Days and Dates of the Year 2025 (new site)
  • Site Link: Magic Squares of Order 8 Representing Days and Dates of the Year 2025 (old site)
• Inder J. Taneja, Magic Squares of Order 9 Representing Days and Dates of the Year 2025, Zenodo, May 09, 2025, pp. 1-132, https://doi.org/10.5281/zenodo.15375349.
  • Site Link: Magic Squares of Order 9 Representing Days and Dates of the Year 2025 (new site)
  • Site Link: Magic Squares of Order 9 Representing Days and Dates of the Year 2025 (old site)
• Inder J. Taneja, Magic Squares of Order 10 Representing Days and Dates of the Year 2025, Zenodo, May 21, 2025, pp. 1-59, https://doi.org/10.5281/zenodo.15481738.
  • Site Link: Magic Squares of Order 10 Representing Dates and Days of the Year 2025 (new site)
  • Site Link: Magic Squares of Order 10 Representing Dates and Days of the Year 2025 (old site)
• Inder J. Taneja, Magic Squares of Order 12 Representing Days and Dates of the Year 2025 Zenodo, June 10, 2025, pp. 1-43, https://doi.org/10.5281/zenodo.15631884.
  • Site Link: Magic Squares of Order 12 Representing Dates and Days of the Year 2025 (new site)
  • Site Link: Magic Squares of Order 12 Representing Dates and Days of the Year 2025 (old site).

Part 2: Reduced Entries Agebraic Magic Squares

• Inder J. Taneja, Reduced Entries Magic and Semi-Magic Squares of order 12, Zenodo, June 18, 2025, pp. 1-57, https://doi.org/10.5281/zenodo.15692014.
• Inder J. Taneja, Reduced Entries Magic and Semi-Magic Squares of Orders 3, 5, 7 and 9, Zenodo, July 01, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.15783321.
  • Site Link: Reduced Entries Algebraic Magic Squares of Orders 3, 5, 7 and 9 (new site)
  • Site Link: Reduced Entries Algebraic Magic Squares of Orders 3, 5, 7 and 9 (old site)
• Inder J. Taneja, Reduced Entries Magic and Semi-Magic Squares of Orders 4, 6, 8 and 10, Zenodo, July 05, 2025, pp. 1-85, https://doi.org/10.5281/zenodo.15814675.
  • Site Link: Reduced Entries Algebraic Magic Squares of Orders 4, 6, 8 and 10 (new site)
  • Site Link: Reduced Entries Algebraic Magic Squares of Orders 4, 6, 8 and 10 (olde site)
• Inder J. Taneja, Reduced Entries Algebraic Magic and PanMagic Squares of Order 12, Zenodo, July 23, 2025, pp. 1-74, https://doi.org/10.5281/zenodo.16370556.
  • Site Link: Reduced Entries Algebraic Magic and Panmagic Squares of Order 12 (new site)
  • Site Link: Reduced Entries Algebraic Magic and Panmagic Squares of Order 12 (old site)
• Inder J. Taneja, Reduced Entries Algebraic Semi-Magic Squares of Order 12, Zenodo, July 23, 2025, pp. 1-60, https://doi.org/10.5281/zenodo.15692014.
  • Site Link: Reduced Entries Algebraic Semi-Magic Squares of Order 12 (new site)
  • Site Link: Reduced Entries Algebraic Semi-Magic Squares of Order 12 (old site)
• Inder J. Taneja – Reduced Entries Algebraic Pandiagonal Magic Squares of Orders 4 to 8, Zenodo, August 12, 2025, pp. 1-63, https://doi.org/10.5281/zenodo.16809756.
  • Site Link: Reduced Entries Algebraic Pandiagonal Magic Squares of Orders 4 to 8 (new site)
  • Site Link: Reduced Entries Algebraic Pandiagonal Magic Squares of Orders 4 to 8 (old site)
• Self-Made Algebraic Magic, Semi-Magic and Pandiagonal Magic Squares of Order 9, Zenodo, August 27, 2025, pp. 1-92, https://doi.org/10.5281/zenodo.16955571.
  • Site Link: Self-Made Algebraic Magic, Semi-Magic and Pandiagonal Magic Squares of Order 9 (new site).
  • Site Link: Self-Made Algebraic Magic, Semi-Magic and Pandiagonal Magic Squares of Order 9 (old site).