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 48 magic squares, out of them 9 are just magic squares, 10 are semi-magic squares and 29 are pandiagonal magic squares. This work is available online 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 10, Zenodo, September 18, 2025, pp. 1-112, https://doi.org/10.5281/zenodo.17149185
See below the details of the work
Self-Made Algebraic Magic and Sem-Magic Squares of Order 10
This part is already discussed before. See the following links:
- Inder J. Taneja, 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 (old site)
Self-Made Algebraic Pandiagonal Magic Squares of Order 10
Below are 29 pandiagonal magic squares of order 10. These are based on the results given above. It is divided in four parts.
Part 1: Four Equal Sums Padiagonal Magic Squares of Order 5
Result 1: 4 Equal Sum Pandiagonal Magic Squares of Order 5.
It is a pandiagonal magic square of order 10 divided in four equal sums blocks of order 5. These blocks are also pandiagonal magic squares of order 5. The magic sum of order 10 is represented by L=2*S, where S is the sum of magic square of order 5. In this case, the magic sums of order 10 is always an even mumber.
There are no conditions on the magic sum of order 5 to bring this pandiagonal magic square of order 10. See below two examples:
Part 2: Double-Digits Bordered Pandiagonal Magic Squares
Result 2: Double-Digits Bordered Pandiagonal Magic Square of Order 10
It is a double-digit bordered pandiagonal magic square of order 10 having pandiagonal magic square of order 6 in the middle. The four magic rectangles of order 2×6 are of equal width and length. We have both the magic squares of orders 10 and 6 pandiagonal. The magic sum of order 10 is given as L=(5/3)*S, where S is the magic sum of order 6.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 6 should be multiple of 6. See below two examples:
Result 3: Double-Digits Bordered Pandiagonal Magic Square of Order 10
It is a double-digit bordered pandiagonal magic square of order 10 having cornered pandiagonal magic square of order 6 in the middle. It contains a pandiagonal magic square of order 4 at the upper-left corner. The magic rectangles of order 2×4 and 2×6 are of equal width and length in each case. All the magic squares of orders 4, 6 and 10 are pandiagonal. The magic sums of orders 10 and 6 are given as L10×10=(5/2)*M and S6×6=(3/2)* M, where M is the magic sum of order 4.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 4 should be multiple of 20. See below two examples:
Result 4: Double-Digits Bordered Pandiagonal Magic Square of Order 10
It is a double-digit bordered pandiagonal magic square of order 10 having single-digit pandiagonal magic square of order 6 with pandiagonal magic square of order 4. The four magic rectangles of order 2×6 are of equal width and length. All the magic squares of orders 4, 6 and 10 are pandiagonal. The magic sums of orders 10 and 6 are given as L10×10=(5/2)*M and S6×6=(3/2)*M, where M is the magic sum of order 4.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 4 should be multiple of 4. See below two examples:
Result 5: Double-Digits Bordered Pandiagonal Magic Square of Order 10
It is a double-digit bordered pandiagonal magic square of order 10 having pandiagonal magic square of order 6 with four equal sums semi-magic squares of order 3. The four magic rectangles of order 2×6 are of equal width and length. The magic squares of orders 6 and 10 are pandiagonal and the blocks of order 3×3 are semi-magic. The magic sums of orders 10 and 6 are given as L10×10 = (10/3)*M and S6×6=2* M, where M is the semi-magic sum of order 3.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 3 should be multiple of 3. See below two examples:
Result 6: Double-Digits Bordered Pandiagonal Magic Square of Order 10
It is a double-digit bordered pandiagonal magic square of order 10 having striped pandiagonal magic square of order 6 with three equal sums magic rectangles of order 2×6. The four magic rectangles of order 2×6 are of equal width and length. The magic squares of orders 6 and 10 are pandiagonal. The magic sums of orders 10 and 6 are given as L10×10=5*m and S6×6=3*m, where magic rectangle of order m x3m.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the width of magic rectangles of order 2×6 should be multiple of 10. See below two examples:
Part 3: Cornered Pandiagonal Magic Squares
Result 7: Cornered Pandiagonal Magic Square of Order 10
It is a cornered pandiagonal magic square of order 10 having pandiagonal magic square of order 8 at the upper-left corner. It contains four equal sums pandiagonal magic squares of order 4. The two magic rectangles of order 2×8 are of equal width and length. The magic squares of orders 4, 8 and 10 are pandiagonal. The magic sums of orders 10 and 8 are given as L10×10=(5/2)*M and S6×6=(3/2)* M, where M is the magic sum of order 4.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 4 should be multiple of 4. See below two examples:
Result 8: Cornered Pandiagonal Magic Square of Order 10
It is a cornered pandiagonal magic square of order 10 having pandiagonal magic square of order 8 at the upper-left corner. It contains eight equal sums magic rectangles of order 2×4. The two magic rectangles of order 2×8 are of equal width and length. The magic squares of orders 8 and 10 are pandiagonal. The magic sums of orders 10 and 8 are given as L10×10=5*m and S8×8=4*m, where m is the width of the magic rectangle of order mx2m.
In order to bring xthis pandiagonal magic square of order 10 without decimal entries, the width magic rectangle of order sum of order 2×4 should be multiple of 2. Also the first entry, i.e., A1, should be multiple of 2. See below two examples:
Result 9: Cornered Pandiagonal Magic Square of Order 10
It is a cornered pandiagonal magic square of order 10 having double-digit pandiagonal magic square of order 8 at the upper-left corner. It contains a pandiagonal magic square of order 4. The two magic rectangles of order 2×8 are of equal length and width . The magic squares of orders 4, 8 and 10 are pandiagonal. The magic sums of orders 10, 8 and 4 are given as L10×10=(5/2)* M and T8×8 = 2* M, where M is the magic sum of magic square of order 4.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 4 should be multiple of 4. See below two examples:
Result 10: Cornered Pandiagonal Magic Square of Order 10
It is a cornered pandiagonal magic square of order 10 having cornered pandiagonal magic square of order 8 at the upper-left corner. It contains a pandiagonal magic square of order 6. The two magic rectangles of orders 2×6 and 2×8 are of equal length and width in each case. The magic squares of orders 6, 8 and 10 are pandiagonal. The magic sums of orders 10, 8 and 6 are given as L10×10=(5/3)* S and T8×8=(4/3)*S, where S is the magic sum of order 6.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 6 should be multiple of 6. See below two examples:
Result 11: Cornered Pandiagonal Magic Square of Order 10
It is a cornered pandiagonal magic square of order 10 having cornered pandiagonal magic square of orders 8 and 6 at the upper-left corners. It contains a pandiagonal magic square of order 4. The magic rectangles of orders 2×4, 2×6 and 2×8 are of equal length and width in each case. The magic squares of orders 4, 6, 8 and 10 are pandiagonal. The magic sums of orders 10, 8, 6 and 4 are given as L10×10=(5/2)* M, T8×8=2*M and S6×6=(3/2)*M, where M is the magic sum of order 4.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 4 should be multiple of 4. See below two examples:
Result 12: Cornered Pandiagonal Magic Square of Order 10
It is a cornered pandiagonal magic square of order 10 having cornered pandiagonal magic square of orders 8 and 6 at the upper-left corners. The magic square of order 6 is single-digit bordered pandiagonal with pandiagonal magic square of order 4 in the inner part. The magic rectangles of orders 2×6 and 2×8 are of equal length and width in each case. The magic squares of orders 4, 6, 8 and 10 are pandiagonal. The magic sums of orders 10, 8, 6 and 4 are given as L10×10=(5/2)*M, T8×8=2*M and S6×6=(3/2)* M, where M is the magic sum of order 4.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 4 should be multiple of 4. See below two examples:
Result 13: Cornered Pandiagonal Magic Square of Order 10
It is a cornered pandiagonal magic square of order 10 having cornered pandiagonal magic square of orders 8 at the upper-left corners. The magic square of order 6 is a striped pandiagonal. The magic rectangles of orders 2×6 and 2×8 are of equal length and width in each case. The magic squares of orders 6, 8 and 10 are pandiagonal. The magic sums of orders 10, 8 and 6 are given as L10×10:=5*m, T8×8:=5*m and S6×6 := 3*m, where m is the width of magic rectangle of order 2×6.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the width of magic rectangles of order 2×6 should be multiple of 10. See below two examples:
Result 14: Cornered Pandiagonal Magic Square of Order 10
It is a cornered pandiagonal magic square of order 10 having single-digit pandiagonal magic squares of orders 6 and 8 at the upper-left corner with pandiagonal magic square of order 4 in the inner part. The magic squares of orders 4, 6, 8 and 10 are pandiagonal. The magic sums of orders 10, 8, 6 and 4 are given as L10×10:=(5/2)*M, T8×8:=2*M and S6×6:=(3/2)*M, where M is the magic sum of order 4.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 4 should be multiple of 4. See below two examples:
Result 15: Cornered Pandiagonal Magic Square of Order 10
It is a cornered pandiagonal magic square of order 10 having single-digit pandiagonal magic squares of orders 8 at the upper-left corner with pandiagonal magic square of order 6 in the inner part. The magic squares of orders 6, 8 and 10 are pandiagonal. The magic sums of orders 10, 8 and 6 are given as L10×10:=(5/3)*S and T8×8:=(4/3)* S, where S is the magic sum of order 6.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 6 should be multiple of 6. See below two examples:
Result 16: Cornered Pandiagonal Magic Square of Order 10
It is a cornered pandiagonal magic square of order 10 having single-digit pandiagonal magic squares of orders 8 at the upper-left corner with pandiagonal magic square of order 6 formed by four equal sums semi-magic squares of order 3 in the inner part . The magic squares of orders 6, 8 and 10 are pandiagonal. The magic sums of orders 10, 8 and 6 are given as L10×10:=(10/3)*M, T8×8:=(8/3)*M and S6×6:=2*M, where M is the semi-magic sum of order 3.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 3 should be multiple of 3. See below two examples:
Result 17: Cornered Pandiagonal Magic Square of Order 10
It is a cornered pandiagonal magic square of order 10 having single-digit pandiagonal magic squares of orders 8 at the upper-left corner with pandiagonal magic square of order 6 formed by three equal sums magic rectangles of order 2×6 in the inner part . The magic squares of orders 6, 8 and 10 are pandiagonal. The magic sums of orders 10, 8 and 6 are given as L10×10:= 5*m, T8×8:=4*m and S6×6:=3*m, where m is the width of magic rectangle of order 2×6.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the width of magic rectangles of order 2×6 should be multiple of 10. See below two examples:
Part 4: Single-Digit Bordered Pandiagonal Magic Squares
Result 18: Single-Digit Bordered Pandiagonal Magic Square of Order 10
It is a single-digit bordered pandiagonal magic square of order 10 embedded with a pandiagonal magic square of order 8. It contains four equal sums pandiagonal magic square of order 4. The magic squares of orders 4, 8 and 10 are pandiagonal. The magic sums of orders 10, 8 and 4 are given as L10×10 :=(5/2)*M and T8×8:=2*M, where M is the magic sum of order 4.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 4 should be multiple of 4. See below two examples:
Result 19: Single-Digit Bordered Pandiagonal Magic Square of Order 10
It is a single-digit bordered pandiagonal magic square of order 10 embedded with a pandiagonal magic square of order 8. It contains eight equal sums magic rectangles of order 2×4. The magic squares of orders 8 and 10 are pandiagonal. The magic sums of orders 10 and 8 are given as L10×10:=5* m and T8×8:=4*m, where m is the width of magic rectangle of order m×2m.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the width of magic rectangle of order m×2m, i.e., m should be multiple of 2. See below two examples:
Result 20: Single-Digit Bordered Pandiagonal Magic Square of Order 10
It is a single-digit bordered pandiagonal magic square of order 10 embedded with a double-digit pandiagonal magic square of order 8. It contains a pandiagonal magic square of order 4 in the middle. The magic squares of orders 4, 8 and 10 are pandiagonal. The magic sums of orders 10, 8 and 4 are given as L10×10 :=(5/2)* M and T8×8:=2* M, where M is the magic sum of order 4.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 4 should be multiple of 8. See below two examples:
Result 21: Single-Digit Bordered Pandiagonal Magic Square of Order 10
It is a single-digit bordered pandiagonal magic square of order 10 embedded with a cornered pandiagonal magic square of order 8. It contains a pandiagonal magic square of order 6 at the upper-left corner. The magic squares of orders 6, 8 and 10 are pandiagonal. The magic rectangles of orders 2×6 are of equal width and length. The magic sums of orders 10, 8 and 6 are given as L10×10 :=(5/3)*S and T8×8:=(4/3)*S, where S is the magic sum of order 6.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 6 should be multiple of 6. See below two examples:
Result 22: Single-Digit Bordered Pandiagonal Magic Square of Order 10
It is a single-digit bordered pandiagonal magic square of order 10 embedded with a cornered pandiagonal magic square of order 8. It contains a pandiagonal magic square of order 6 at the upper-left corner. It is composed of 4 equal sum semi-magic squares of order 3. The magic squares of orders 6, 8 and 10 are pandiagonal. The magic rectangles of orders 2 × 6 are of equal width and length. The magic sums of orders 10, 8, 6 and 3 are given as L10×10:=(10/3)*M, T8×8:=(8/3)*M and S6×6:=2*M, where M is the semi-magic sum of order 3.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 3 should be multiple of 3. See below two examples:
Result 23: Single-Digit Bordered Pandiagonal Magic Square of Order 10
It is a single-digit bordered pandiagonal magic square of order 10 embedded with a cornered pandiagonal magic square of order 8. It contains a single-digit bordered pandiagonal magic square of order 6 at the upper-left corner having pandiagonal magic square of order 4 in the middle. The magic squares of orders 4, 6, 8 and 10 are pandiagonal. The magic rectangles of orders 2×6 are of equal width and length. The magic sums of orders 10, 8, 6 and 4 are given as L10×10:=(5/2)*M, T8×8:=2*M and S6×6 =(3/2)* M, where M is the magic sum of order 4.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 4 should be multiple of 4. See below two examples:
Result 24: Single-Digit Bordered Pandiagonal Magic Square of Order 10
It is a single-digit bordered pandiagonal magic square of order 10 embedded with a cornered pandiagonal magic square of order 8 having pandiagonal magic square of order 6 at the upper-left corner. It is formed by three equal sums magic rectangles of order 2×6. The magic squares of orders 6, 8 and 10 are pandiagonal. The magic sums of orders 10, 8 and 6 are given as L10×10:=5*m, T8×8:=4*m and S6×6:=3*m, where m is the width of magic rectangle of order m×3m.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the width of magic rectangle of order m×3m, i.e., m should be multiple of 10. See below two examples:
Result 25: Single-Digit Bordered Pandiagonal Magic Square of Order 10
It is a single-digit bordered pandiagonal magic square of order 10 embedded with a cornered pandiagonal magic square of orders 8 and 6. It contains a pandiagonal magic square of order 4 at the upper-left corner. The magic squares of orders 4, 6, 8 and 10 are all pandiagonal. The magic rectangles of orders 2×4 and 2×6 are of equal width and length in each case. The magic sums of orders 10, 8, 6 and 4 are given as L10×10:=(5/2)*M, T8×8:=2* M and S6×6 :=(3/2)* M, where M is the magic sum of order 4.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 4 should be multiple of 4. See below two examples:
Result 26: Single-Digit Bordered Pandiagonal Magic Square of Order 10
It is a single-digit bordered pandiagonal magic square of order 10 and 8 embedded with a pandiagonal magic square of order 6. The magic squares of orders 6, 8 and 10 are pandiagonal. The magic sums of orders 10, 8 and 6 are given as L10×10:=(5/3)*S and T8×8:=(4/3)*S, where S is the magic sum of order 6.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 6 should be multiple of 6. See below two examples:
Result 27: Single-Digit Bordered Pandiagonal Magic Square of Order 10
It is a single-digit bordered pandiagonal magic square of order 10 and 8 embedded with a pandiagonal magic square of order 6. It contains 4 equal sums magic squares of order 3. The magic squares of orders 6, 8 and 10 are pandiagonal. The magic sums of orders 10, 8, 6 and 3 are given as L10×10 :=(10/3)*M, T8×8 :=(8/3)*M and S6×6:=2*M where M is the magic sum of order 3.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 3 should be multiple of 3.
See below two examples:
Result 28: Single-Digit Bordered Pandiagonal Magic Square of Order 10
It is a single-digit bordered pandiagonal magic square of order 10 and 8 embedded with a pandiagonal magic square of order 6. It contains 3 equal sums magic rectangles of orders 2×6. The magic squares of orders 6, 8 and 10 are pandiagonal. The magic sums of orders 10, 8, 6 are given as L10×10:=5*m, T8×8 :=4*m and S6×6 :=3*m where m is the width of magic rectangle of order m×3m.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the width of magic rectangle of order m × 3m should be multiple of 10. See below two examples:
Result 29: Single-Digit Bordered Pandiagonal Magic Square of Order 10
It is a single-digit bordered pandiagonal magic square of order 10, 8 and 6 embedded with a pandiagonal magic square of order 4. The magic squares of orders 4, 6, 8 and 10 are pandiagonal. The magic sums of orders 10, 8, 6 and 4 are given as L10×10:=(5/2)*M, T8×8:=2*M and S6×6:=(3/2)*M, where M is the magic sum of order 4.
In order to bring this pandiagonal magic square of order 10 without decimal entries, the magic sum of order 4 should be multiple of 4. See below two examples:
References
Part 1: Representing Days and Date
- 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 11 Representing Days and Dates of the Year 2025, Zenodo, May 31, 2025, pp. 1-94, https://doi.org/10.5281/zenodo.15564676
- 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.
Part 2: Revised with Examples
- 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.
- 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.
- 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.
- 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.
- 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.
- Inder J. Taneja. Self-Made Algebraic Magic, Semi-Magic and Pandiagonal Magic Squares of Order 10, Zenodo, September 18, 2025, pp. 1-112, https://doi.org/10.5281/zenodo.17149185
- Site Link: Self-Made Algebraic Magic, Semi-Magic and Pandiagonal Magic Squares of Order 10 (new site)
- Site Link: Self-Made Algebraic Magic, Semi-Magic and Pandiagonal Magic Squares of Order 10 (old site)