This work brings magic, panmagic and semi-magic squares of order 12 for the reduced entries. By reduced or less entries we understand that instead of considering 144 entries in a sequential way, we are using non-sequential entries in less numbers . These non-sequential entries may be positive and/or negative numbers. In some cases, these may be decimal or fractional values. It depends on the type of magic squares. Initially, the work is written in terms of letters instead of numbers and then are followed by examples. These kind of magic squares sometimes we call as algebraic magic squares. The complete work is composed of 57 different types of magic squares of order 12. Out of them 28 are magic, 4 are panmagic or pandiagonal and 25 are semi-magic squares. By panmagic we understand that the magic squares are pandiagonal. We have divided this work in two parts. This part is composed of magic and panmagic squares. The second part is with semi-magic squares of order 12. In case of semi-magic squares some conditions are also explained to change them in magic squares. These are based on four types of magic squares, i.e., pandiagonal, cornered, single-digit bordered and double-digit bordered magic squares. This work also include the idea of magic rectangles. In each magic square, the magic rectangles of same order are equal in width and length. For similar kind of work for the orders 3 to 10 see below the reference list. Previously, the author also brought similar kind of work for the orders 3 to 12 for the dates and days of the year 2025, where the dates are few entries and days are the sums of magic squares. For this kind of work also see the reference list. This is extended and enlarged version of author’s previous works. This is second part on semi-magic squares of order 12. For the first part on magic and panmagic squares refer the link.
Below are the links to download both the works:
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.
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.
This work on semi-magic squares. Every semi-magic squrare is brought to a magic square based on certain conditions. These conditions given by
1. L = 5*R/6
2. T = 4*L/5
3. S = 3*T/4
4. M = 2*S/3
The letter M, S, T, L and R represents the magic or semi-magic sums of orders 4, 6, 8, 10 and 12. All the reduced entries semi-magic appearing in this work are semi-magic only in one diagonal, i.e., upper-diagonal.
Based on above conditions for semi-magic squares to bring as magic squares, the work is divided in four sub-groups:
1. Single-Condition Magic Squares.
2. Double-Conditions Magic Squares.
3. Three-Conditions Magic Squares .
4. Four-Conditions Magic Squares.
Single Condition Algebraic Semi-Magic Squares of Order 12
Below are few examples of reduced entries algebraic semi-magic squares of order 12. In order to bring them as magic squares we applied one of the above four conditions.
Result 1: Single Condition Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of order 12 embedded with a pandiagonal magic square of order 10 composed by four equal sums pandiagonal magic squares of order 5. The letters S, L and R represents respectively the magic or semi-magic sums of orders 5, 10 and 12 respectively. Below are two examples. First is semi-magic and second is magic satisfying the condition 1 as given above, i.e., L=5*R/6, where L and R are magic sums of orders 10 and 12 respectively.
Result 2: Single Condition Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of order 12 embedded with a double-digit bordered magic square of order 10 having a magic square of order 6 in the middle. The letters S, L and R represents respectively the magic or semi-magic sums of orders 6, 10 and 12 respectively. Below are two examples. First is semi-magic and second is magic satisfying the condition 1 as given above, i.e., L=5*R/6, where L and R are magic sums of orders 10 and 12 respectively.
Result 3: Single Condition Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of order 12 embedded with a double-digit bordered magic square of order 10 having a pandiagonal magic square of order 6 composed of four equal sums semi-magic squares of order 3. The letters M, S, L and R represents respectively the magic or semi-magic sums of orders 3, 6, 10 and 12 respectively. Below are two examples. First is semi-magic and second is magic satisfying the condition 1 as given above, i.e., L=5*R/6, where L and R are magic sums of orders 10 and 12 respectively.
Result 4: Single Condition Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of order 12 embedded with double-digit magic square of order 10. The inner part is a cornered magic square of order 6 with magic square of order 4. The letters M, S, L and R represents respectively the magic or semi-magic sums of orders 4, 6, 10 and 12. Below are two examples. First is semi-magic and second is magic satisfying the condition 1 as given above, i.e., L=5*R/6, where L and R are magic sums of orders 10 and 12 respectively.
Result 5: Single Condition Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of order 12 embedded with cornered magic squares of orders 6, 8 and 10. The letters M, S, T, L and R represents respectively the magic or semi-magic sums of orders 4, 6, 8, 10 and 12. Below are two examples. First is semi-magic and second is magic satisfying the condition 1 as given above, i.e., L=5*R/6, where L and R are magic sums of orders 10 and 12 respectively.
Result 6: Single Condition Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of order 12 embedded with cornered magic square of orders 8 and 10, where the magic square of order 6 is in the upper-left corner. The letters S, T, L and R represents respectively the magic or semi-magic sums of orders 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic and second is magic satisfying the condition 1 as given above, i.e., L=5*R/6, where L and R are magic sums of orders 10 and 12 respectively.
Result 7: Single Condition Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of order 12 embedded with cornered magic square of orders 8 and 10. It contains a pandiagonal magic square of order 6 at upper-left corner composed of four equal sums semi-magic squares of order 3. The letters M, S, T, L and R represents respectively the magic or semi-magic sums of orders 3, 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic and second is magic satisfying the condition 1 as given above, i.e., L=5*R/6, where L and R are magic sums of orders 10 and 12 respectively.
Result 8: Single Condition Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of order 12 embedded with a cornered magic square of order 10, where in the upper-left corner we have a double-digit bordered magic square of order 8 having magic square of order 4 at the center. The letters M, S, T, L and R represents respectively the magic or semi-magic sums of orders 4, 6, 8, 10 and 12. Below are two examples. First is semi-magic and second is magic satisfying the condition 1 as given above, i.e., L=5*R/6, where L and R are magic sums of orders 10 and 12 respectively.
Result 9: Single Condition Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of order 12 embedded with a cornered magic square of order 10, where in the upper-left corner we have a striped pandiagonal magic square of order 8. The letters T, L and R represents respectively the magic or semi-magic sums of orders 8, 10 and 12 respectively. Below are two examples. First is semi-magic and second is magic satisfying the condition 1 as given above, i.e., L=5*R/6, where L and R are magic sums of orders 10 and 12 respectively.
Result 10: Single Condition Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of order 12 embedded with a cornered magic square of order 10, where in the upper-left corner we have a pandiagonal magic square of order 8 composed by four equal sums pandiagonal magic squares of order 4. The letters S, T, L and R represents respectively the magic or semi-magic sums of orders 4, 8, 10 and 12 respectively. Below are two examples. First is semi-magic and second is magic satisfying the condition 1 as given above, i.e., L=5*R/6, where L and R are magic sums of orders 10 and 12 respectively.
Result 11: Single Condition Algebraic Semi-Magic Squares of Order 12
It is a double-digit bordered semi-magic square embedded with single-digit bordered semi-magic square of order 8. The inner block is a magic square of order 6. It is semi-magic only due to order 8 being a semi-magic. Below are two examples. First is semi-magic and second is magic satisfying the condition 3 as given above, i.e., S=3*T/4, where S and T are magic sums of orders 6 and 8 respectively.
Result 12: Single Condition Algebraic Semi-Magic Squares of Order 12
It is a double-digit bordered semi-magic square embedded with single-digit bordered semi-magic square of order 8. The inner block is a pandiagonal magic square of order 6 composed with four equal sums semi-magic squares of order 3. It is semi-magic only due to order 8 being a semi-magic. Below are two examples. First is semi-magic and second is magic satisfying the condition 3 as given above, i.e., S=3*T/4, where S and T are magic sums of orders 6 and 8 respectively.
Result 13: Single Condition Algebraic Semi-Magic Squares of Order 12
It is a double-digit bordered semi-magic square embedded with single-digit bordered semi-magic square of order 8. The inner block is a cornered magic square of order 6 with magic square of order 4 at the upper-left corner. It is semi-magic only due to order 8 being a semi-magic. Below are two examples. First is semi-magic and second is magic satisfying the condition 3 as given above, i.e., S=3*T/4, where S and T are magic sums of orders 6 and 8 respectively.
Double Conditions Algebraic Semi-Magic Squares of Order 12
Below are few examples of reduced entries algebraic semi-magic squares of order 12. In order to bring them as magic squares we applied two of the above four conditions.
Result 14: Double Conditions Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic squares of orders 10 and 12 embedded with a pandiagonal magic square of order 8 composed by four equal sums magic squares of order 4. The letters S, T, L and R represents respectively the magic or semi-magic sums of orders 4, 8, 10 and 12 respectively. Below are two examples. First is semi-magic and second is magic satisfying the conditions 1 and 2 as given above, i.e., L=5*R/6 and T=4*L/5.
Result 15: Double Conditions Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of orders 10 and 12 embedded with a striped pandiagonal magic square of order 8. It is composed by 8 equal sums magic rectangles of order 2×4. The letters T, L and R represents the magic or semi-magic sums of orders 8, 10 and 12 respectively. Below are two examples. First is semi-magic and second is magic satisfying the conditions 1 and 2 as given above, i.e., L=5*R/6 and T=4*L/5.
Result 16: Double Conditions Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of orders 10 and 12 embedded with a cornered magic square of order 8 having a magic square of order 6 at upper-left corner. The letters S, T, L and R represents respectively the magic or semi-magic sums of orders 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic and second is magic satisfying the conditions 1 and 2 as given above, i.e., L=5*R/6 and T=4*L/5.
Result 17: Double Conditions Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of orders 10 and 12 embedded with a cornered magic square of order 8 with a pandiagonal magic square of order 6 at upper-left corner. This magic square of order 6 is composed of four equal sums semi-magic squares of order 3. The letters M, S, T, L and R represents respectively the magic or semi-magic sums of orders 3, 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic and second is magic satisfying the conditions 1 and 2 as given above, i.e., L=5*R/6 and T=4*L/5.
Result 18: Double Conditions Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square embedded with a cornered magic squares of orders 6 and 8 having magic square of order 4 at upper-left corner. The letters M, S, T, L and R represents respectively the magic or semi-magic sums of orders 4, 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic and second is magic satisfying the conditions 1 and 2 as given above, i.e., L=5*R/6 and T=4*L/5.
Result 19: Double Conditions Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of order 10 and 12 embedded with cornered magic squares of order 8 with single-digit bordered magic square of order 6 at the upper-left corner having magic square of order 4. The letters M, S, T, L and R represents respectively the magic or semi-magic sums of orders 4, 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic and second is magic satisfying the conditions 1 and 2 as given above, i.e., L=5*R/6 and T=4*L/5.
Result 20: Double Conditions Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of order 12 embedded with double-digit bordered magic square of order 10. The inner part is again a single-digit bordered magic square of order 6 with magic square of order 4. The letters M, S, L and R represents respectively the magic or semi-magic sums of orders 4, 6, 10 and 12. Below are two examples. First is semi-magic and the second is magic satisfying the conditions 1 and 4 as given above, i.e., L=5*R/6 and M=2*S/3.
Result 21: Double Conditions Algebraic Semi-Magic Squares of Order 12
It is a double-digit bordered semi-magic square embedded with a single-digit bordered semi-magic squares of orders 6 and 8. The inner block is a magic square of order 4. The letters M, S, T and R represents respectively the magic or semi-magic sums of orders 4, 6, 8 and 12 respectively. Below are two examples. First is semi-magic and the second is magic satisfying the conditions 3 and 4 as given above, i.e., S = 3*T/4 and M=2*S/3.
Triple Conditions Algebraic Semi-Magic Squares of Order 12
Below are few examples of reduced entries algebraic semi-magic squares of order 12. In order to bring them as magic squares we applied three of the above four conditions.
Result 22: Triple Conditions Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of orders 8, 10 and 12 embedded with a magic square of order 6. The letters S, T, L and R represents the magic or semi-magic sums of orders 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic and the second is magic satisfying the conditions 1, 2 and 3 as given above, i.e., L = 5*R/6, T = 4*L/5 and S = 3*T/4.
Result 23: Triple Conditions Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square of orders 8, 10 and 12 embedded with a pandiagonal magic square of order 6. This magic square of order 6 is again composed of four equal sums semi-magic squares of order 3. The letters M, S, T, L and R represents respectively the magic or semi-magic sums of orders 3, 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic and the second is magic satisfying the conditions 1, 2 and 3 as given above, i.e., L = 5*R/6, T = 4*L/5 and S = 3*T/4.
Result 24: Triple Conditions Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic square so order 8, 10 and 12 embedded with a cornered magic square of order 6 having magic square of order 4 at upper-left corner. The letters M, S, T, L and R represents respectively the magic or semi-magic sums of orders 4, 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic and the second is magic square satisfying the conditions 1, 2 and 3 as given above, i.e., L = 5*R/6, T = 4*L/5 and S = 3*T/4.
Triple Conditions Algebraic Semi-Magic Squares of Order 12
Below is a single example of a reduced entries algebraic semi-magic squares of order 12. In order to bring it as magic square we applied all the four conditions given above.
Result 25: Four Conditions Algebraic Semi-Magic Squares of Order 12
It is a single-digit bordered semi-magic squares of orders 6, 8, 10 and 12 having magic square of order 4 at the inner part. The letters M, S, T, L and R represents the magic or semi-magic sums of orders 4, 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic and the second is magic satisfying the conditions 1, 2, 3 and 4 as given above, i.e., L = 5*R/6, T = 4*L/5, S = 3*T/4 and M=2*S/3.
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 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.
- 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 Magic and PanMagic Squares of Order 12, Zenodo, July 23, 2025, pp. 1-74, https://doi.org/10.5281/zenodo.16370556.
- 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.