Reduced Entries Algebraic Pandiagonal Magic Squares of Orders 4 to 8

This work brings algebraic pandiagonal magic squares of orders 4 to 8 for the reduced entries. By reduced or less entries we understand that instead of normal n² entries of a magic square of order n, we are using less numbers. In these cases, 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 type of the magic square. By algebraic, we understand that the work is not only in numbers but in letters followed by numerical examples. For each order, there are more than one result. The general results for the reduced entries algebraic magic squares of orders 3 to 12 refer authors work see the reference list below. These reference also include few results on algebraic pandiagonal magic squares.

Sometimes, we may refer to these magic squares as self-made magic squares. 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 – 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

Summary of the work is given below for each order of magic squares

Reduced Entries Algebraic Magic Squares of Order 3

Below are two magic squares of order 3. One is complete magic square while another is a semi-magic square.

Result 1: Magic Square of Order 3

It is a magic square of order 3. It can be seen in F. Gaspalou’s webs-site http://www.gaspalou.fr/magic-squares/. Below are two examples with even and odd number magic sums

Result 2: Semi-Magic Square of Order 3

It is a semi-magic square of order 3. Below are two examples.

We shall used frequently these two magic squares in construction of pandiagonal magic squares such as of orders 5, 6, 7, etc.

Reduced Entries Algebraic Magic Squares of Order 4

Below are two type of algebraic magic squares of order 4. One is normal and another is striped.

Part 1: Reduced Entries Normal Algebraic Magic Squares of Order 4

Result 1: Magic Square of Order 4

It is a magic square of order 4. It can be seen in F. Gaspalou’s webs-site http://www.gaspalou.fr/magic-squares/. Below are two examples with even and odd number magic sums:

Result 2: Striped Magic Square of Order 4

The algebraic magic square given above is not a pandiagonal. In this case the entries are always integers, while magic sum is always an even numbers. The magic sums of magic rectangles may be even or odd. In this case, S = 2*m, where S is the magic sum of order 4 and m is the width of the magic rectangle of order 2×4. See below two examples:

Part 2: Reduced Entries Algebraic Pandiagonal Magic Squares of Order 4

Result 2: Pandiagonal Magic Square of Order 4

It is a pandiagonal magic square of order 4 for reduced entries. In this case the magic sum should be multiple of 2 otherwise we get fractional values. It is also can also be seen in F. Gaspalou’s web-site http://www.gaspalou.fr/magic-squares/. Below are two examples with even and odd number magic sums:

Result 4: Pandiagonal Striped Magic Square of Order 4

It is a pandiagonal striped magic square of order 4 for reduced entries. In this case the entries are always integers, while magic sum is always an even numbers. In this case, S=2*m, where S is the magic sum of order 4 and m is the width of the magic rectangles of order 2×4. See below two examples:

Reduced Entries Algebraic Pandiagonal Magic Squares of Order 5

Below are three results giving algebraic pandiagonal magic squares of order 5 based on reduced number of entries.

Result 1: Pandiagonal Magic Square of Order 5

It is an algebraic pandiagonal magic square of order 5. It uses only few entries to bring magic squares of order 5. It can given seen in F. Gaspalou’ s webs-site http://www.gaspalou.fr/magic-squares/. It appears in the site without general sums, i.e, instead of 65 is considered. Here, we worked with general sum, i.e., S.

See below two examples of even and odd magic sums:

Result 2: Cornered Algerbraic Pandiagonal Magic Square of Order 5

It is cornered algebraic pandiagonal magic square of order 5, where there is a magic square of order 3 at the upper-left corner. The magic rectangles of orders 2×3 are of equal width and length. In order to get a magic square without decimal entries or fractional entries, we must consider magic sum of magic square of order 3 as multiple of 3. The letters M and S represents the magic sums of orders 3 and 5, where S=5*M/3. See below two examples with even and odd magic sums:

Result 3: Single-Digit Bordered algebraic pandiagonal Magic Square of Order 5

It is a single-digit bordered algebraic pandiagonal magic square of order 5, where there is a magic square of order 3 in the middle. The letters M and S represents the magic sums of orders 3 and 5, where S=5*M/3. See below two examples.

Reduced Entries Algebraic Pandiagonal Magic Squares of Order 6

Below are four results giving algebraic pandiagonal magic squares of order 6 based on reduced entries.

Result 1: Algebraic Pandiagonal Magic Square of Order 6

It is an algebraic pandiagonal magic square of order 6 for reduced entries without any block. The letter S represents the magic sum. In this case the entries are non sequential. We observe that the magic sum is divided by 2, 3 and 6. It requires that the magic sum should be multiple of 6, otherwise some of the entries may be either decimal or fractional numbers. See below two examples:

Result 2: Algebraic Pandiagonal Magic Square of Order 6

It is an algebraic pandiagonal magic square of order 6 for reduced entries. It is composed of four equal sums semi-magic squares of order 3. The letter M represent the semi-magic sum of order 3, and S=2*M is the magic sum of order 6. In order to get non-decimal entries, the magic sum of order 3 should be multiple of 3. See below two examples:

Result 3: Algebraic Cornered Algebraic Pandiagonal Magic Square of Order 6

It is an algebraic cornered pandiagonal magic square of order 6, where the pandiagonal magic square of order 4 is at the upper-left corner. The magic rectangles of order 2 x4 are of equal width and length. Let M represents the magic sum of order 4, then S=3*M/2 represents the magic sums of order 6. Moreover, both the magic squares of orders 4 and 6 are pandiagonal. In order avoid decimal or fractional entries the magic sum of order 4 should be multiple of 4. See below two examples:

Result 4: Algebraic Single-Digit Bordered Pandiagonal Magic Square of Order 6

It is an algebraic single-digit bordered pandiagonal magic square of order 6 embedded with a pandiagonal magic square of order 4. Let M represents the magic sum of order 4 then S=3*M/2 represents the magic sums of order 6. Moreover, both the magic squares of orders 4 and 6 are pandiagonal. In order avoid decimal or fractional entries the magic sum of order 4 should be multiple of 4. See below two examples:

Result 5: Algebraic Pandiagonal Stripled Magic Square of Order 6

It is an algebraic pandiagonal striped magic square of order 6 for reduced entries. It is constructed based three equal sums magic rectangles or strips of order 2×6. The letter m represents the width of each magic rectangle. The magic sum of order 6 is S=3*m. In order to avoid decimal or fractional entries, the magic sum of width of magic rectangle should be multiple of 10. See below two examples:

Reduced Entries Algebraic Pandiagonal Magic Squares of Order 7

Below are four results giving algebraic pandiagonal magic squares of order 7 for reduced entries.

Result 1: Algebraic Pandiagonal Magic Square of Order 7

It is a algebraic pandiagonal magic square of order 7 without any block. In this case, similar to pandiagonal magic square of order 5, don’t require any condition to bring magic square. See below two examples with even and odd magic sums:

Result 2: Algebraic Double-Digit Bordered Pandiagonal Magic Square of Order 7

It is an algebraic double-digit bordered pandiagonal magic square of order 7 for reduced entries embedded with a magic square of order 3. Let the letter M respresents the magic sum of order 3, then T= 7*M/3 is the magic sum of order 7. The four magic rectangles of order 2×3 are of equal sums, i.e., equal width and equal length. To avoid decimal or factional entries, the magic sum of order 3 should be multiple of 3. See below two examples.

Result 3: Algebraic Cornered Pandiagonal Magic Square of Order 7

It is an algebraic cornered pandiagonal magic square of order 7 with pandiagonal magic square of order 5 at upper-left corner. The letter S represents the magic sum of order 5, and T = 7*S/5 is the magic sum of order 7. To avoid decimal or factional entries, the magic sum of order 5 should be multiple of 5. In this case, both the magic squares of orders 5 and 7 are pandiagonal. See below two examples:

Result 3: Algebraic Single-Digit Bordered Pandiagonal Magic Square of Order 7

It is an algebraic single-digit bordered pandiagonal magic square of order 7, where pandiagonal magic square of order 5 is in the inner part. The letter S represents the magic sum of order 5, and T=7*S/5 is the magic sum of order 7. To avoid decimal or fractional entries, the magic sum of order 5 should be multiple of 5. In this case, both the magic squares of orders 5 and 7 are pandiagonal. See below two examples:

Reduced Entries Algebraic Pandiagonal Magic Squares of Order 8

Below are 10 results giving algebraic pandiagonal magic squares of order 8 for the reduced entries.

Result 1: Algebraic Block-Wise Pandiagonal Magic Square of Order 8

It is an algebraic block-wise pandiagonal magic square of order 8 composed of four equal sums pandiagonal magic squares of order 4. The letter S represents the magic sums of magic squares of order 4. In this case, T=2*S is the magic square of order 8. In order to avoid decimal or fractional entries the magic sums of order 4 should be multiple of 2. The magic squares of orders 4 and 8 are pandiagonal. See below two examples:

As written above each example is composed of four equal sums pandiagonal magic squares of order 4.

Result 2: Algebraic Striped Pandiagonal Magic Square of Order 8

It is an algebraic striped pandiagonal magic square of order 8, where the magic rectangles of order 2 x4 are of equal width and length, i.e., mx2*m. In this case the magic sum of order 8 is T = 4*m, where m is the width of the magic rectangle. It also includes 5 magic squares of order 4. See below two examples:

As written above, each example includes five magic squares of order 4. See below:

Result 3: Algebraic Double-Digit Pandiagonal Magic Square of Order 8

It is an algebraic double-digits bordered pandiagonal magic square of order 8 where the magic rectangles of order 2×4 are of equal width and length. The internal magic square of order 4 is also a pandiagonal. The letter M represents the magic sums of magic squares of order 4. In this case, T = 2*M, where T is the magic sums of order 8. Both the magic squares of orders 4 and 8 are pandiagonal.

There is interesting observation that the first two entries i.e., A1 and A2 should be an even number to avoid decimal entries. See below two examples:

Result 4: Algebraic Cornered Pandiagonal Magic Square of Order 8

It is an algebraic cornered pandiagonal magic square of order 8 for reduced entries, where the pandiagonal magic square of order 6 is at the upper-left corner. The magic rectangles of order 2×6 are of equal length and width. The letters T and S represents the magic sums of orders 6 and 8, where T=$*S/3. In order to get integer values entries, the magic sum of order 6 must be a multiple of 6. Here both the magic squares of orders 6 and 8 are pandiagonal. See below two examples:

Result 5: Algebraic Cornered Pandiagonal Magic Square of Order 8

It is an algebraic cornered pandiagonal magic square of order 8, where the magic square of order 4 is at the upper-left corner. The magic square of order 6 is also a cornered magic square. The magic rectangles of orders 2×4 and 2×6 are of equal length and width. The letters M, S and T represents the magic sums of orders 4, 6 and 8, where T=4*S/3 and S=3*M/2, i.e., T=2*M. In order to avoid decimal entries, the magic sum of order 4 must be a multiple of 6. See below two examples:

Result 6: Algebraic Cornered Pandiagonal Magic Square of Order 8

It is an algebraic cornered pandiagonal magic square of order 8, where the pandiagonal magic square of order 6 is at the upper-left corner. It is composed of four equal sums semi-magic squares of order 3. The magic rectangles of order 2×6 are of equal length and width. The letters M, S and T represents the magic sums of orders 4, 6 and 8, where T=4*S/3 and S=2*M, i.e., T=8*M/3. In order to avoid decimal entries, the semi-magic sum of order 3 must be a multiple of 3. See below two examples:

Result 7: Algebraic Single-Digit Bordered Pandiagonal Magic Square of Order 8

It is an algebraic cornered pandiagonal magic square of orders 8 and having pandiagonal magic square of order 6 at the upper-left corner. It contains pandiagonal magic square of order 4 in the middle. The letters M, S and T represents the magic sums of orders 4, 6 and 8 with T = 2* M and S = 3*M/2. This means that the magic sum of order 8 depends on the choice of magic square of order 4, i.e., T=2*M. To avoid decimal entries the magic sum of order 4 should be multiple of 4. In this case, all the three magic squares of orders 4, 6 and 8 are pandiagonal. See below two examples:

Result 8: Algebraic Single-Digit Bordered Pandiagonal Magic Square of Order 8

It is an algebraic cornered pandiagonal magic square of orders 8 and having pandiagonal magic square of order 6 as the upper-left corner. It is composed of three equal sums magic rectangles of order m×3m, where m is the width of the magic rectangle. The letters S and T represents the magic sums of orders 6 and 8. In this case, both the magic squares of orders 6 and 8 are pandiagonal. See below two examples:

Result 9: Algebraic Single-Digit Bordered Pandiagonal Magic Square of Order 8

It is an algebraic single-digit bordered pandiagonal magic square of order 8 embedded with a pandiagonal magic square of order 6. The letters S and T represents the magic sums of orders 6 and 8, where T=4*S/3. Both the magic squares of orders 6 and 8 are pandiagonal. In order de avoid decimal entries, we must have magic sum of order 6 as multiple of 6. See below two examples:

Result 10: Algebraic Single-Digit Bordered Pandiagonal Magic Square of Order 8

It is an algebraic single-digit bordered pandiagonal magic square of order 8 embedded with a pandiagonal magic square of order 6. This magic square of order 6 is composed of four equal sums magic squares of order 3. The letters M, S and T represents the magic sums of orders 3, 6 and 8, where T=4*S/3 and S=2*M, i.e., T=8*M/3. In order de avoid decimal entries, we must have magic sum of order 3 as multiple of 3. See below two examples

Result 11: Algebraic Single-Digit Bordered Pandiagonal Magic Square of Order 8

It is an algebraic single-digit bordered pandiagonal magic square of order 8 embedded with a magic square of order 6. This magic square of order 6 is again a cornered magic square having magic square of order 4 at the upper-left corner. The letters M, S and T represents the magic sums of orders 4, 6 and 8, where T=4*S/3. In order de avoid decimal entries, we must have magic sums of orders 4 and 6 as multiples of 2 and 6 respectively. In this case, only the magic square of order 8 is pandiagonal, while the orders 4 and 6 are just magic squares. See below two examples:

Result 12: Algebraic Single-Digit Bordered Pandiagonal Magic Square of Order 8

It is an algebraic single-digit bordered pandiagonal magic square of order 8 embedded with a pandiagonal magic square of orders 6 and 4. The pandiagonal magic square of order 4 is the inner block. The letters M, S and T represents the magic sums of orders 4, 6 and 8 with T=4*S/3 and S=3*M/2, i.e., T=2*M . This means that the magic sum of order 8 depends on the choice of magic square of order 4, i.e., T = 2*M. To avoid decimal entries the magic sum of order 4 should be multiple of 4. In this case, all the three magic squares of orders 4, 6 and 8 are pandiagonal. See below two examples:

Result 13: Algebraic Single-Digit Bordered Pandiagonal Magic Square of Order 8

It is an algebraic single-digit bordered pandiagonal magic square of orders 8 and having pandiagonal magic square of order 6 in the inner block. It is composed of three equal sums magic rectangles of order m×3m, where m is the width of the magic rectangle. The letters S and T represents the magic sums of orders 6 and 8 with the condition T :=(4/3)* S. To avoid decimal entries the width of magic rectangle of order m×3m should be multiple of 10. In this case, the magic squares of orders 6 and 8 are pandiagonal. See below two examples: