Reduced Entries Algebraic Magic Squares of Orders 4, 6, 8 and 10

This work brings algebraic magic and semi-magic squares of orders 4, 6, 8 and 10 for reduced entries. By reduced or less entries, we understand that instead of normal n² entries of a magic square order n, we are using less numbers. 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 ractional values depending on the orders of magic squares. In this work we bring different ways of writing magic and semi-magic squares of orders 4, 6, 8 and 10. These are based on three types of magic squares, i.e., cornered, single-digit bordered and double-digit bordered magic squares. except for the magic square of order 4. In all orders we constructed pandiagonal magic squares for reduced entries. It is not necessary, but we worked with magic rectangles with equal widths and lengths of the same category within a magic square. 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. For this kind of work refer the first part of the reference list given below. This work we work is revised and enlarged version of previous works.

For more details see the link given below:
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.

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

Reduced Entries Algebraic Magic Squares of Order 6

Result 1: Magic Square of Order 4

It is 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: Pandiagonal Magic Square of Order 4

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

Reduced Entries Algebraic Magic Squares of Order 6

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

Result 1: Cornered Magic Square

It is a cornered magic square of order 6 having magic square of order 4 at upper-left corner. The two magic rectangles of orders 2×4 are of equal width and length. The letters M and S represents the magic squares of orders 4 and 6 respectively. In order to avoid decimal entries, the magic sums M and S should be of same type, i.e., either even or odd numbers.

Result 2: Normal Magic Square

It is an algebraic magic square of order 6. The letter S represents sum of a magic square of order 6. See below two examples with even and odd magic sums:

Result 3: Pandiagonal Magic Square

It is a pandiagonal magic square of order 6. The letter S represent the magic sum of order 6. In order to get non-decimal entries, the magic sum should be multiple of 6. If the entries in a magic square of order 6 are sequential numbers, then it is impossible to get pandiagonal magic square. In this case the entries are non-sequential type. See below two examples with even and odd magic sums:

Result 4: Pandiagonal Magic Square

It is a pandiagonal magic square of order 6. The letter S represent the semi-magic sum of order 3. In order to get non-decimal entries, the magic sum of order 6 should be multiple of 6. If the entries in a magic square of order 6 are sequential numbers, then it is impossible to get pandiagonal magic square. In this case the entries are non sequential. See an example below.

Result 5: Sem-Magic Square

It is a single-digit bordered semi-magic square of order 6 embedded witha magic square of orders 4. It is semi-magic only at one diagonal. In order to bring it as a magic square we need a condition M=(2/3)*S, where M and S are magic sums of magic squares of orders 4 and 6 respectively. Here M=A7 and S=A15. See below two examples with semi-magic and magic squares of order 6:

Reduced Entries Algebraic Magic Squares of Order 8

Below are few results giving algebraic magic squares of order 8 based on reduced entries.

Part 1: Reduced Entries Magic Squares of Order 8

Result 1: Double-Digit Bordered Magic Square

It is a double-digit bordered magic square of order 8 embedded a magic square of order 4. The four magic rectangles of orders 2×4 are of equal width and length. The letters M and T represents the magic squares of orders 4 and 8 respectively. In order to avoid decimal entries, the magic sums M and T should be of same type, i.e., either even or odd numbers. See below two examples with even and odd numbers magic sums:

Result 2: Cornered Magic Square

It is a cornered magic square of order. The magic rectangles of orders 2×4 and 2×6 are of equal width and length in each case. The letters M, S and T represents the magic squares of orders 4, 6 and 8 respectively. In order to avoid decimal entries, the magic sums M, S and T should be of same type, i.e., either even or odd numbers. Below are two examples with even and odd numbers magic sums:

Result 3: Cornered Magic Square

It is a cornered magic square of order embedded with a single-digit bordered magic square of order 6. It has magic square of order 4 as an inner part. The magic rectangles of orders 2×6 are of equal width and length. The letters M, S and T represents the magic squares of orders 4, 6 and 8 respectively. In order to avoid decimal entries, the magic sums M, S and T should be of same type, i.e., either even or odd numbers. Below are two examples with even and odd numbers magic sums:

Result 4: Cornered Magic Square

It is a cornered magic square of order embedded with a magic square of order 6. The magic rectangles of orders 2×6 are of equal width and length. The letters S and T represents the magic squares of orders 6 and 8 respectively. In order to avoid decimal entries, the magic sums S and T should be of same type, i.e., either even or odd numbers. Below are two examples with even and odd numbers magic sums:

Part 2: Reduced Entries Striped Magic Squares of Order 8

Result 5: Striped Magic Square

It is a striped magic square of order 8 where the magic rectangles of order 2×4 are of equal width and length, i.e., mx2m. In this case the magic sum of order 8 is T=2m. It includes five magic squares of order 4 also. These are specified in an example below:

The magic square sum is T=160 and each magic rectangle of order 2×4 is with magic sum 40×80. It c contains five magic squares as given below:

Part 3: Reduced Entries Pandiagonal Magic Squares of Order 8

Result 6: Pandiagonal Magic Square

It is a pandiagonal magic square of order 8, where the four blocks of order 4×4 are of equal magic sums magic squares. See below an example.

It is a pandiagonal magic square of order 8 with four equal sums magic squares of order 4. The magic square sum is T_8×8 = 200 and each magic square of order 4 is with magic sum S_4×4 = 100. See below these magic squares of order 4.

Part 4: Reduced Entries Semi-Magic Squares of Order 8

Result 7: Single-Digit Bordered Semi-Magic Square

It is a single-digit bordered semi-magic square of order 8 embedded with cornered magic squares of orders 6. It is semi-magic only at one diagonal. In order to bring it as a magic square we need the condition S=(3/4)*T, where S and T are magic sums of magic squares of orders 6 and 8 respectively. See below two examples with semi-magic and magic sums:

Result 8: Single-Digit Bordered Semi-Magic Square

It is a single-digit bordered semi-magic square of order 8 embedded with single-digit bordered semi-magic square of order 6. It is semi-magic only at one diagonal. In order to bring it as a magic square we need the conditions, M=(2/3)*S and S=(3/4)*T, where M, S and T are magic sums of magic squares of orders 4, 6 and 8 respectively. See below two examples with semi-magic and magic sums:

Result 9: Single-Digit Bordered Semi-Magic Square

It is a single-digit bordered semi-magic square of order 8 embedded with a magic square of order 6. I It is semi-magic only at one diagonal. In order to bring it as a magic square we need a condition S=(3/4)*T, where S and T are magic sums of magic squares of orders 6 and 8 respectively. See below two examples with semi-magic and magic sums:

Reduced Entries Algebraic Magic Squares of Order 10

Below are few results giving algebraic magic squares of order 10 based on reduced entries.

Part 1: Reduced Entries Magic Squares of Order 10

Result 1: Double-Digit Bordered Magic Square

It is a double-digit bordered magic square of order 10 embedded with a cornered magic square of order 6 having magic square of order 4 at the upper-left corner. The two magic rectangles of orders 2×4 are of equal width and length. Also the four magic rectangles of orders 2×6 are of equal width and length. The letters M, S and L represents the magic squares of orders 4, 8 and 10 respectively. In order to avoid decimal entries, the magic sums M, S and L should be of same type, i.e., either even or odd numbers. See below two examples with even and odd magic sums:

Result 2: Cornered Magic Square

It is a cornered magic square of order 10 with double-digit bordered magic square of order 8 with magic square of order 4 in the middle. The four magic rectangles of orders 2×4 are of equal width and length. Also the two magic rectangles of orders 2×8 are of equal width and length. The letters M, T and L represents the magic squares of orders 4, 8 and 10 respectively. In order to avoid decimal entries, the magic sums M, T and L should be of same type, i.e., either even or odd. See below two examples with even and odd magic sums:

Result 3: Cornered Magic Square

It is a cornered magic square of order 10 having the magic squares of orders 6 and 8 also cornered magic squares. The two magic rectangles of orders 2×4, 2×6 and 2×8 are of equal width and length in each case. The letters M, S, T and L represents the magic squares of orders 4, 6, 8 and 10 respectively. In order to avoid decimal entries, the magic sums M, S, T and L should be of same type, i.e., either even or odd numbers. See below two examples with even and odd magic sums:

Result 4: Cornered Magic Square

It is also a cornered magic square of order 10 with single-digit bordered magic square of order 6 at the upper left corner containing magic square of order 4. The magic rectangles of orders 2×6 and 2×8 are of equal width and length in each case. The letters M, S, T and L represents the magic squares of orders 4, 6, 8 and 10 respectively. In order to avoid decimal entries, the magic sums M, S, T and L should be of same type, i.e., either even or odd numbers. See below two examples with even and odd magic sums:

Result 5: Cornered Magic Square

It is also a cornered magic square of order 10 with single-digit bordered magic square of order 8 at the upper left corner. The inner part is again a cornered magic square of order 6 embedded with magic square of order 4. The magic rectangles of orders 2×4 and 2×8 are of equal width and length in each case. The letters M, S, T and L represents the magic squares of orders 4, 6, 8 and 10 respectively. In order to avoid decimal entries, the magic sums pairs (M, S) and (T, L) should be of same type, i.e., either even or odd numbers. See below two examples with even and odd magic sums:

Result 6: Cornered Magic Square

It is also a cornered magic square of order 10 with single-digit bordered magic square of order 8 at the upper left corner. The inner part is magic square of order 4. The magic rectangles of order 2×8 are of equal width and length in each case. The letters M, S, T and L represents the magic squares of orders 4, 6, 8 and 10 respectively. In order to avoid decimal entries, the magic sums pairs (T, L) should be of same type, i.e., either even or odd numbers. See below two examples with even and odd magic sums:

Result 7: Cornered Magic Square

It is also a cornered magic square of order 10 with magic square of order 6 at the upper left corner It includes magic square order 8 also as a cornered magic square . The magic rectangles of order 2×6 and 2×8 are of equal width and length in each case. The letters S, T and L represents the magic squares of orders 6, 8 and 10 respectively. In order to avoid decimal entries, the magic sums pairs (T, L) and (S, T) should be of same type, i.e., either even or odd numbers. See below two examples with even and odd magic sums:

Result 8: Cornered Magic Square

It is also a cornered magic square of order 10 with striped magic square of order 8 at the upper left corner. The letters T and L represents the magic squares of orders 8 and 10 respectively. Here the magic sum of order 8 is always an even numbers. To get the integer entries, the magic sum of order 10 should also be an even number, otherwise some of the entries may be decimal numbers. See below two examples with even and odd magic sums. The second example is with decimal entries.

Result 9: Cornered Magic Square

It is also a cornered magic square of order 10 with pandiagonal magic square of order 8 at the left-upper corner. The letters T and L represents the magic squares of orders 8 and 10 respectively. The magic sum of order 8 is always an even numbers as it is formed by four equal sums magic squares of order 4. To get the integer entries, the magic sum of order 10 should also be an even number, otherwise some of the entries may be decimal numbers. See below two examples with even and odd magic sums. The second example is with decimal entries.

Part 2: Reduced Entries Pandiagonal Magic Squares of Order 10

Result 10: Pandiagonal Magic Square

It a pandiagonal magic square of order 10 divided in four equal sums blocks of order 5. These blocks are pandiagonal magic squares of order 5. The magic sum of order 10 is represented by letter T=2*S, where S is the sum of magic squares of order 5. In this case we always have even number magic sums of order 10. See below an example of a pandiagonal magic square of order 10.

The above magic square is formed by four equal sums pandiagonal magic squares of order 5. See below:

Part 3: Reduced Entries Semi-Magic Squares of Order 10

Result 11: Double-Digit Bordered Semi-Magic Square

It is a double-digit bordered semi-magic square of order 10 embedded with a single-digit bordered semi-magic squares of orders 6. It is semi-magic only at one diagonal. In order to bring it as a magic square we need a condition, i.e., M=(2/3)*S, where M and S are magic sums of orders 4 and 6 respectively. See below two examples with semi-magic and magic sums: