Reduced Entries Algebraic Pandiagonal Magic Squares of Orders 4 to 8

Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 Reduced Entries Algebraic Pandiagonal Magic Squares of Orders 4 to 8 Inder J. Taneja1 The whole work as pdf files is available at author’s sites: http://bit.ly/45zFk8R http://bit.ly/4mGXsVf This work is without use of any kind of programming language Abstract 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 n2entries 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. In each case, there are more than one results for each order. The general results for the reduced entries algebraic magic squares of orders 3 to 12 refer authors work [37, 38, 39, 41]. These reference also includes few results on algebraic pandiagonal magic squares. Sometimes, we may refer to these magic squares as self-made magic squares. Self-made implies 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. 1Formerly, Professor of Mathematics, Federal University of Santa Catarina, Florianópolis, SC, Brazil (1978-2012). E-mail: [email protected]; Twitter: @IJTANEJA. Web-sites: http://inderjtaneja.wordpress.com; http://numbers-magic.com 1 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 Contents 1 Introduction 3 2 Magic Square of order 3 4 2.1 Reduced Entries Magic and Semi-Magic Squares of Order 3 . . . . . . . . . . . . . . . . . . . . . . . 4 3 Magic Squares of order 4 6 3.1 Reduced Entries Magic Squares of order 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Reduced Entries Pandiagonal Magic Squares of order 4 . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Summary.................................................... 10 4 Magic Squares of order 5 10 4.1 Reduced Entries Pandiagonal Magic Squares of Order 5 . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Summary.................................................... 14 5 Magic Squares of order 6 14 5.1 Reduced Entries Magic Squares of Order 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.2 Reduced Entries Pandiagonal Magic Squares of Order 6 . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.3 Summary.................................................... 23 6 Magic Squares of Order 7 24 6.1 Reduced Entries Pandiagonal Magic Squares of Order 7 . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.2 Summary.................................................... 33 7 Magic Squares of Order 8 34 7.1 Reduced Entries Pandiagonal Magic Square of Order 8 . . . . . . . . . . . . . . . . . . . . . . . . . 36 7.2 Summary.................................................... 58 2 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 1 Introduction In this paper, the author worked with algebraic pandiagonal magic squares of orders 4 to 8 for the reduced entries. By reduced or less entries we understand that instead of normal n2entries of a magic square of order n, we are using less numbers. Moreover, the entries are no more sequential numbers. These entries are nonsequential positive and negative numbers. In some cases, these may be decimal or fractional values depending on the nature the magic squares. The results are based on five types of magic squares, i.e., pandiagonal cornered, single-digit bordered,double-digit bordered and striped magic squares. It is not necessary, but we worked with magic rectangles with equal widths and lengths of the same category within a magic square. If we relax this condition, i.e., by considering only equality in width, still we have good results. Sometimes, we may refer to these magic squares as self-made magic squares. Self-made implies 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 refer author’s work [31]. The general results for the reduced entries algebraic magic squares of orders 3 to 12 refer authors work [37, 38, 39, 41]. These reference also include few results on pandiagonal algebraic magic squares. From order 5 onwards, there are more than one result of reduced entries pandiagonal magic squares. See the table below: Orders Pandiagonal Magic Squares 42 53 65 76 811 The author [31, 32, 33, 34, 35, 36] also worked on similar kind of work but from different point of view. This work is for the magic squares of 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 the magic squares. 3 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 2 Magic Square of order 3 Example 2.1. A magic square of order 3 is given by 15 81615 35715 49215 15 15 15 15 2.1 Reduced Entries Magic and Semi-Magic Squares of Order 3 Result 2.1. A magic square of order 3 with reduced entries algebraic is given by •Details Knowing only two entries A1 and A2 and the magic sum M, we can construct a magic square of order 3. The some of the entries are decimal numbers. To avoid this we may consider the magic sum as multiple of 3. The Result 2.1 shall be used frequently in works on reduced magic squares of orders 5, 7 and 9. This magic square can seen in the web-site of F. Gaspalou [3]. Below are two examples. One is with decimal entries and another is with magic sums as multiple of 3. These are based on 2.1. Example 2.2. A magic square of order 3 with reduced entries is given by 4 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 The first example is with decimal entries because the magic sum is not a multiple of 3. The second example is with integer values as the sum of magic square is a multiple of 3. The magic square of order 3 given in Result 2.1 is with decimal entries except the magic sum multiple of 3. To get normal entries, below is another result giving reduced entries semi-magic square of order 3 Result 2.2. Asemi-magic square of order 3 with reduced entries algebraic semi-magic is given by •Details Knowing only four entries a1, a2, b1 and b2 and the magic sum M, we can construct a semi-magic square of order 3. Below are two examples of semi-magic squares of order 3 Example 2.3. A magic square of order 3 with reduced entries is given by We shall frequently apply the magic and semi-magic squares of order 3 given in 2.1 and 2.2. 5 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 3 Magic Squares of order 4 Below is an example of a pandiagonal magic square of order 4. It for the sequential entries from 1 to 16. Example 3.1. The famous Khajuraho magic square of order 4 is given by 34 34 34 34 7 12 1 14 34 34 2 13 8 11 34 34 16 3 10 5 34 34 9 6 15 4 34 34 34 34 34 34 3.1 Reduced Entries Magic Squares of order 4 We shall present two different forms of magic squares of order 4 for reduced entries. One is a normal and second is striped magic squares of order 4. Result 3.1. Let’s consider the following magic square of order 4 •Details It is a normal magic square. In this case the entries are always integers independent of magic sum even or odd numbers. The letter S represents the magic sum of order 4. This magic square is also due to F. Gaspalou [3]. See the examples below. 6 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 Example 3.2. Let’s consider following two examples of reduced entries magic squares of order 4 based on the Result 3.1 Result 3.2. Let’s consider following striped magic square of order 4 for reduced entries: •Details The magic square given in Result 3.2 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 m×2mSee below two examples: Example 3.3. Let’s consider two examples of stripedmagic squares of order 4 based on the Result 3.2: 7 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 3.2 Reduced Entries Pandiagonal Magic Squares of order 4 We shall present two different forms of pandiagonal magic squares of order 4 for reduced entries. One is normal magic square while second one is striped magic square. Result 3.3. Apandiagonal magic square of order 4 with reduced entries is given by •Details The magic square given in Result 3.3 is pandiagonal. We observe that, in the last two lines of this result the sum S is divided by 2. In order to avoid decimal or fractional entries, the magic sum should be multiple of 2. This magic square of order 4 is due to F. Gaspalou [3]. See below two examples: Example 3.4. Let’s consider following two examples of pandiagonal magic squares of order 4 based on the Result 3.2: 8 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 Result 3.4. Let’s consider following striped pandiagonal magic square of order 4 for reduced entries: •Details The magic square given in Result 3.4 is pandiagonal. In this case the entries are always integers, while magic sum is always an even numbers. While 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 rectangles of order m×2m. See below two examples: Example 3.5. Let’s consider following two examples of pandiagonal magic squares of order 4 based on the Result 3.2: 9 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 •Details It is an algebraic pandiagonal magic square of order 6 for reduced entries. It constructed based three equal sums magic rectangles or strips of order 2×6. The letter m represents width of each magic rectangle. The magic sum of order 6 is S=3×m. See below two examples: Example 5.2. Let’s consider following two examples representing pandiagonal magic squares of order 6 16 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 5.2 Reduced Entries Pandiagonal Magic Squares of Order 6 Below are four results with examples on reduced entries pandiagonal magic squares of order 6. Result 5.2. Let’s consider a following magic square of order 6 with reduced entries: •Details It is an algebraic pandiagonal magic square of order 6 for reduced entries. 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. The represents 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: Example 5.3. Let’s consider following two examples representing pandiagonal magic squares of order 6 17 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 In Result 5.3, we observe that the magic sum should be multiple of 6, otherwise we may have decimal or fractional numbers entries. This is the reason both the examples are with even number magic sums. Result 5.3. Let’s consider a following magic square of order 6 with reduced entries: •Details It is an algebraic pandiagonal magic square of order 6 with four equal sums semi-magic squares of order 3. The letter M represent the semi-magic sum of order 3, and S=2×Mis the magic sum of order S. In order to get non-decimal entries, the magic sum of order 3 should be multiple of 3. See below two examples: 18 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 Example 5.4. Let’s consider following two examples representing pandiagonal magic squares of order 6 In Result 5.5, we observe that the semi-magic of order 3 should be multiple of 3, otherwise we may have decimal or fractional numbers entries. This is the reason both the examples are with even number magic sums. Any way the semi-magic of order 3 are 48 and 45 respectively. Result 5.4. Let’s consider a following magic square of order 6 with reduced entries: •Details It is an algebraic pandiagonal cornered magic square of order 6, where pandiagonal magic square of order 4 is at the upper-left corner. The magic rectangles of order 2×4are of equal width and length. The letter M 19 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 represents the magic sum of order 4 and S=3 2×Mis the magic sums of order 6. Moreover, both the magic squares of orders 4 and 6 are pandiagonal. Moreover, in order avoid decimal or fractional entries, we have magic sum of order 4 as multiple of 4. See below two examples: Example 5.5. Let’s consider following two examples representing pandiagonal magic squares of order 6 In this case the magic sums of order 6 are 66 and 72 respectively. The magic sums of order 4 are 44 and 48. As written above, we have the magic sums of order as multiple of 4. Result 5.5. Let’s consider a following magic square of order 6 with reduced entries: 20 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 •Details It is an algebraic pandiagonal single-digit bordered magic square of order 6, where the inner is also a pandiagonal magic square of 4. In this case also to avoid decimal or fractional entries we must as magic square of order 4 as multiple of order 4. The letter M represents the magic sum of order 4 and S=3 2×Mis the magic sums of order 6. See below two examples Example 5.6. Let’s consider following two examples representing pandiagonal magic squares of order 6 21 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 In this case the magic sums of order 6 are 72 and 78 respectively. The magic sums of order 4 are 48 and 54. As written above, we have the magic sums of order as multiple of 4. Result 5.6. Let’s consider a following magic square of order 6 with reduced entries: •Details It is an algebraic pandiagonal 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 width of each magic rectangle. The 22 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 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: Example 5.7. Let’s consider following two examples representing pandiagonal magic squares of order 6 5.3 Summary In this section, there are 4 results on pandiagonal magic squares of order 5. Each result is of different type or nature. These results summarizing below along with conditions to avoid decimal or fractional entries: 1. Result 5.2: It is an algebraic pandiagonal magic square composed of 6 with no block. In this case, the magic sum should multiple of 6. 2. Result 5.3:It is an algebraic pandiagonal magic square of order 6 composed with four equal sums semimagic squares of order 3. a magic squares of order 3. It requires magic sum as mutiple of 3. 3. Result 5.4:It is an algebraic cornered pandiagonal magic square formed by 6 having pandiagonal magic square of order 4 at the upper-left corner. The magic rectangles of order 2 ×4 are of equal sums. It requires the magic sum as a multiple of 4. 4. Result 5.5: It is an algebraic single-digit bordered pandiagonal magic square of order 6 having a pandiagonal magic square of order 4 in the middle. It also requires the magic sum as a multiple of 4. 5. Result 5.6: It is an algebraic striped pandiagonal magic square of order 6. The 3 strips are of equal width and length. It also requires that the magic strips should be multiple of 10. 23 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 6 Magic Squares of Order 7 Below are four examples of magic squares of order 7. These are pandiagonal,cornered,single-digit bordered and double-digit bordered magic squares. All the four examples are for the sequential entries from 1 to 49. Example 6.1. Let’s consider following four examples of magic squares of order 7: •Details The first example is a pandiagonal magic square. The second magic square is single-digit bordered magic square embedded with magic squares of orders 5 and 3. The third example is double-digit bordered magic 24 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 square with magic square of order 3 in its inner part. In this case the four magic rectangles of order 2×3are of equal sums. The forth example is cornered magic square having magic squares of orders 3 and 5 at upper-left corner. In this case the each two magic rectangles of order 2×3and 2×5are of equal sums. Based on above four examples, we shall work with reduced entries algebraic pandiagonal magic squares. 6.1 Reduced Entries Pandiagonal Magic Squares of Order 7 Below are four results with examples on reduced entries algebraic pandiagonal magic squares of order 7. Result 6.1. Let’s consider following reduced entries algebraic magic square of order 7: •Details It is an algebraic pandiagonal magic square of order 7 without any block. See below two examples. 25 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 Thus, the magic sums for the orders 7, 5 and 3 of first example are 105, 75 and 27 and for the second example are 112, 80 and 30 respectively. For the order 3 these values are semi-magic Result 6.6. Let’s consider following reduced entries algebraic magic square of order 7: 32 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 •Details It is an algebraic pandiagonal cornered magic square of order 7, where a pandiagonal magic square of order 5 is in the upper-leftcorner, and It is an algebraic single-digit bordered magic square with magic square of order 3 in the middle. The letter M, S and T represents the magic sum of orders 3, 5 and 7, where T=7 3×M and S=5 3×M. To avoid decimal or factional entries, the magic sum of order 3 should be multiple of 3. In this case, both the magic squares of orders 5 and 7 are pandiagonal, while the order 3 is a magic square. See below two examples. Example 6.7. Let’s consider following two examples representing pandiagonal magic squares of order 7 Thus, the magic sums for the orders 7, 5 and 3 of first example are 63, 45 and 27 and for the second example are 77, 55 and 33 respectively. 6.2 Summary In this section, there are 4 results on pandiagonal magic squares of order 7. Each result is of different type or nature. These results are summarizing below along with conditions to avoid decimal or fractional entries: 1. Result 6.1: It is an algebraic pandiagonal magic square composed of 7 with no block. It don’t require any condition to choose the magic sum. 33 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 2. Result 6.2:It is an algebraic double-digit pandiagonal magic square of order 7 having a magic squares of order 3 in the middle. The magic rectangles of order 2 ×3 are of equal sums. It requires the magic sum as a multiple of 3. 3. Result 6.3:It is an algebraic cornered pandiagonal magic square formed by 7 with pandiagonal magic square of order 5 at the upper-left corner. The magic rectangles of order 2 ×5 are of equal sums. It requires magic sums as multiple of 5. 4. Result 6.4: It is an algebraic single-digit bordered pandiagonal magic square of order 7 having a pandiagonal magic square of order 5 in the middle. It also requires magic sums as multiple of 5. 5. Result 6.5: It is an algebraic cornered pandiagonal magic square of order 7 having a magic square of orders 5 and 3 at the upper-left corner. The blocks of orders 5 and 5 are magic and semi-magic squares. It also requires magic sum of order 5 as multiple of 5. 6. Result 6.6: It is an algebraic single-digit bordered pandiagonal magic square of order 7 having a pandiagonal magic square of order 5 in the middle. Both the magic squares orders 5 and 7 are pandiagonal. It requires multiple 3 as the magic sum of order 3. 7 Magic Squares of Order 8 Below are five examples of magic squares of order 8. These are pandiagonal,striped,cornered,single-digit bordered and double-digit bordered magic squares. All the four examples are for the sequential entries from 1 to 64. Example 7.1. Let’s consider following five magic squares of order 8 for the sequential entries from 1 to 64 34 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 •Details The first magic square is pandiagonal with four equal sum pandiagonal magic squares of order 4. The second magic square cornered magic square. The third magic square is single-digit bordered magic squares. The fourth magic square is double-digit bordered magic square. The internal magic squares are of orders 4 and 6. In case of forth example, there is only one internal magic square of order 4. The fifth example is striped magic square of order 8. More details on different kinds of magic squares of order 8 refer author’s work [11, 16, 25, 27, 28]. Based on above magic squares we shall construct reduced entries pandiagonal magic squares of order 8. 35 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 7.1 Reduced Entries Pandiagonal Magic Square of Order 8 Below are ten results on algebraic pandiagonal magic squares of order 8 for the reduced entries. These are of different types, such as, single-digit bordered,double-digit bordered, striped,block-wise,cornered, etc. Result 7.1. Let’s consider following pandiagonal square of order 8 with reduced entries: Details: It is an algebraic 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×Sis the magic square of order 8. In order to avoid fractional entries the magic sums of order 4 should be multiple of 2. Moreover, the magic squares of orders 4 and 8 are pandiagonal See below two examples: Example 7.2. Let’s consider following two examples representing pandiagonal magic squares of order 8. 36 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 According to Result 7.1, the magic sums are 96 and 68, where 48 and 34 are the sums of the magic squares of order 4 respectively. Moreover, the magic squares of orders 4 and 8 are pandiagonal. See below magic squares of order 4: Result 7.2. Let’s consider following pandiagonal square of order 8 with reduced entries: 37 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 Details: It is an algebraic pandiagonal striped magic square of order 8 where the magic rectangles of order 2×4are of equal width and length, i.e., m×2m. In this case the magic sum of order 8 is T=4×m. It includes five magic squares of order 4 also. These are specified below in examples. Example 7.3. Let’s consider following two examples representing pandiagonal magic squares of order 8. 38 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 According to Result 7.2, the magic sums are 100 and 96 with sums of magic rectangles 100 =4×25 and 100 =4×24 respectively, where 25 and 24 are the sums of magic rectangles of order 2×4. Moreover each magic square of order 8 is with five magic squares of order 4. See below More precisely, these are 39 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 Result 7.3. Let’s consider following pandiagonal square of order 8 with reduced entries: 40 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 Details: It is an algebraic double-digits bordered pandiagonal magic square of order 8 where the magic rectangles of order 2×4are 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, otherwise we shall get decimal entries. See below two examples. Example 7.4. Let’s consider following two examples representing pandiagonal magic squares of order 8. 41 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 The respective magic sums of orders 3, 6 and 8 are 48, 96 and 128, and 51, 102 and 136 for the first and second examples. The magic rectangles sums are 30 ×60 and 24 ×72 for the first example and 32 ×96 and 34 ×102 for the second example. Result 7.7. Let’s consider following pandiagonal square of order 8 with reduced entries: 48 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 Details: 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 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×Mand S=3 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: Example 7.8. Let’s consider following two examples representing pandiagonal magic squares of order 8. 49 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 The magic sums of first example of order 4, 6 and 8 are 48, 72 and 96 respectively. The magic sums of the second example of orders 4 6, and 8 are 64, 96 and 128 respectively. Result 7.8. Let’s consider following pandiagonal square of order 8 with reduced entries: 50 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 Details: 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 3×S. 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: Example 7.9. Let’s consider following two examples representing pandiagonal magic squares of order 8. 51 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 The magic sums order 6 and 8 of the first example are 48 and 64, and the second example are 54 and 72 respectively. In both the examples, the magic sums 6 and 8 are pandiagonal. Result 7.9. Let’s consider following pandiagonal square of order 8 with reduced entries: 52 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 Details: 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 composed with 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 3×Sand S=3 2×M. In order de avoid decimal entries, we must have magic sum of order 3 as multiple of 3. See below two examples: Example 7.10. Let’s consider following two examples representing pandiagonal magic squares of order 8. 53 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 The the respective magic sums for the first example for the magic squares of orders 3, 6 and 8 are given as 42, 84 and 112 respectively. For the second example the respective magic sums for the orders 3, 6 and 8 are given as 45, 90 and 120. Result 7.10. Let’s consider following pandiagonal square of order 8 with reduced entries: 54 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 Details: It is an algebraic single-digit bordered pandiagonal magic square of order 8 embedded with a cornered 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 3×S. 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: Example 7.11. Let’s consider following two examples representing pandiagonal magic squares of order 8. 55 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 The the respective magic sums for the first example for the magic squares of orders 4, 6 and 8 are given as 26, 96 and 128 respectively. For the second example the respective magic sums for the orders 4, 6 and 8 are given as 30, 90 and 120. The magic rectangle sums are 70 ×140 for the first example and 60 ×120 for the second example. Result 7.11. Let’s consider following pandiagonal square of order 8 with reduced entries: 56 Inder J. Taneja https://inderjtaneja.com; https://numbers-magic.com; 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 Details: It is an algebraic single-digit bordered pandiagonal magic square of orders 8 and having pandiagonal magic square of order 4 as inner block. The letters M, S and T represents the magic sums of orders 4, 6 and 8 with T=4 3×Sand S=3 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: Example 7.12. Let’s consider following two examples representing pandiagonal magic squares of order 8. 57