Abstract Normal magic square is a square composed of consecutive natural numbers starting at 1 and each number appears exactly once, and the number of elements in each row, column, and diagonal is the same. Examples of normal magic squares are Lo-Shu, Agrippa, Duhrer and Khajuraho magic squares. In Javanese tradition, a magic square is also found. The existence of this magic square is found in a book that collects Javanese knowledge, called primbon. The elements contained in the magic square in the primbon are written in Arabic numerals and Arabic characters (Hijaiyah). The numerical value of the magic square can be determined because each Arabic character is assigned a specific numeric value. This research was conducted with a literature study. The objectives of this research are describe some of the magic squares by order 4 that found in the primbon (book of Javanese knowledge) used by the Javanese who inhabit the Java Island in Indonesia, and produces some theorems about magic square that found in primbon. In my research, we obtain a variation of the magic square in the Javanese tradition. From this variation we come up with several lemmas, theorems and conjectures. Keywords: Magic square, Javanese, primbon. 1. Introduction
A magic square of order n is a square matrix of nn× or array that consists of consecutive natural numbers of 1, 2, 3, ..,
2n such that the sum of the elements of each row, column, main diagonal and main back diagonal is the
Proceedings of the 11th Annual International Conference on Industrial Engineering and Operations Management Singapore, March 7-11, 2021
same number, called the magic square, magic sum or line-sum, denoted by ρ or ( )Mσ where ..... 5, 4, 3,n = (Lallo, 2018; Taneja, 2010). The characteristics of a magic square are
∑∑∑∑=
+−===
=====n
iin,i
n
iii
n
jji
n
jij n,...,,iaaaa
11
11121 ; ; ; ; ρρρρ
The sequence of numbers that make up the magic square forms an arithmetic sequence with the first term 1, the
last term 2n , by a difference 1. The sum of the numbers that make up the magic square can be determined by a formula
( )( )bnanS 122
−+= (1)
where a is the first term, n is orde and b is difference. Magic square nn× has n rows and n colums. If all the numbers contained in the magic square nn× are
added up, by equation (1) we have (Teixeira, 2018; Ashhab, 2016):
( )( ) ( )2
111122
32122
22
2 +=⋅−+⋅=++++=
nnnnn....S (2)
Since the magic square has n columns, the sum of each column, row and diagonal can be equalized by dividing the result in equation (2) by the number of rows. The results are given in Equation (3) (Teixeira, 2018; Fahimi and Javadi, 2016; Ashhab, 2016):
( )2
12 +=
nnT (3)
According to equation (3), the sum of each row, column and two diagonals (main diagonal and back main diagonal) of the magic square of size 1010 ....., 44 33 ××× ,, is 15, 34, 65, 111, 175, 260, 369 and 505. This number is called magic constant or magic sum, denoted by σ or ( ).Mσ Characteristics that must be fulfilled by the magic square are
1. the sum of numbers on each row is the same; 2. the sum of numbers on each column is the same; 3. the sum of numbers on each diagonal is the same; 4. the sum of numbers on each row = the sum of numbers on each column = the sum of numbers on each
diagonal.
A magic square by order n is a square matrix M which is an arrangement of 2n numbers such that the number of elements in each row, column and both diagonals is the same (Sidharan and Srinivas, 2012). In general, the
numbers that make up the magic square are the natural numbers 1, 2, ..., 2n where each number that occurs exactly once. This magic square is called normal magic square. Based on the type of magic square, different symbols are used, namely:
1. ( )Mtr states the sum of elements on the main diagonal;
2. ( )Mbtr states the sum of elements on the main back diagonal;
3. ( )M&σ states the number of elements in each row and column in the square matrix M& where M& is obtained
from the square matrix M& by omiting the first and last rows and the first and last columns; 4. ( )Mπ states the product of the elements in each row, column and diagonals; and
5. ( )MC states the element at the center of a square matrix with odd order.
The history of the magic square has started a long time ago. China started with a magic square of size 33×discovered by Lo Shu in 2200 BC (Rupali and Sabharwal, 2015). The magic square of size 44× was discovered by Agrippa, a philosopher in 1510 AD and Albrecht Duhrer an artist in 1514. Furthermore, Benjamin Franklin who lived between 1706 and 1790 found the magic square by order 88× and 1616× (Pasles, 2008).
Proceedings of the 11th Annual International Conference on Industrial Engineering and Operations Management Singapore, March 7-11, 2021
The magic squares that are widely known such as the Lo-Shu Magic square, Duhrer magic square, Agrippa magic square, and Khajuraho magic square (Taneja, 2010). In this article, we will describe some of the magic squares found in the primbon (book of Javanese knowledge) used by the Javanese who inhabit the Java Island in Indonesia. The use of the magic square in Javanese is almost similar to Indian society. It is called a magic square because the numbers written in the box are believed to have magical powers. According to Lallo (2018), magic square by order 3 in India (like Lo-Shu magic square) has the power to find missing people. The 4th order magic square with magic constant 34 is useful for safety during travels; and to reassure a fussy baby, you can use magic square 4th order with magic constant 84.
Magic square was initially viewed as recreational mathematics (Pasles, 2008; Pickover, 2002), but according to Neraadha et al. (2017) has now become a serious study in mathematics. In Javanese tradition as found in primbon, magic square is more used as a talisman/amulet. Aliviana and Abdussakir (2012) consider magic square as a numeric amulet and classify them based on the type of magic square. Aliviana and Abdussakir (2012) found in their research that the types of magic square used in Javanese tradition are (1) perfect magic square, (2) semi-magic square, (3) symmetric magic square, (4) addition- multiplication magic square, and (5) other variations. More details are given in Table 1.
Table 1. Types of magic squares found in Javanese tradition, use and magic constant No Type of
Magic Square
Use Order Magic Constant ( )Mσ
( )Mtr ( )Mbtr ( )M&σ
( )Mπ ( )MC
1 Perfect magic square
Amulet to add blessings
33× 15 15 15 - - -
Amulet for various
properties
33× 1638 1638 1638 - - -
Amulet to find a mate
33× 630 630 630 - - -
Amulet makes speech easier
35× 39 39 39 - - -
2 Semimagic square
Amulet of a prosperous life
66× 1002 1002 730 - - -
3 Symmetric magic square
Amulet for gaining
compassion
44× 177 177 177 177 - -
Amulet to torments the
enemy
44× 250 250 250 250 - -
Amulet for eliminating fear
44× 478 478 478 478 - -
Amulet of various benefits
44× 110 110 110 110 - -
Amulet to attract people
44× 1020 1020 1020 1020 - -
4 Addition-multiplication magic square
Amulet of strength
55× 25 25 25 - 945 -
Amulet of salvation
44× 786 786 786 - 640,085,292 -
5 Other variation
Amulet against lust
55× 250 250 250 - - 0
Amulets increase fortune
55× 65 65 65 - - 0
Amulet against lust
55× 2267 2267 2267 - - 0
Amulet revenge 55× 585 585 585 - - 0
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Mathematically, the magic square is not something extraordinary because it can be traced to how it was creation. However, from the point of view of the Javanese who believe it, each magic square has different uses.
2. Methods This research was conducted with a literature study. The stages in this research include: 1. Describe the types of magic square
This section describes how to classify magic square based on the magic constant type. Examples are given for each type of magic square.
2. Describe Lo-Shu, Agrippa, Duhrer and Khajuraho magic square. This section describes four popular magic squares. The Lo-Shu magic square has an order of 3 while the other three magic squares have an order of 4. This review is necessary because the focus of this article is to discuss the magic square order 4 which is found in the primbon and is used by the Javanese people in their daily life.
3. Give examples of magic squares obtained in Javanese tradition This section describes various examples of magic squares of the order 3, 4, 5 and others. The use of the magic square as a talisman provides information that the Javanese are more familiar with the magic square as a square that has magical and supernatural powers.
4. Analysis of the variant magic square order 4 in the Javanese tradition This section provides an analysis of magic square by order 4 and the theorems obtained from the analysis.
3. Results and Discussion 3.1. Types of Magic Square This definition does not require that the constituent numbers be consecutive natural numbers, even negative numbers are possible. Therefore, the number system used is integers. The following are the types of magic squares obtained from various sources, such as Rupali and Sabharwal 2015; Aliviana and Abdussakir, 2012; Stephen, 1993; and Ward, 1980. For more details, see Fahimi and Javadi (2016) and Ashhab (2016).
1. Perfect magic square or pandiagonal or diabolic: matrix of size nn× so that the number of elements in each row, column and diagonal is the same. An example of the perfect magic square is the Lo-Shu magic square (Figure 1).
=
618753294
M
Figure 1. Lo-Shu magic square with ( ) 15=Mσ
If the numbers 1, 2,…, 2n appear exactly once, then the square is called a normal magic square. The sum of the diagonal elements in a magic square M is called the trace of the magic square, and is written by ( ).Mtr Meanwhile, the number of elements on the main back diagonal is called the back trace of magic square M , and is written by ( ).Mbtr
2. Semi-magic square: matrix of size nn× so that the number of elements in each row and column is the same.
An example of a magic square of size 55× where the sum of each element in each row and column is 65 (Figure 2). In this example, the number of elements on the main back diagonal is ( ) .Mbtr 75=
=
91821125312211019
226204131651423715248171
M
Figure 2. Semi-magic square with ( ) 65=Mσ and ( ) .Mbtr 75=
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3. Symmetric magic square: a matrix of size nn× so that the number of elements in each row, column and diagonal is the same and the sum of all two cells including the cell in the center is the same. This magic square is also called an associative magic square. Agrippa, Duhrer, and Khajuraho magic squares are symmetric magic squares (Figure 3).
=
133216810115
12679115144
1M ,
=
11415412769811105
132316
2M ,
=
41569510316
118132141127
3M ,
=
16231351110897612414151
4M
Figure 3. Symmetric magic square with ( ) 34=iMσ for 4321 ,,,i = The symmetrical magic square is the perfect magic square as long as the additional sum of any two cells including the cell in the center is the same. It can be concluded that the perfect magic square is not necessarily a symmetric magic square. View the Agrippa magic square measuring 44× . The magic square
can be divided into squares measuring 22× (Figure 4) where the number of elements ( )M&σ is equal to the
number of elements in each row, column and two diagonals of the magic square i.e. ( ) 34=Mσ
79
144
126115
216
115
133810
101167
Figure 4. The square obtained from the magic square is symmetrical with ( )=M&σ ( ) 34=Mσ Symmetric magic square is also called associative magic square.
4. Concentric magic square (bordered): matrix of size nn× , such that if we delete the top and bottom rows, as well as the first and last columns, then we get another magic square regardless of type. From the example below, a magic square of size 77× , eliminating the first and last rows, and the first and last columns, we get the perfect magic square of size 55× , where the sum of main diagonal and main back diagonal is 125 (Figure 5).
Figure 5. Concentric magic square with ( ) 1757 =Mσ and ( ) ( ) 125== MbtrMtr
5. Zero magic square: a matrix of size nn× such that the number of elements in each row, column and diagonal is zero. This magic square contains negative numbers. An example of a magic square of this type is given in Figure 6.
−−
−−
143202341
Figure 6. Zero magic square
6. Geometric/multiplication magic square: matrix of size nn× such that the product of the elements in each row, column and diagonal is the same. The product of elements in each row, column and diagonal is
Proceedings of the 11th Annual International Conference on Industrial Engineering and Operations Management Singapore, March 7-11, 2021
( ) 496746.M =π (Aliviana. And Abdussakir, 2012; Rupali and Sabharwal, 2015). An example of a geometric magic square is given in Figure 7.
21444854216123681082472416186432
Figure 7. Geometric magic square with ( ) 496746.M =π It is possible to have a geometric magic square where the product of the elements in one or both diagonals is
different from the product of the elements in a row and column.
7. Addition-multiplication magic sqaure: a matrix of size nn× such that the sum of the numbers in each row, column and diagonal is the same and the multiplication of numbers in each row, column and diagonal is the same (Stephens, 1993: 6-7). An example of this square is Figure 8. In this example, the magic sum is 840 and the magic product is 2,058,068,231,856,000
Figure 8. Addition-multiplication magic sqaure with ( ) 840=Mσ and ( ) 0008562310680582 ,,,,,M =π
8. IXOHOXI magic square: matrix of size nn× such that the number of elements in each row, column and
diagonal is 19998. The interesting thing is that if it is reversed (the top one below and vice versa) it still reads as a magic square. An example of the IXOHOXI magic square in Figure 9 (Aliviana and Abdussakir, 2012):
3.2. Lo-Shu, Agrippa, Duhrer and Khajuraho Magic Square In 1510, Heinrich Cornelius Agrippa produced seven magic squares sizes 33× to .99× These magic squares are shown in Figure 10.
618753294
133216810115
12679115144
1521962322141181092113517
1682512432072411
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Figure 10. Agrippa magic square by order 3 to 9 Look at the magic square with n is odd number. The mean values for 9 7 5 3 ,,,n = are 5, 13, 25 and 41. These values are obtained from the formula
2
1 lueCenter va2n+
= ; .... 9 7 5 3 ,,,,n =
Another thing to conclude for a magic square with n an odd is that the number 1 is just below the middle value and the largest number 2n is just above the middle value. In 1514, Duhrer discovered the magic square contained in his book Melencolia I (Figure 11). The order of this magic square is 44× , consists of 16 numbers from 1, 2,…, 16, and the sum of each row, column and diagonal is 34. On the fourth line is the number of the year the magic square was found, which is 1514.
Figure 11 Melencolia I by Albrecht Duhrer and the magic square contained therein
magic square that Agrippa found by swapping columns 1 and 4 obtained
16321351011896712415141
. Rotating 180 degrees
counterclockwise we get the Duhrer's magic square. Indians keep a magic square called Khajuraho magic square. Its form is given in Figure 13.
41569510316
118132141127
Figure 13. Khajuraho magic square (Taneja, 2010) Agrippa, Duhrer and Khajuraho magic square are symmetric magic square because:
1. 1. The sum of each row, column and diagonal is 34, that is ( ) ( ) ( ) .MbtrMtrM 34===σ
2. 2. There are five 22× squares whose total is 34, that is ( ) .M 34=&σ In addition, there are other features found in the Agrippa and Duhrer magic square.
Table 2. Properties of Agrippa and Duhrer magic square Properties Agrippa magic square Duhrer magic square
1. The sum of each row, column and diagonal is 34
133216810115
12679115144
11415412769811105
132316
2. There are four squares, each consisting of four digits totaling 34
79
144
126115
216
115
133810
105316
811
132
15469
114
127
Proceedings of the 11th Annual International Conference on Industrial Engineering and Operations Management Singapore, March 7-11, 2021
1. Pattern 1 - omit rows 3 and 4 as well as columns 3 and 4, obtained
41569510316
118132141127
=
132127
- remove rows 3 and 4 and columns 1 and 2, - remove rows 1 and 2 and columns 3 and 4, - remove rows 1 and 2 and columns 1 and 2,
118141
69316
415510
2. Pattern 2
- remove rows 3 and 4 and columns 1 and 2, - remove rows 3 and 4 and columns 3 and 4, - remove rows 3 and 4 and columns 3 and 4, - remove rows 3 and 4 and columns 3 and 4,
813112
510
118
316
132
156103
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3.3 Magic Square in Javanese Tradition Javanese culture stores and preserves mathematical knowledge in the form of ready-to-use information. This knowledge is written in primbon and is in the form of knowledge that is still used in everyday life today. Prabowo and Sidi (2014) reconstruct this mathematical knowledge and write the results in a book in Indonesian. Previously, Prabowo (2010) had published the results of his research on numbers and numbers in Javanese culture. Two years later, Prabowo (2016) summarized all of his research on Javanese mathematics in the form of articles published through seminars. Other results regarding the contribution of Javanese culture to mathematics can be found in Prabowo et al. (2017), Prabowo et al. (2019) and Prabowo et al. (2020). In this article, we describe the magic square found in the primbon. Primbon is a collection of Javanese knowledge that is applicable. The use of primbon by Javanese people has been going on for generations. As knowledge, the truth of the information presented in the primbon is absolute-intuitive, for those who believe it. Scientifically, the truth is difficult to prove, but Javanese people believe it and use it as a practical guide in navigating their lives. Knowledge in primbon is not theoretical. There is no theory about something that is presented in primbon. Information in primbon is presented as ready-made information. The correctness of the information has been proven beforehand, and after being presented in the primbon, the information becomes true forever. However, that truth is the kind of truth that is believed. One person may not believe the correctness of the information in the primbon, but that person also cannot pinpoint the error. If there is information that is not in accordance with reality, that fact does not make the primbon wrong for those who practice it. So, once put in the primbon, it is believed to be true forever, until the people who own the knowledge no longer use it. Even so, not all Javanese people believe in and use primbon in living their lives. One of the information contained in the primbon is the magic square. Primbon does not explain where the numbers on the magic square came from. Primbon only provides information regarding the benefits of this magic square. For example, in primbon there is a magic square which is a symmetric magic square in Figure 14. The numbers on the magic square are written in Arabic numerals. Primbon does not mention that this shape is an example of a magic square. Primbon simply states that a square box with 16 cells filled with Arabic numerals is a talisman for various properties. Primbon also does not explain why the numbers were chosen, how to place each number, even primbon does not even say that the square is a magic square.
Figure 14. Amulets for various kinds of properties which are one type of magic square
For those who believe it, the magic square in Figure 14 has supernatural powers so it is used as amulet/talisman. As an amulet, the magic square is written in Arabic numerals and Arabic script (Hijaiyah). The following is a table of the letters al Jumal which gives numeric values to Arabic letters (Aliviana and Abdussakir, 2012)
Table 3. Numeric values of Arabic letters (Hijaiyah) Letters Numeric
Value Letters Numeric
Value
Proceedings of the 11th Annual International Conference on Industrial Engineering and Operations Management Singapore, March 7-11, 2021
The following are examples of magic squares whose original form is written in Arabic numerals (Hijaiyah). 3.3.1 Perfect Magic Square Welfare-Enhancing Amulet
Figure 15. Magic square as a prosperity-enhancing talisman
(Aliviana and Abdussakir, 2012)
618753294
( )Mσ = ( )Mtr = ( )Mbtr = 15
Amulet various kinds of efficacy
Figure 16. Magic square as a talisman of various properties
(Aliviana dan Abdussakir, 2012)
543550545548546544547542549
( )Mσ = ( )Mtr = ( )Mbtr = 1638
Amulet speeding up a mate In this talisman there is an error. Written the number 306 should be 206.
Figure 17. Magic square as a talisman to speed up getting a mate
Amulets to Attract People The amulet is a combination of Arabic numerals and Arabic letters.
Figure 24. Magic square as a talisman attracts people
(Aliviana dan Abdussakir, 2012)
9017337938890272113971899709001040
( )Mσ = ( )Mtr = ( )Mbtr = ( )M&σ = 1020
3.3.4 Addition-Multiplication Magic Square Amulet of strength
Figure 25. Magic square as a talisman of strength
(Aliviana dan Abdussakir, 2012)
5791391357357917913513579
( )Mσ = ( )Mtr = ( )Mbtr = 25
( )Mπ = 945 Amulet of Salvation
Figure 26. Magic square as a safety talisman
(Aliviana dan Abdussakir, 2012)
66102289329289329661023292891026610266329289
( )Mσ = ( )Mtr = ( )Mbtr = 786
( )Mπ = 640.085.292 3.3.5 Variation Magic Square A variation of the magic square found in the Javanese tradition is that the numbers in the middle matrix cells are omitted. The magic square is an odd square, involving the number 0. The position 0 is in the middle of the matrix. Mathematically, we can enter any positive number in the blank cell. Amulets increase fortune
Figure 27. Magic square as a talisman increases fortune
(Aliviana dan Abdussakir, 2012)
42221720724236525140111531319219
261211016
( )Mσ = ( )Mtr = ( )Mbtr = 65 ( )MC = 0
Proceedings of the 11th Annual International Conference on Industrial Engineering and Operations Management Singapore, March 7-11, 2021
Figure 28. Magic square as a talisman against lust
(Aliviana dan Abdussakir, 2012)
422217205720923652514019615
188131921926121861016
( )Mσ = ( )Mtr = ( )Mbtr = 250
( )MC = 0 The Amulet to Overcoming Lust
Figure 29. Magic square as a talisman against lust
(Aliviana dan Abdussakir, 2012)
422217222272226236525140221315
22051319219261222031016
( )Mσ = ( )Mtr = ( )Mbtr = 2267
( )MC = 0 The Amulet of Revenge
Figure 30. Magic square as a talisman of revenge
(Aliviana dan Abdussakir, 2012)
422217540754423652514053115
523131921926125211016
( )Mσ = ( )Mtr = ( )Mbtr = 585
( )MC = 0 3.4 Magic Square Order 4 in Javanese Tradition In the previous section, we explained the types of magic squares used by Javanese people along with examples. In this section we will explore the order 4 magic square found in primbon. There are five magic squares in the form of a symmetric magic square and an Addition-Multiplication magic square, namely
The original form of the magic square used in the primbon in Figures 21 - 23 is the magic square in Figure 31. This magic square is a normal symmetric magic square, but has a different arrangement from the Agrippa, Duhrer, and Khajuraho magic square. Table 4 illustrates the formation of a magic square in Figure 21-23 based on Figure 31.
Proceedings of the 11th Annual International Conference on Industrial Engineering and Operations Management Singapore, March 7-11, 2021
Figure 31. Normal symmetric magic square which is the reference in making magic square in Primbon The sum of the numbers that make up the normal symmetric magic square satisfies Equation (2), and the number of rows, columns, and diagonals satisfies Equation (3). For a normal symmetric magic square with the order of
44× , the sum of all the constituent numbers is 136 with the sum of each row, column and diagonal being 34. The magic square used in the primbon (Figure 21-23) is still a symmetric magic square, composed of consecutive natural numbers, but starting with the first term greater than 1. Lemma 1:
Let M be a matrix that states a symmetric magic square with the building blocks 221 na,....,a,a +++ ; Ν∈a
and Ν∈a . The number of elements that make up M is ( )
2122
2 ++=
nnanS and the number of elements in
each row, column and diagonal of M is ( )
212 +
+=nnanT .
Proof. It is known that M is a symmetric magic square, thus the elements that make up M are arranged so that the number of each row, column and diagonal is the same. Symmetric magic square M is forme by the elements
221 na,....,a,a +++ . The sum of all these elements is
( ) ( ) ( ) ( )( ) ( )
( )2
1
(2)Persamaan dari ; 21
321
222
2
2
++=
+++++++=
++++++++=
nnan
n....a....aa
na....aaaS
(4)
Furthermore, the sum of each row, column and diagonal is the sum of all elements divided by the number of rows, i.e.
( )
212 +
+==nnan
nST (5)
By using Equations (4) and (5), it is obtained the number of numbers that build a symmetric magic square and the number of each row, column and diagonal in the symmetric magic square (Table 4).
Table 4. The sum of the numbers and the sum of each row, column and diagonal for symmetric magic square 44× found on primbon
A variant of the symmetric magic square is the elimination of one number in the symmetric magic square and replacing it with one number which is a continuation of the last number.
For example, a symmetric magic square is built by 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51. However, the magic square found in the primbon was constructed by 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, by removing the number 40 and replacing it with figure 33. The number chosen as a substitute is the continuation of the last digit. It is important to pay attention to this removal and replacement which causes a shift in location.
The Magic Square builder for the sequence of numbers 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51 is the normal symmetric magic square in Figure 30. If we take k = 37, we get the sum of the builders S = 696 and the sum of each row, column and diagonal T = 174, calculated by equations (4) and (5), respectively. The symmetric magic square obtained is available in Figure 31.
50394045414451384742374836494643
Figure 32. Symmetric magic square for numbers 36, 37, 38, …, 51. If you omit the number 40 in the symmetric magic square (Figure 32), a symmetric magic square is created in Figure 33 by adding the number 52. The positions of the 36, 37, 38 and 39 in the new magic square are the same as the positions in the old magic square. The position occupied by the number 40 in the old magic square will be occupied by 41 people in the new magic square. Etc. The resulting symmetrical magic square is shown in Figure 33. In this magic square, the number of builders is S = 708 and the sum of each row, column and diagonal is T = 177.
If n odd, then 12 +n even. As a result ( )12 +nn even dan ( )
212 +nn
even.
Case 1.1: a odd If a odd and n odd, then an odd. As a result T is odd. Case 1.2: a even If a even and n odd, then an even. As a result T is even. Case 2: n is even
If n even, then 12 +n odd. As a result ( )12 +nn even and ( )
212 +nn
even.
Case 2.1: a odd If a odd and n even, then an even. As a result T is even. Case 2.2: a even If a even and n even, then an even. As a result T is even. Furthermore, it is known that M by order 44× , means n is even. Base don case 2, T is even. Theorem 1: Suppose M is a matrix with the order nn× which is a normal symmetric magic square. Let f be a number or
numbers which is omitted by 21 nf ≤≤ . Let g be a number or numbers which is omitted. If M~ is a symmetric
magic square obtained from M with these omissions and substitutions, then the sum of numbers that make up M~
is ( ) ( )knmnnS~ n −⋅++= 1222
; n,...,,m 21= represents the number of digits omitted and ,...,,k 210= represents the group numbering. Before being given proof, based on Theorem 1, the numbers in M which are less than f occupy the same position
in M~ and the numbers in M which are greater than or equal to f take the same position in M~ but the number
increases by .m . The sum of the numbers per row, column and diagonal is ( ) ( )knmnT~ n −⋅++= 122
Proof:
Let f be a number or numbers omitted, where 21 nf ≤≤ . The number of omitted is 1, 2, or n , so n,...,,m 21= represents the of numbers omitted. Furthermore, f can be denoted by
∑=
+=m
iiknf
1
where n represents the order matrix M and k denotes the group numbering, ,...,k 10= The explanation of the group k numbering is as follows. For example, in normal symmetric magic square M with the order 44× which is built by the numbers 1, 2,…, 16, the first group is for 0=k so that the number or numbers that are omitted is one of ( ) ( ) ( ) ( ){ }4321321211 ,,,,,,,., . The second group is for 1=k where the number or
digits omitted is one of ( ) ( ) ( ) ( ){ }8765765655 ,,,,,,,., . Etc. Basically, when referring to the orde M , then the largest
k value i 1−n , but any value can be chosen for k .
Suppose g is a number or numbers that replace f . Of course, the value 2ng ≥ so
Proceedings of the 11th Annual International Conference on Industrial Engineering and Operations Management Singapore, March 7-11, 2021
Theorem 2: Suppose M is a matrix of order nn× in the form of a symmetric magic square with the elements
221 na,....,a,a +++ where Ν∈> a,a 0 . Let f be a number or numbers which is omitted where 21 nafa +≤≤+ . Let g is the number or numbers inserted instead of the omitted numbers. If M~ is the
symmetric magic square obtained from M with these omissions and substitutions, then the number of numbers that
make up M~ is ( ) ( )knmnnanS~ n −⋅+++= 122
2 2; n,...,,m 21= is the number of digits omitted and
,...,,k 210= is for the group numbering. Furthermore, the numbers on M that are less than f occupy the same position on M~ and the numbers on M
that are greater than or equal to f occupy the same position on M~ , but the numbers increase by .m . The sum of
the numbers per row, column and diagonal is ( ) ( )knmnanT~ n −⋅+++= 122
The use of Theorem 2 is found in the magic square for the azimuth of mercy (Figure 20) in the form of a symmetric magic square, composed of numbers 36 to 52 by eliminating the number 40. Using Theorem 2 with
35=a , 4=n , 1=m and 1=k is obtained
( ) ( ) 70812136560122
2 2=++=−⋅+++= knmnnanS~ n
( ) ( ) 177334140122 =++=−⋅+++= knmnanT~ n
For this case, the normal symmetric magic square that builds is the same normal symmetric magic square in Table 5.Thus, the symmetric magic square is
50394045414451384742374836494643
Figure 34 Symmetric magic square for the arrangement of numbers 36, 37, …, 51
where ( )Mσ = ( )Mtr = ( )Mbtr = ( )M&σ = 174
If the number 40 is removed and the number 52 is added, we get a symmetric magic square found in the primbon and in this article it is available as Figure 20. Furthermore, Figure 35 is the symmetric magic square.
Proceedings of the 11th Annual International Conference on Industrial Engineering and Operations Management Singapore, March 7-11, 2021
Figure 35 Symmetric magic square for the arrangement of numbers 36 - 52 without the number 40,
where ( )Mσ = ( )Mtr = ( )Mbtr = ( )M&σ = 177
In the previous section, we explained the normal symmetric magic square and symmetric magic square. In both magic squares, the numbers that compose them are sequential, or in the form of arithmetic sequences with a difference of 1. Several questions can be asked, for example (1) what if the difference between the two consecutive numbers is greater than, (2) what if the numbers used form a group, and (3) in number 1, what if a number is missing. From these questions, several lemmas or theorems can be constructed.
Table 6. Mathematical results obtained from the type and shape of the magic square Types of
Magic Square Conditions Mathematical
Results Normal Symmetric Magic Square (NSMS)
The constituent numbers start at 1 and are consecutive. The composing number starts at 1, consecutive and there are numbers missing.
Persamaan 2 Teorema 1
Symmetric Magic Square (SMS)
The building blocks do not start at 1, but are consecutive. The building blocks do not start with 1, are consecutive and there are numbers missing.
Lemma 1 Teorema 2
Varian dari NSMS The constituent number starts at 1, with a difference of more than 1. The constituent number starts at 1, with a difference of more than 1 and there are numbers missing.
-
Varian dari SMS The building block does not start at 1, with a difference of more than 1. The building block does not start at 1, with a difference of more than 1 and there are numbers missing.
-
Varian dari NSMS The constituent numbers start at 1, and the numbers form groups. The constituent numbers start from 1, the numbers form groups and there are numbers missing.
-
Varian dari SMS The constituent numbers do not start with 1, and the numbers form groups. The composing number does not start from 1, the numbers form groups and there are numbers that are missing.
-
The next variant of the symmetric magic square is the use of numbers that do not form an arithmetic sequence
(Figure 23). The numbers used in constructing the magic square are 8, 9, 10, 11, 37, 38, 39, 40, 70, 71, 72, 73, 899, 900, 901, and 902. The numbers can be grouped into
Group 1 8 9 10 11 Group 2 37 38 39 40 Group 3 70 71 72 73 Group 4 899 900 901 902
Suppose that the first number of each group is expressed as d,c,b,a . The symmetric magic square in Figure 24 can be represented by
Proceedings of the 11th Annual International Conference on Industrial Engineering and Operations Management Singapore, March 7-11, 2021
Figure 36. The representation of the symmetric magics square in Figure 23 Based on Figure 36 on the right, the sum of each row, column and diagonal is 6++++ dcba . This means
that by choosing any number d,c,b,a , a symmetric magic square will be obtained with a pattern or arrangement of numbers as in Figure 36. Conjecture 1: Suppose M is an 44× -order magic square made up of 4 groups of numbers, with the first number in each group being different. Suppose that each number in M is different. If the sum of numbers in each row, column and diagonal is 6++++ dcba with d,c,b,a is the first number in each group of numbers, then M is a symmetric magic square. The Ramanujan magic square (Figure 37 left) uses the same pattern in its construction. Ramanujan's magic square is composed of four groups of numbers, namely [9, 10, 11, 12], [16, 17, 18, 19], [22, 23, 24, 25], and [86, 87, 88, 89]. Ramanujan's magic square has the sum of each row, column and diagonal is 139. The first line of Ramanujan's magic square is the birth date of Srinivasa Ramanujan (Sridharan and Srinivas, 2012; Manattu, 2016). A similar result is the magic square at Figure 36 right. Both are made by applying Conjecture 1.
Figure 37. Ramanujan's magic square and Prabowo's 1 magic square
Prabowo's 1 magic square is a perfect magic square with the condition that the additional number of every two cells including the cell in the middle is the same.
We find one more variant, namely the use of four numbers, each of which appears four times and forms the addition-multiplication magic square (AMMS). Figure 27 is an example of this case. Suppose the numbers are
d,c,b,a such that for each row, column and diagonal the number appears only once. Consider Figure 38. In Figure 38, the left is the magic square found in the primbon and is presented as Figure 27. The image on the right is a representation.
66102289329289329661023292891026610266329289
→
abcdcdabdcbabadc
Figure 38. The representation of the symmetric magics square in Figure 25
Based on Figure 38 on the right, the sum of each row, column and diagonal is dcba +++ . This means that by choosing any number for d,c,b,a , a symmetric magic square will be obtained with a pattern or arrangement of numbers as in Figure 38. Next, the product of each row, column and diagonal is abcd . Thus, the magic square that is formed is addition-multiplication magic square, applicable to each selected d,c,b,a value. Conjecture 2: Let M be a magic square of order 44× constructed by 4 numbers where dcba ≠≠≠ . If the sum of the numbers in each row, column and diagonal is dcba +++ , then M is the addition-multiplication magic square.
Proceedings of the 11th Annual International Conference on Industrial Engineering and Operations Management Singapore, March 7-11, 2021
An example of applying Conjecture 2 is Prabowo's 2 magic square (Figure 39). 23 2 19 71 19 71 23 2 71 19 2 23 2 23 71 19
Figure 39. Prabowo’s 2 magic square
4. Conclusions The magic square with the nn× order found was a symmetric magic square and addition-multiplication magic square. We find four types of magic squares.
1. Symmetric magic square which is equivalent to Agrippa, Duhrer and Khajuraho magic square. The numbers in this magic square are arranged sequentially with a difference of 1 and the first number is any natural number other than 1. The sum of each row, column and both diagonals is an even number.
2. Symmetric magic square which is equivalent to Agrippa, Duhrer and Khajuraho magic square. The numbers in this magic square start with the first number any real numbers other than 1 and the next number is different from 1 but there are numbers omitted and so the last digit is added by 1.
3. Symmetric magic square with four groups of numbers where each group is four consecutive numbers. 4. Addition-multiplication magic square composed of 4 numbers, each of which appears four times.
Acknowledgments The authors would like to thank Jenderal Soedirman University (UNSOED) and the Ministry of Research, Technology and High Education of Republic of Indonesia. This work was an individual research. References Aliviana, R. dan Abdussakir, Analisis matematik terhadap azimut numerik, Jurnal Cauchy, vol. 2, no 2, pp. 105-
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Proceedings of the 11th Annual International Conference on Industrial Engineering and Operations Management Singapore, March 7-11, 2021
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50-63, 2018. Ward III, J.E., Vector space of magic square, Mathematics Magazine, vol. 53, no. 2, pp. 108-111, 1980. Biographies Agung Prabowo is a lecturer in the Department of Mathematics, Universitas Jenderal Soedirman, with the field of research are: financial mathematics, survival model analysis and ethnomathematics. Hasriati is a lecturer in the Department of Mathematics, Universitas Riau, with the field of research are actuarial sciences and survival model analysis. Sukono is a lecturer in the Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran. Currently as Chair of the Research Collaboration Community (RCC), the field of applied mathematics, with a field of concentration of financial mathematics and actuarial sciences. Mustafa Mamat is a lecturer in Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Malaysia. The research field of applied mathematics, with a field of concentration of optimization. Abdul Talib Bon is a professor of Production and Operations Management in the Faculty of Technology Management and Business at the Universiti Tun Hussein Onn Malaysia since 1999. He has a PhD in Computer Science, which he obtained from the Universite de La Rochelle, France in the year 2008. His doctoral thesis was on topic Process Quality Improvement on Beltline Moulding Manufacturing. He studied Business Administration in the Universiti Kebangsaan Malaysia for which he was awarded the MBA in the year 1998. He’s bachelor degree and diploma in Mechanical Engineering which his obtained from the Universiti Teknologi Malaysia. He received his postgraduate certificate in Mechatronics and Robotics from Carlisle, United Kingdom in 1997. He had published more 150 International Proceedings and International Journals and 8 books. He is a member of MSORSM, IIF, IEOM, IIE, INFORMS, TAM and MIM.
Proceedings of the 11th Annual International Conference on Industrial Engineering and Operations Management Singapore, March 7-11, 2021
