Using Magic Borders to Generate Magic Squares

Using Magic Borders to Generate Magic SquaresAuthor(s): BETTY CLAYTON LYONSource: The Mathematics Teacher, Vol. 77, No. 3 (March 1984), pp. 223-226Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27963972 .

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Using Magic Borders to

Generate Magic Squares By BETTY CLAYTON LYON, Kansas State University, Manhattan, KS 66506

Magic

squares and their properties have fascinated people for centuries.

Those who are challenged by magic squares are always searching for new ways to gen erate them. A novel approach to their crea tion is to take an existing magic square and surround it with a magic border. For exam

ple, one can change a 3 3 magic square into a 5 5 magic square by surrounding the smaller square with a magic border. Some examples of fifth-order magic borders

appear in figure 1. Please note that on the

magic borders the opposite entries horizon

tally and vertically add up to 26, except for the corner points, where the diagonal cor ners sum to 26. The number 26 comes quite naturally from the fact that the opposite border entries add up to 52 4-1 for a 5 5

magic square. Any one of the fifth-order magic borders

in figure 1 is easily constructed by a trial and-error method. Since it has been deter mined that the sum of opposite entries and the sum of diagonals is 26, one can write

pairs of numbers whose sum is 26. Select the top row and the right-hand column to

give the magic constant (in this case, 65), then fill in the complementary pairs on the bottom and left-hand side. Sixteen different

configurations are possible for this border.

Now it is time to put in the magic square insert. The insert must be a magic square that sums to 65 26, or 39. Noting the numbers on the border, it is evident that the insert square contains numbers from 9 to 17. A magic square of this type can be obtained by adding 8 to each entry in the 3x3 magic square shown in figure 2. To determine the amount to be added to each entry in the original square to get the insert square, one must consider how many numbers are in the border. There are six teen numbers in the border square: 16 di vided by 2 is 8; so add 8 to each entry in the 3x3 magic square.

Take another look at the new 5x5

magic square. Not only do the opposite sides and diagonal corners of the border

3

20

2

3

1

1 21 1 4 ]... :.-4

24 1f

It

12

13

17

14

15

10

22 2S Vi

Fig. 2. Add 8 to each entry in the magic square, then use a magic border.

March 1984 223



TABLE 1

Sum of opposite Size of border sides and corners Numbers used

Fifth order 26 1-8, 18-25

Seventh order 50 1-12,38-49 Ninth order 82 1-16,66-81

Eleventh order 122 1-20,92-121 nth order n2 + 1 1 to In - 2, n2 - 2n + 3 to n2

add to 26 but the opposite sides and diago nal corners of the inside square total 26, and the center added to itself gives 26. This

particular insert square has twelve differ ent arrangements.

It is possible to expand our construction of magic borders to magic squares of higher order. To continue with magic squares of odd order, a magic border of seventh order is illustrated in figure 3. Note that in the seventh-order border, the sum of opposite sides and the sum of diagonal corners is 50 = 72 + 1.

46

45

44

7

12

11

1 2 3 41 42 40 5 2d 36 13 20 27

35 17 19 26 28

16 18 25 32 34

22 24 31 33 15

23 30 37 14 21

6

43

38 39 4 10 49 48 47 9 8

Fig. 3. A seventh-order magic border in use

Border squares of higher order can be

constructed with a single border or with

several borders. Examples of seventh-order

magic squares and ninth-order magic squares appear in figure 4.

From the previous examples, one could

generalize about the set of numbers that would appear on a magic border. Let be the order of the magic border. The sum of

opposite sides and the sum of diagonal cor ners will always be n2 + 1. The system for

generating magic-square borders for odd

squares could be summarized as shown in table 1.

Another thought about magic borders for odd squares comes from observing the concentric magic squares of seventh order and ninth order that appear in figure 4. If is the order of the magic border, then it is known that the sum of the opposite sides and the sum of diagonal corners of the

magic border will be n2 +1. Place the number (n2 + l)/2 in the center of the insert

square and call it C. We shall now border the center number, C, with a series of con

centric borders. The first border will have

eight numbers : C 4 to C 1 and C + 1 to C + 4. Make a magic square of them by trial and error. The next magic border will

77 1 2 3 4 72 71 70

46 1 45 44 7

12 11

34 17 20

2 3 41 42 40 5 35 13 14 32 31

16 28 21 26 23 25 27 24 29 22

33 30

19 37 36 18 15

6 43 38 39

10 49 48 47 9 8 4

76 75

74 9

16 15 14

62 17 18 19 58 57 56 21 61

60 23 28 27

51 29 30 48 47 32 50

33 36

44 37 42 39 41 43

40 45 38 49 46

35 53 52 34 31

22 59 54 55

26 65 64 63 24 25 20

6 7

8 73 66 67

13 81 8b 79 78 10 11 12 5

Fig. 4. Seventh-order and ninth-order magic squares

224 Mathematics Teacher



have 52 32 = 16 numbers. These numbers range from C 12 to C 5 and from C + 5 to C + 12. The next magic border will have 72 52 = 24 numbers, and so on.

The method for expanding concentric borders from the center number can be con densed by using a formula that gives the number of the upper bound and lower bound for each border. If re is again the order of the magic border and k is the ordi nal number of the border from the center number (first, second, third, etc.), then the

upper bound for each Mi border can be found by

n2 + (2k + l)2 2

and the lower bound for each Arth border can be computed by

n2 + 2 - (2k + l)2 2

Nothing has been said about even magic squares can they also be expanded by a

magic border? Yes, even-order magic squares of sixth order and larger can be

generated by using magic borders. Making use of the 4x4 magic square invented by D rer in 1514 (Kenney 1982) and a magic border of order six, a new 6x6 magic square can be generated (fig. 5). Note that the new sixth-order magic border contains

twenty numbers. Since 20 divided by 2

1 35 34 5 30 6

1 15 14 4 33 11 25 24 14 4

12 6 7 9 8 22 16 17 19 29

8 10 11 ' 5

: 28 If 20 21 15 9

13 3_ 10 ja_ 13 12 26 27 31 2 3 32 7 36

Fig. 5. Add 10 to each entry in the magic square, then use a magic border.

1 143 142

132

14

130

129

17

127

19

125

21

22

4 5 139 138 8 9 135 134 12

13 24122120 26 27 28116115114 33| 113

111

35

109

37

38

106

40

41 103 102 44 45 99 98 48

49 96

95] 51

'93

53

54

55 89 88 59 84 60

87

62

82

64

80 66 67 77

69 75 74 72

73 71 70 76

68 78 79 65

58

83

63

81 85 56 57 86 61 90

50

94

52

92

91

97 42 43 101 100 46 47 104

32

34

110

36

108

107

39

105

112 23 25119118117 29 30 31 121

131

15

16

128

18

126

20

124

123

133 2 3141 140 6 7 137 136 10 11 144

Fig. 6. A twelfth-order magic square with concentric magic borders

March 1984 225



TABLE 2

Sum of opposite Size of border sides and corners Numbers used

Sixth order 37 1-10, 27-36

Eighth order 65 1-14, 51-64

Tenth order 101 1-18, 83-100

Twelfth order 145 1-22, 123-144

nth order n2 + 1 1 to In - 2, n2 - In + 3 to n2

equals 10, one must add 10 to each entry in the D rer square before inserting it into the sixth-order border.

In the new sixth-order border, the op posite pairs and the diagonal corners total 37 = 62 + 1. Also, the 4x4 insert has its op posite sides and diagonal corners adding to

37, and the four center numbers add to 37. Borders for larger even magic squares

can be generated in the same manner as for odd magic squares. The final example in this group of even border squares shows a twelfth-order magic square with concentric

magic borders (fig. 6). The numbers to be placed on a single

even magic border can be determined in a manner similar to that for odd magic bord ers. Table 2 gives the details for this pro cedure.

One last question remains to be con sidered: Can one make an even magic square with a series of concentric borders that begin at the center of the square ? Un

fortunately, there is no center position to

begin with there are four center positions. Their sum is 2{n2 + 1). Because of the nature of even magic squares, the real trick is to determine what will be the numbers in the inner 4x4 magic square. Take

(n2 + l)/2, which will be a fraction. Use the

integers on both sides of the fraction and call them and /+. There will be sixteen consecutive integers whose lower bound will be /_ 7 and whose upper bound will be f+ + 7. For example, in figure 6, 145/2 =

72^; so use the numbers 72 and 73. The lower bound will be 72 7 = 65, and the

upper bound will be 73 + 7 = 80. The six teen numbers will be 65 through 80. Choose four of these integers whose sum is 290 =

2(n2 + 1), place them in the four center

positions, and you are on your way.

BIBLIOGRAPHY

Ackerson, Paul. "Everything You Want to Know about Magic Squares." Unpublished document, Uni

versity of Nebraska at Omaha, 1975.

Ball, W. W. Rouse, and H. S. M. Coxeter. Mathemat ical Recreations & Essays, 12th ed. Buffalo: Univer

sity of Toronto Press, 1974.

El-Zaidi, S. M. "Constructing Magic Squares" (Reader Reflections). Mathematics Teacher 75 (November 1982) :637.

Kenney, Margaret J. "An Artful Application Using Magic Squares." Mathematics Teacher 75 (January 1982):83-89.

Kraitchik, Maurice. Mathematical Recreations. New York: W. W. Norton & Co., 1942.

Swetz, Frank. "Mysticism and Magic in the Number

Squares of Old China." Mathematics Teacher 71

(January 1978) :50~56.

SABBATICAL LEAVE PROGRAM Salary stipends and study opportunities provided for h.s. teachers, one each in Bio, Chem, Geo, Math/Comp Sci, Phys for 1984-85 school year as half-time instructors. Send curriculum vitae, three reference letters, and letter giving reasons for applying to Myron Ander son, Gustavus Adolphus Coll, St. Peter, MN 56082 before Mar 15. Master's preferred. 507-931-7328

MATHEMATICS Responsibilities include teaching assigned courses, assisting with review and develop ment of mathematics curriculum, advising students, and providing general service to the college. MINIMUM QUALIFICATIONS: Master's degree in mathematics or mathematics education, ability to instruct mathematics courses ranging from developmental to differential

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30, 1984: letter of application, official college applica tion form, copies of college transcripts, resume to: Personnel Division, Lakeland Community College, Mentor, Ohio 44060. An Equal Opportunity/Affirmative Action Employer.

226 Mathematics Teacher



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Using Magic Borders to Generate Magic Squares