Magic Squares

Andrew Yuwen

Types of Magic Squares● odd vs even● singly even (6x6 or 10x10)● doubly even (4x4 or 8x8)●

History● Discovered in China, c. 650BCE● First magic squares (orders 5 and 6) found in

Baghdad, Encyclopedia of the Brethren of Purity

● used with “magic letters” in Arabic spirituality, to assist Sufi illusionists

Lo Shu Square● flooding of the river Lo● a turtle emerges with a magic square on his

shell● sum was 15, number of days of a cycle in the

solar year● Afterwards, people could predict floods and

protect themselves

Lo Shu Square● unique 3x3 magic square

with 1 on bottom and 2 in upper-right corner

● all other 3x3 magic squares are reflections and rotations of this

4 9 23 5 78 1 6

Chautisa Yantra

7 12 1 142 13 8 1116 3 10 59 6 15 4

Chautisa Yantra● 10th century,

Parshvanath Jain temple● rows, columns,

diagonals, 2x2 subsquares, corners of 3x3 and 4x4 squares, middle two entries of outer rows and columns all sum to 34

7 12 1 142 13 8 1116 3 10 59 6 15 4

Chautisa Yantra● every second diagonal

number sums to 17 (including short diagonals)

● 8 trapeziums sum to 34● 4 triangles with three

numbers and a corner sum to 34

7 12 1 142 13 8 1116 3 10 59 6 15 4

Melencolia I (Abrecht Durer)

Melencolia I (Abrecht Durer)

Melencolia I (Abrecht Durer)gnomon magic square: sum of quadrants and middle 2x2 square is 34

16 3 2 135 10 11 89 6 7 124 15 14 1

Melencolia I (Abrecht Durer)artist’s name and year

16 3 2 135 10 11 89 6 7 124 15 14 1

Sagrada Familia● significance of

magic constant● can also be

expanded to a magic cube

1 14 14 411 7 6 98 10 10 513 2 3 15

● Heinrich Cornelius Agrippa● magic squares (kameas) represented

astrological planets● highly influential until 16th century counter-

reformation● still used in ceremonial magic

De Occulta Philosophia

Sator Square● sator - progenitor, originator● arepo - trust● tenet - holds, comprehends,

preserves● opera - work, care, aid● rotas - wheel, rotate● “The sower works for mastery

by turning the wheel.”● method to overcome envy

S A T O R

A R E P O

T E N E T

O P E R A

R O T A S

Sator Square and ChristianityS A T O R

A R E P O

T E N E T

O P E R A

R O T A S

● associated with cross of Pater Noster (Our Father, from the Lord’s Prayer)

● A’s and O’s represent Alpha and Omega

● way for early Christians to signal each other to avoid persecution

● the names of the five nails with which Jesus was crucified

● palindromes could not be tampered with by the devil, who would be confused by the repetition of letters

Construction of 3x3 Squares● Edward Lucas● Every 3x3 magic square

is of this formc-a c-a-b c+b

c-a+b c c+a-b

c-b c+a+b c-a

Construction of Squares● Magic squares exist

for all values of n except 2

● Odd and doubly even are easy

● Singly even is more difficult

● LUX Method (John Horton Conway)

● Strachey Method

Siamese Method● French diplomat de la

Loubere (1693)● A New Historical

Relation of the Kingdom of Siam

1

Mystic Squares● Start with a

numbered square

1 2 3 45 6 7 89 10 11 1213 14 15 16

Mystic Squares● Generate truth

table1 0 0 10 1 1 00 1 1 01 0 0 1

Mystic Squares● Create mystic

square1 15 14 412 6 7 99 10 11 513 3 2 16

LUX Method● (2n+1) x (2n+1)● n + 1 rows of L’s● 1 row of U’s● n - 1 rows of X’s

● Expand letters to sequenced 2x2 numbers using the Siamese method

LUX MethodL L L L LL L L L LL L U L LU U L U UX X X X X

68 65 96 93 4 1 32 29 60 57

66 67 94 95 2 3 30 31 58 59

92 89 20 17 28 25 56 53 64 61

90 91 18 19 26 27 54 55 62 63

16 13 24 21 49 52 80 77 88 85

14 15 23 24 50 51 78 79 86 87

37 40 45 48 76 73 81 84 9 12

38 39 46 47 74 75 82 83 10 11

41 44 69 72 97 100 5 8 33 36

43 42 71 70 99 98 7 6 35 34

Strachey Method

A CD B

17 24 1 8 15 67 74 51 58 65

23 5 7 14 16 73 55 57 64 66

4 6 13 20 22 54 56 63 70 72

10 12 19 21 3 60 62 69 71 53

11 18 25 2 9 61 68 75 52 59

92 99 76 83 90 42 49 26 33 40

98 80 82 89 91 48 30 32 39 41

79 81 88 95 97 29 31 38 45 47

85 87 94 96 78 35 37 44 46 28

86 93 100 77 84 36 43 50 27 34

Strachey MethodExchange the leftmost n columns of A with D

92 99 1 8 15 67 74 51 58 65

98 80 7 14 16 73 55 57 64 66

79 81 13 20 22 54 56 63 70 72

85 87 19 21 3 60 62 69 71 53

86 93 25 2 9 61 68 75 52 59

17 24 76 83 90 42 49 26 33 40

23 5 82 89 91 48 30 32 39 41

4 6 88 95 97 29 31 38 45 47

10 12 94 96 78 35 37 44 46 28

11 18 100 77 84 36 43 50 27 34

Strachey MethodExchange the rightmost n-1 columns in C with B

92 99 1 8 15 67 74 51 58 40

98 80 7 14 16 73 55 57 64 41

79 81 13 20 22 54 56 63 70 47

85 87 19 21 3 60 62 69 71 28

86 93 25 2 9 61 68 75 52 34

17 24 76 83 90 42 49 26 33 65

23 5 82 89 91 48 30 32 39 66

4 6 88 95 97 29 31 38 45 72

10 12 94 96 78 35 37 44 46 53

11 18 100 77 84 36 43 50 27 59

Strachey MethodExchange the middle cell of the leftmost column of A with D. Also, exchange the central cell of A with D.

92 99 1 8 15 67 74 51 58 40

98 80 7 14 16 73 55 57 64 41

4 81 88 20 22 54 56 63 70 47

85 87 19 21 3 60 62 69 71 28

86 93 25 2 9 61 68 75 52 34

17 24 76 83 90 42 49 26 33 65

23 5 82 89 91 48 30 32 39 66

79 6 13 95 97 29 31 38 45 72

10 12 94 96 78 35 37 44 46 53

11 18 100 77 84 36 43 50 27 59

Panmagic Square20 8 21 14 211 4 17 10 237 25 13 1 193 16 9 22 1524 12 5 18 6

Skalli Multiplicative Square27 50 66 84 13 2 32

24 52 3 40 54 70 11

56 9 20 44 36 65 6

55 72 91 1 16 36 30

4 24 45 60 77 12 26

10 22 48 39 5 48 63

78 7 8 18 40 33 60

Skalli Multiplicative Square21+14i -70+30i -93-9i -105-217i 16+50i 4-14i 14-8i

63-35i 28+114i -14i 2+6i 3-11i 211+357i -123-87i

31-15i 13-13i -103+69i -261-213i 49-49i -46+2i -6+2i

102-84i -28-14i 43+247i -10-2i 5+9i 31-27i -77+91i

-22-6i 7+7i 8+14i 50+20i -525-492i -28-42i -73+17i

54+68i 138-165i -56-98i -63+35i 4-8i 2-4i 70-53i

24+22i -46-16i 6-4i 17+20i 110+160i 84-189i 42-14i