Magic Squares Report

Introduction

Chapter 1: The History of Magic Squares

Chapter 2: What is a Magic Square? 2.1: Odd Order 2.2: Doubly Even 2.3: Singly Even

Chapter 3: Variations on Magic Squares 3.1: Panmagic 3.2: Anti-Magic Squares 3.3: Franklin Squares 3.4: Latin Squares 3.4.1: Graeco-Latin Squares

Chapter 4: Applications of Magic Squares 4.1: Music 4.2: Sudoku

Conclusion

References Introduction

This report aims to explore the possible applications of magic squares in everyday life. In doing this, different types of magic squares will be investigated and the methods used to construct them. How magic squares have evolved and where they originally came from should also be considered. Also, variations on the basic magic square will be looked at to see if these have any practical applications.

Chapter 1: The History of Magic Squares

The earliest magic square known dates from around 2800 B.C. in China. A Chinese myth says that Emperor Yu found a tortoise with a pattern on its shell while walking along the Yellow River. He called this unique diagram, Loh-Shu (Anderson, 2001).

The Loh-Shu Tortoise (Fig 1.1)

However, the first recorded magic square was described as the scroll of the river Loh or Loh-Shu by Fuh-Hi (Farrar, 1997 & Grogono, 2004). It is a 3x3 magic square with symbols rather than numbers (See fig 1.2). The Chinese scholars of today have only managed to trace the Loh-Shu back as far as the fourth century B. C. and from then until the tenth century it was seen as a symbol of great significance. This Loh-Shu was numerical, with the number of dots in each symbol representing a whole number (See fig 1.2). The even numbers were thought to represent the female principle, yin, and the odd numbers the male principle, yang. The 5 in the middle was thought to be the earth, around which lie the other four elements; metal, 4 and 9, fire, 2 and 7, water, 1 and 6 and wood 3 and 8 (Gardner, 1988).

The Loh-Shu Magic Square (Fig 1.2) There are also Greek writings relating to magic squares from around 1300 B.C. (Farrar, 1997). It is thought that from China, magic squares were introduced to Indian culture, and it was there that the first magic square of order four was discovered (Swaney, 2000). In India, magic squares were used not only in the traditional mathematical context, but also for other applications such as in recipes for making perfume and also in medical work, with a third order magic square appearing as a means of easing childbirth (Anderson, 2001).

Islamic and Arabic mathematicians were aware of magic squares, probably from the Indians, by about the fifth century A.D. and are often attributed to using them in astrology and predictions. Their magic squares were of larger order & they compiled a list of magic squares up to order nine (Ballew, 2006). It was Islamic mathematicians who first made simple rules for creating magic squares. In around 1300, the Byzantine, Manual Moschopoulos, wrote a book based on the findings of Al-Buni, an Arab mathematician, about magic squares. It was Moschopoulos who introduced magic squares to Europe, where they were associated with divination, alchemy and astrology (Anderson, 2001).

Since then, magic squares have been looked at in relation to planets and the sun, art and religion. Also in the past, magic squares were important in African culture. They held spiritual importance and were often inscribed on masks, clothes and religious artefacts and were influential in house design and building (Anderson, 2001). Chapter 2: What is a Magic Square?

A basic magic square of order n can be defined as an arrangement of numbers 1 to n2 in an n n matrix, such that every row, column and diagonal add up to the same number (Adler, 1996). The magic sum, or the number that each row, column and

1 diagonal add up to can be found by the formula nn2 1 (Ball, 1959). In general, 2 magic squares remain magic if the same positive integer is added to each number in the square or each number in the original square is multiplied by the same number (Kraitchik, 1960).

A basic 4x4 magic square (Fig 2.1)

2.1: Odd Order

An odd order magic square is of the form, n  2m 1. There are several methods of generating such magic squares for m 1. With the most common being the known as the Siamese or staircase method (see fig 2.1.1). In this method the numbers are written in ascending numerical order as an upward diagonal to the right. When a filled square is reached the next number is placed vertically below its predecessor. This method was devised by De la Loubère when the 1 is placed in the middle column of the top row (Ball, 1959). If the 1 lies in the middle column on the row directly above the middle row it is known as the method of Bachet de Méziriac (Kraitchik, 1960). The Siamese or staircase method for generating odd order magic squares (Fig 2.1.1)

2.2: Doubly Even

A doubly even magic square is in the form n  4m . One method of constructing this type of magic square, for m 1, is the cross method (See fig 2.2.1). By writing all the numbers in order from the top left of a square to the bottom right, then drawing a cross through every 4x4 square, or sub-square of a larger square, and swapping the numbers along the diagonals of the cross, will yield a magic square.

The cross method for generating doubly even magic squares (Fig 2.2.1) 2.3: Singly Even A singly even magic square is of the form 4m  2 , when m 1. One method of construction is that of Ralph Strachey, to divide the square up into equal quarters. For example, in a 6x6 square, this will give four 3x3 squares. Each of these can then be formed using De la Loubère’s method for odd order squares (Ball, 1959).

8 1 6 26 19 24 3 5 7 21 23 25 4 9 2 22 27 20 35 28 33 17 10 15 30 32 34 12 14 16 31 36 29 13 18 11 Strachey Method (Fig 2.3.1)

Another method for generating singly even magic squares was found by J. H. Conway and is called the LUX method. Create m 1 rows of L followed by one row of U and then m 1 rows of X at the bottom. Then swap the middle U with the L directly above it. The rows of letters form an odd order square, so starting at the top middle L, put in the numbers working through the letters using the De la Loubère method (Weisstein, 2003).

LUX method (Fig 2.3.2) Chapter 3: Variations on Magic Squares There are many variations of magic squares such as border squares, magic stars, cubes, rectangles and other shapes, alphamagic squares, reversible and complimentary magic squares among others. It was felt that looking into Franklin squares, Latin squares and panmagic squares would give the most insight to possible applications.

3.1: Panmagic Squares

Panmagic squares, also known as pandiagonal and diabolic squares, have the same properties as normal magic squares except that all the broken diagonals of the square must also equal the magic sum (Ball, 1959), therefore the square must be magic along all rows, columns, the two full diagonals and all broken diagonals (See Fig 3.1.1). Panmagic squares do not exist for order 3 or for order 4m  2 , where m is any integer. The Siamese method for generating odd order magic squares will produce panmagic squares for order 6m 1 when using vector (2, 1) and break vector (1, -1) (Weisstein, 2006). Where the vector is the number of cells moved across and down respectively. The break vector is how the pattern changes when a filled cell is reached again.

Broken diagonals (Fig 3.1.1)

3.2: Antimagic Squares

An antimagic square is the complete opposite of a magic square in that all the rows, columns and diagonals equal different values. They contain the same numbers, 1 to n2 , just in a different arrangement. Antimagic squares of order 1, 2 and 3 are impossible to create (Weisstein, 2002). Examples of antimagic squares (Fig 3.2.1)

3.3: Franklin Squares

Benjamin Franklin produced several magic squares during the mid 1700’s, many of which were only partial Latin squares since their diagonal totals did not add up to the magic sum. However, they had other properties which made them of interest to mathematicians.

260 260 292 52 61 4 13 20 29 36 45 260 14 3 62 51 46 35 30 19 260 53 60 5 12 21 28 37 44 260 11 6 59 54 43 38 27 22 260 55 58 7 10 23 26 39 42 260 9 8 57 56 41 40 25 24 260 50 63 2 15 18 31 34 47 260 16 1 64 49 48 33 32 17 260 260 260 260 260 260 260 260 260 228 Franklin’s original 8x8 square (Fig 3.3.1)

130 130 52 61 4 13 20 29 36 45 260 14 3 62 51 46 35 30 19 260 53 60 5 12 21 28 37 44 260 11 6 59 54 43 38 27 22 260 55 58 7 10 23 26 39 42 260 9 8 57 56 41 40 25 24 260 50 63 2 15 18 31 34 47 260 16 1 64 49 48 33 32 17 228 130 Franklin’s original 8x8 square (Fig 3.3.2)

As fig 3.3.1 shows, Franklin was interested in looking at bendy rows which also sum to the same constant as each row and column. From fig 3.3.2, it is clear that in an 8x8 Franklin square, all the half rows and half columns and any 2x2 sub-square within it total half of the magic sum (Morris, 2005).

When considering Franklin’s 16x16 square (see fig 3.3.3), the bendy rows and columns and any half rows or colums add up to half the row or column total, but there are no sub-squares which do. This is because half a row would be 8 cells, no sub- square can contain 8 cells because there is no value for n such that n2  8 .

2168 2040 199 216 231 248 7 24 39 56 71 88 103 120 135 152 167 184 2040 57 38 25 6 249 230 217 198 185 166 153 134 121 102 89 70 2040 197 218 229 250 5 26 37 58 69 90 101 122 133 154 165 186 2040 59 36 27 4 251 228 219 196 187 164 155 132 123 100 91 68 2040 200 215 232 247 8 23 40 55 72 87 104 119 136 151 168 183 2040 54 41 22 9 246 233 214 201 182 169 150 137 118 105 86 73 2040 202 213 234 245 10 21 42 53 74 85 106 117 138 149 170 181 2040 52 43 20 11 244 235 212 203 180 171 148 139 116 107 84 75 2040 204 211 236 243 12 19 44 51 76 83 108 115 140 147 172 179 2040 50 45 18 13 242 237 210 205 178 173 146 141 114 109 82 77 2040 206 209 238 241 14 17 46 49 78 81 110 113 142 145 174 177 2040 2040 48 47 16 15 240 239 208 207 176 175 144 143 112 111 80 79 2040 195 220 227 252 3 28 35 60 67 92 99 124 131 156 163 188 2040 61 34 29 2 253 226 221 194 189 162 157 130 125 98 93 66 2040 193 222 225 254 1 30 33 62 65 94 97 126 129 158 161 190 2040 63 32 31 0 255 224 223 192 191 160 159 128 127 96 95 64 2040 2040 2040 2040 2040 2040 2040 2040 2040 2040 2040 2040 2040 2040 2040 2040 2040 1912 Franklin’s original 16x16 square (Fig 3.3.3)

3.4: Latin Squares

A Latin square differs from a normal magic square in that it is an n n matrix containing only the numbers 1 to n rather than 1 to n2 . They are written in such a way that each row and column contain every number only once. Latin squares can be formed for any order, for example there is one Latin square of order 1, two of order 2 (See fig 3.4.1), twelve of order 3 (See fig 3.4.2) and 576 of order 4 (Weisstein, 2006).

Latin squares of order 2 (Fig 3.4.1)

Latin squares of order 3 (Fig 3.4.2)

3.4.1: Graeco Latin Squares Graeco Latin squares are so called because it is customary to use Latin letters as the symbols in one square and Greek letters for the symbols of a second square. They are also known as Euler squares and exist for all n , except n  2 and n  6 (Beezer, 1995).

A Graeco-Latin square of order 10 (Fig 3.4.3)

Chapter 4: Applications of Magic Squares

Modern day applications of magic squares are difficult to find. There seems to be some sort of link between magic squares and music and the Latin squares along with the Greaco Latin squares are used in the popular puzzle, Sudoku. Apart from that, other applications found were from mathematicians in history which no longer apply.

4.1: Music

The main area of the application of magic squares to music is in rhythm, rather than notes. Indian musicians seem to have applied them to their music and they seem to be useful in time cycles and additive rhythm. In this case it is not the usual magic properties of a square that are important, but the relationship of the central number to the total sum of all the numbers in the magic square. This is because for rhythm, consecutive numbers 1 to n2 are not used to fill the cells of the n n magic square. This relationship is: The total sum of the magic square’s numbers = central number x 9. This is important to music as it shows the size of the magic square, which is how many pulses or sub-divisions there are in the sequence, this will indicate how and where to apply it.

3 5 7 5 8 11 7 11 15 Magic Square for Rhythm (Fig 4.1.1)

Using fig 4.1.1 as an example, 8x9=72 gives the size of the magic square. This can therefore be applied to a piece of music with 18 crotchet beats since 18x4=72. Rests can also be added between the first and second or second and third rows to create a feeling of the music building towards a cadence. By choosing different values for the rests, the same magic square can create many different musical passages (Dimond, 2006).

4.2: Sudoku

Sudoku was first introduced in 1979 and became popular in Japan during the 1980’s (Pegg & Weisstein, 2006). It has recently become a very popular puzzle in Europe, but it is actually a form of Latin square. A Sudoku square is a 9x9 grid, split into 9 3x3 sub-squares. Each sub-square is filled in with the numbers 1 to n where n  9 , so that the 9x9 grid becomes a Latin square. This means each row and column contain the numbers 1 to 9 only once. Therefore each row, column and sub-square will sum to the same amount. An Example of a Sudoku Square (Fig 4.2.1)

Conclusion

Mathematicians today do not need to speculate and attach meaning to magic squares to make them important, as has been done in the past with Chinese and other myths. The squares were thought to be mysterious and magic, although now it is clear that they are just ways of arranging numbers and symbols using certain rules. They can be applied to music and Sudoku as has been discussed but are mainly of interest in mathematics for their “magic” properties rather than their practical applications.

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Beezer, R. (1995). Graeco-Latin squares [online]. Available from http://buzzard.ups.edu/squares.html [Accessed 5th December 2006].

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Farrar, M. S. (1997). History of magic squares [online]. Available from http://www.markfarrar.co.uk/msqhst01.htm [Accessed 6th December 2006].

Gardner, M. (1988). Time travel and other mathematical bewilderments. New York. W. H. Freeman and Company.

Grogono, A. W. (2004). A mini-history of magic squares [online]. Available from http://www.grogono.com/magic/history.php [Accessed 6th December 2006].

Kraitchik, M. (1960). Mathematical Recreations. London. George Allen & Unwin Ltd. Morris, D. (2005). Franklin squares primer [online]. Available from http://www.bestfranklinsquares.com/franklinsquaresprimer2.html [Accessed 8th December 2006].

Pegg, E. Jnr. & Weisstein, E. W. (2006). Sudoku [online]. Available from http://mathworld.wolfram.com/Sudoku.html [Accessed 5th December 2006].

Swaney, M. (2000). Mark Swaney on the history of magic squares [online]. Available from http://www.ismaili.net/mirrors/Ikhwan_08/magic_squares.html [Accessed 7th December 2006].

Weisstein, E. W. (2002). Antimagic Square [online]. Available from http://mathworld.wolfram.com/AntimagicSquare.html [Accessed 8th December 2006].

Weisstein, E. W. (2003). Magic Square [online]. Available from http://mathworld.wolfram.com/MagicSquare.html [Accessed 7th December 2006].

Weisstein, E. W. (2006). Latin square [online]. Available from http://mathworld.wolfram.com/LatinSquare.html [Accessed 5th December 2006].

Weisstein, E. W. (2006). Panmagic square [online]. Available from http://mathworld.wolfram.com/PanmagicSquare.html [Accessed 8th December 2006].