Magic Square By Andrea Schweim

What is a magic square? Magic squares are amazing things. If you aren't sure what a magic square is, it's a square grid of numbers containing all the numbers 1, 2, 3 and so on, up to the number of grids within the square, each one exactly once. What's more, the sum of each row, column and diagonal of a magic square must equal the same number! This is the magic constant (MC). Where did the magic square come from? Let’s find out… Magic Square 9/1/2011

Where is the Magic Square from? Magic squares have a history dating so far back they disappear into the boundary between history and myth. From ancient Chinese literature we have the following story: At one time, there was a huge flood. The people tried to offer sacrifices to the god of one of the flooding rivers, the Lo river, to calm his anger. As they were doing this, a turtle emerged from the water with a curious pattern on its shell, with patterns of circular dots arranged in a three-by-three grid on the shell, such that the sum of the numbers in each row, column and diagonal was the same: 15. The people were able to use this magic square to control the river and reduce the flood. A magic square on the Sagrada Família church façade This same square inspired the floor plan of the Ming'Tang palace, a mythical ancient Chinese palace. In fact, ancient Chinese literature is peppered with references to this square from 2800BC to 570AD. Magic Square 9/1/2011

Where is the Magic Square from? In ancient Greek writing, references are sparse. It is said that Greek mathematicians as far back as 1300BC wrote about magic squares, but I could not find any further details about that. Original script from Shams Al-ma'arif. In first century Turkey (in a town called Smyrna) there was born a man called Theon. He would explore mathematical concepts that are still interesting today, including square numbers, triangular numbers, and many others. It is often said that he also wrote about the 3 by 3 magic square, but in fact he did not. He did write about ways to arrange the numbers 1 to 9 in a grid, but not in such a way that the rows, columns and diagonals would all have the same sum. Magic Square 9/1/2011

Where is the Magic Square from? Amongst those ancient mathematicians who knew about magic squares were the Arabs. The 3x3 magic square was used as a lucky charm, and larger squares were also known. In fact, by the 13th century, the Arabs had produced a 10 by 10 magic square. Some say the Arabs discovered magic squares, others say they learned them from the Indian mathematicians of the 7th and 8th centuries. In any case, it is the Arabs who are first known to have developed recipes for building magic squares. Magic Square of the San Pietro’s Church Interestingly, the Indians seemed to know about order 4 squares before order 3 squares. As far back as 550AD, Varahamihira used a 4 by 4 magic square to describe a perfume recipe, but the earliest known Indian writings about an order 3 square comes from 900AD, as a medical treatment! Magic Square 9/1/2011

Where is the Magic Square from? Dürer's engraving titled Melencolia or Melancholia is one of his best-known, and most enigmatic, works. Part of this picture is the 4-by-4 square on the wall behind the angel. Magic squares were introduced into Europe in 1300AD by Manuel Moschopoulos, who probably learned about them from the Arabs. He wrote a number of works, with his treatise on magic squares being his only mathematical work. The most famous European work involving magic squares is perhaps Albrecht Durer's engraving 'Melancolia', from 1514. The magic square in his artwork is shown below. To see the full engraving, hosted at the University of Hamburg, click here. The engraving is "an allegorical self-portrait" showing the melancholy of the artist. The year of the engraving is, in fact, cleverly hidden in the bottom row of the magic square! Magic Square 9/1/2011

How do you make this “magic” square? Decide on the size Setup the square Expand the square Fill in the blanks Check your work Amaze your friends! Magic Square 9/1/2011

Step 1 Rows & Numbers n = the number of rows and columns where Total number of spaces in the square = n² n² is also the largest number in the square. Magic Square 9/1/2011

Step 1 Examples Examples n = 3 n²=9 n = 5 n²=25 n = 7 n²=49 Magic Square 9/1/2011

Step 2 Middle Number & Constant To figure out the number that goes in the middle of the magic square, we write: To figure out the magic constant (the row/column/diagonal sum), we take the formula for the middle number and multiply it by n: n²+1 2 = middle number n²+1 2 n = magic constant Magic Square 9/1/2011

Step 3 Setting up the Square Take your square and stair-step the outside Magic Square 9/1/2011

Step 4 Filling the Square Start by filling in the numbers (through n²) for each square starting on top of the stair-step Example n=3 We have to fill in a total of 9 numbers (3²) 1 2 3 Move to the next point on the left and so on, until all diagonal squares are filled. 1 2 3 4 7 5 8 6 9 Magic Square 9/1/2011

Step 4 Filling the Square N = 5: we need to n = 7: we need to fill fill 5²=25 numbers in 7²=49 numbers 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 5 4 15 16 17 18 19 20 21 10 9 8 7 6 22 23 24 25 26 27 28 11 12 13 14 15 29 30 31 32 33 34 35 16 17 18 19 20 36 37 38 39 40 41 42 21 22 23 24 25 43 44 45 46 47 48 49 Magic Square 9/1/2011

Step 5 Filling in the Blanks Now we move the numbers from the outside (stair-step) of the original square (outlined in dark) to the empty spaces in the original square 35 41 42 47 48 49 29 36 37 43 44 45 1 2 3 8 9 15 5 6 7 13 14 21 22 23 24 25 26 27 28 16 10 4 17 11 1 2 5 4 16 20 21 22 24 25 10 6 30 18 12 31 19 11 12 13 14 15 7 3 38 32 20 8 39 33 17 9 46 40 34 18 1 3 7 9 23 19 4 2 5 8 6 Magic Square 9/1/2011

Step 5 Filling in the Blanks We take the row of numbers that is closest to the original square… 1 8 2 1 3 9 15 4 2 22 16 10 4 7 5 3 29 37 45 23 17 11 5 13 21 1 8 6 36 30 24 18 12 6 2 6 9 43 31 25 19 7 11 7 3 44 38 32 26 20 14 16 22 12 8 4 10 39 33 27 21 17 13 9 5 46 40 34 28 18 14 35 41 47 23 19 15 48 42 20 24 49 25 Magic Square 9/1/2011

Step 5 Filling in the Blanks … and move it to the opposite side in the opening in the original square. 1 8 2 1 3 9 15 4 2 22 16 10 4 7 5 3 29 37 45 23 17 11 5 13 21 1 8 6 36 30 24 18 12 6 2 6 9 43 31 25 19 7 11 7 3 44 38 32 26 20 14 16 22 12 8 4 10 39 33 27 21 17 13 9 5 46 40 34 28 18 14 35 41 47 23 19 15 48 42 20 24 49 25 Magic Square 9/1/2011

Step 6 Moving Numbers Now, we take the next row and move it to the next open space on the opposite side of the magic square. 1 2 8 4 1 2 3 5 6 7 8 9 22 35 41 47 16 10 4 5 13 21 23 17 11 29 37 45 1 36 44 30 24 18 12 6 14 43 31 25 19 7 11 20 24 7 3 38 32 26 20 4 10 12 8 16 22 39 33 27 21 17 13 9 5 46 3 9 15 40 34 28 18 14 23 2 6 19 15 42 48 49 25 Magic Square 9/1/2011

Step 6 Moving Numbers Repeat step 6 until all empty squares in the original magic square are filled. 1 4 1 2 3 5 6 7 8 9 22 35 41 47 16 10 4 5 13 21 23 42 48 17 11 29 37 45 30 6 14 24 18 36 44 12 43 31 25 19 7 11 20 24 7 3 38 32 26 20 4 10 12 25 8 16 22 39 2 8 33 27 17 5 13 21 9 46 3 9 15 40 34 28 18 1 14 23 2 6 19 15 49 Magic Square 9/1/2011

Step 7 Checking the Numbers Let’s check if we have our magic constant for all rows, columns and diagonals. 4 1 2 3 5 6 7 8 9 22 35 41 47 16 10 4 5 13 21 23 42 48 17 11 29 37 45 30 6 14 24 49 18 36 44 12 31 7 25 43 19 11 20 24 7 3 38 32 1 26 20 4 10 12 25 8 16 22 39 2 8 33 27 17 21 13 5 9 46 3 9 15 40 34 28 18 1 14 23 2 6 19 15 Magic Square 9/1/2011

Step 7 Checking the Numbers n = 3 magic square: MC is 15 Row 1: 4+9+2=15 Row 2: 3+5+7=15 Row 3: 8+1+6=15 Column 1: 4+3+8=15 4 2 9 4 9 Column 2: 9+5+1=15 3 5 7 7 Column 3: 2+7+6=15 1 6 8 1 6 Diagonal 1: 4+5+6=15 Diagonal 2: 2+5+8= 15 Magic Square 9/1/2011

Step 7 Checking the Numbers n=5 magic constant: MC is 65 Row 1: 11+24+7+20+3=65 Row 2: 4+12+25+8+16=65 Row 3: 17+21+13+5+9=65 Row 4: 10+18+1+14+22=65 Row 5: 23+6+9+2+15=65 11 24 7 20 3 4 12 25 8 16 17 21 13 5 9 10 18 1 14 22 Column 1: 11+4+17+10+23=65 Column 2: 24+12+21+18+8=65 Column 3: 7+25+13+1+19=65 Column 4: 20+8+5+14+2=65 Column 5: 3+16+9+22+15=65 23 6 19 2 15 Diagonal 1: 11+12+13+14+15=65 Diagonal 2: 3+8+13+18+23 Magic Square 9/1/2011

Step 7 Checking the Numbers N=7 magic constant: MC is 175 Row 1: 22+47+16+41+10+35+4=175 Row 2: 5+23+48+17+42+11+29=175 Row 3: 30+6+24+49+18+36+12=175 Row 4: 13+31+7+25+43+19+37=175 Row 5: 38+14+32+1+26+44+20=175 Row 6: 21+39+8+33+2+27+45 Row 7: 46+15+40+9+34+3+28 1 4 7 2 8 10 11 12 6 14 3 9 15 16 17 18 19 20 5 13 21 22 23 24 25 26 27 28 30 31 32 33 34 38 39 40 43 36 44 29 37 45 46 35 41 47 42 48 49 Column 1: 22+5+30+13+38+21+46=175 Column 2: 47+23+6+31+14+39+15=175 Column 3: 16+48+24+7+32+8+40=175 Column 4: 41+17+49+25+1+33+9=175 Column 5: 10+42+18+43+26+2+34=175 Column 6: 35+11+36+119+44+27+3=175 Column 7: 4+29+12+37+20+45+28=175 Diagonal 1: 22+23+24+25+26+27+28=175 Diagonal 2: 46+39+32+25+18+11+4=175 Magic Square 9/1/2011

Credits History: http://www.dr-mikes-math-games-for-kids.com/magic-square- history.html Picture 1 (page 3): http://farm1.static.flickr.com/201/451292369_e045e0e03a_z.jpg Picture 2 (page 4): http://ts2.mm.bing.net/images/thumbnail.aspx?q=1131698335873 Picture 3 (page 5): http://santatatiana.files.wordpress.com/2010/05/d0bcd0b0d0b3d0b8d 187d0b5d181d0bad0b8d0b9- d0bad0b2d0b0d0b4d180d0b0d182.jpg?w=840 Picture 4 (page 6): http://userpages.monmouth.com/~chenrich/MagicSquares/melencolia.jpg Link on Page 6: http://userpages.monmouth.com/~chenrich/MagicSquares/melencolia.jpg Magic Square 9/1/2011