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MAGIC SQUARE
By Andrea Schweim
2
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).
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Where did the magic square come from? Let’s find out…
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Where is the Magic Square from?
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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. 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.
A magic square on the Sagrada Família church façade
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Where is the Magic Square from?
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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.
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.
Original script from Shams Al-ma'arif.
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Where is the Magic Square from?
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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.
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 of the San Pietro’s Church
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Where is the Magic Square from?
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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!
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 Square
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How do you make this “magic” square?
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Decide on the size Setup the square Expand the square Fill in the blanks Check your work
Amaze your friends!
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Step 1 Rows & Numbers
n = the number of rows and columns where
n = {3, 5, 7, …..}
Total number of spaces in the square = n²
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n² is also the largest number in the square.
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Step 1 Examples
Examples
n = 3n²=9 n = 5
n²=25 n = 7 n²=49
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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
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Step 3 Setting up the Square Take your square and stair-step the
outside
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Step 4 Filling the Square
Move to the next point on the left and so on, until all diagonal squares are filled.
12
3
41
235
67
89
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)
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Step 4 Filling the Square
N = 5: we need to n = 7: we need to fill
fill 5²=25 numbers in 7²=49 numbers
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9
8
7
6
1
2
3
5
4
11
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15
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25
1
2
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48
49 9/1/2011Magic Square
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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 square311
12
13
14
15
17
18
1923
9
8
7
1
2
5
416
20
21
22
24
25
10
6
410
11
12
16
17
18
19
20
22
23
24
25
26
27
28
30
31
32
33
34
38
39
4046
3541
42
47
48
49
29
36
3743
44
45
1
2
3
8
915
5
6
713
14
21
4 2
5
68
1
37
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Step 5 Filling in the Blanks
We take the row of numbers that is closest to the original square…
4
1
2
35
6
7
8
9
1
3
5
11
12
13
14
15
17
18
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16
22
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2024
25
4
10
9
8
7
26
1
2
4
6
7
8
10
11
12
14
3915
16
17
18
19
20
5
13
21
22
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25
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354147
48
49
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Step 5 Filling in the Blanks
… and move it to the opposite side in the opening in the original square.
4
1
2
35
6
7
8
9
1
3
5
11
12
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16
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2024
25
4
10
9
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7
26
1
2
4
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7
8
10
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3915
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5
13
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354147
48
49
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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.4
1
2
3 5
6
7
8
91
3
5
11
12
13
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15
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22
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2024
25
4
10
9
8
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1
4
7
28
10
11
12 6
14
3915
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5
13
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354147
4248
49
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Step 6 Moving Numbers
Repeat step 6 until all empty squares in the original magic square are filled.
4
1
2
3 5
6
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9
1
3
5
11
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254
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9
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1
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28
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11
126
14
3915
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354147
4248
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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
1
3
5
11
12
13
14
15
17
18
19
21
16
22
23
2024
254
10
9
8
7
26
1
4
7
28
10
11
126
14
3915
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17
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19
20
5
13
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25
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36
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354147
4248
49
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Step 7 Checking the Numbers n = 3 magic square: MC is 15
3 5 7
1 68
4 29
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Row 1: 4+9+2=15 Row 2: 3+5+7=15 Row 3: 8+1+6=15
Column 1: 4+3+8=15 Column 2: 9+5+1=15 Column 3: 2+7+6=15
Diagonal 1: 4+5+6=15
Diagonal 2: 2+5+8= 15
7
1 6
4 9
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Step 7 Checking the Numbers
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18
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16
22
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2024
254
10
9
1
3
5
11
12
17
8
7
26
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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
n=5 magic constant: MC is 65
Diagonal 1: 11+12+13+14+15=65 Diagonal 2: 3+8+13+18+23
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
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Step 7 Checking the Numbers N=7 magic constant: MC is 175
1
4
7
28
10 1
1 12
6
14
3915
16 1
7 18 1
9 20
5
13
21
22 2
3 24 2
5 26 2
7 28
30 3
1 32 3
3 34
38 3
9 40
43
36
44
29
37
454
6
35
41
47 4
248 4
9
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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
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
Diagonal 1: 22+23+24+25+26+27+28=175 Diagonal 2: 46+39+32+25+18+11+4=175
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Credits
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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/d0bcd0b0d0b3d0b8d187d0b5d181d0bad0b8d0b9-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
![The projective geometry of Freudenthal’s magic square · PDF filearXiv:math/9908039v1 [math.AG] 10 Aug 1999 The projective geometry of Freudenthal’s magic square J.M. Landsberg](https://img.pdfslide.us/doc/110x75/5a7951757f8b9ac3268bfbd1/the-projective-geometry-of-freudenthals-magic-square-math9908039v1-mathag.jpg)
