Transcript Magic Squares!!! - Oldham Sixth Form College

Magic Squares!!!
By John Burton
What Are Magic Squares?
A Magic Square is a grid (or matrix) containing numbers from 1
to n (where n is any number), and where every row, column and
diagonal adds to the same number
The most basic example is the 1 x 1 square, shown below
Examples of Magic Squares
One of the most famous magic squares is that of
Albrecht Dürer. It was created in 1514 and is shown below
16 3
2 13
5 10 11
9
6
8
7 12
4 15 14 1
Dürers Square
Here is the
year
As you can see from the square, the total
of each row, column, diagonal and small
square is 34.
You can also see that the year it was
made (1514) appears in the square
Some More Magic Squares
8 1 6
3 5 7
4 9 2
20
9
13
8
6
15
11
18
7
14
10
19
17
12
16
5
The total for each row,
column etc. is 15
The total for each row,
column here is 50
How do magic squares work?
Magic squares can be made to work in
several ways. A 3x3 magic square is made
in a different way to a 4x4 magic square
For example, magic squares can be
constructed in the following way:
Constructing magic squares
To construct 4 x 4 magic squares,
we need the following basic squares:
These squares identify where the numbers
(these can be any numbers) go
Constructing magic squares 2
1
0
0
0
0
0
0
1
0
1
0
0
0
0
1
0
1
0
0
0
B1
0
0
1
0
0
1
0
0
0
0
0
1
B4
1
0
0
0
0
0
1
0
0
0
0
1
0
1
0
0
0
1
0
0
B2
0
1
0
0
0
0
1
0
1
0
0
0
B5
0
0
0
1
0
0
0
1
0
0
1
0
1
0
0
0
B3
0
0
1
0
1
0
0
0
0
1
0
0
B6
0
0
0
1
0
0
0
1
0
1
0
0
B7
1
0
0
0
0
0
1
0
Constructing magic squares 3
From these squares, (any) numbers are
chosen and put into the following formula:
9B1+7B2+6B3+7B4-B5+2B6+3B7
The numbers are then put into a magic square,
corresponding with were the 1s’ are, in the above
basic squares
To see the basic squares again, click here
Constructing the Dürer magic square
Start in the bottom right corner of the grid, and count along,
but only put the numbers you count on the diagonal lines.
i.e. follow the path marked out below, putting numbers on the
diagonals
16
13
10 11
6
4
7
1
As you can see, these numbers lie on the diagonal lines
Constructing the Dürer magic square 2
Next, starting at the bottom left, count backwards
from 16, putting the numbers in the blank spaces.
16 3
2 13
5 10 11
8
6
7
12
4 15 14
1
9
Constructing the Dürer magic square 3
As you can see, the Dürer magic square
has now been constructed
16 3
2 13
5 10 11
9
6
8
7 12
4 15 14 1
The Magic Formula
To find out what the magic total is, we can
use a formula, which will tell us the total of
the rows, columns, diagonals etc.
The formula is ½n(n²+1)
Where n is the number of rows
Deriving the formula
To derive the formula for the magic
square, we must first assign the magic
total. Let’s call this x.
We must then assign the number of rows
(i.e. the size of the square). As you may
have gathered, this will be called n
Deriving the formula 2
We can write this out as
1+2+3+4+…+n²=n.x
From Pure maths, this can be written as:
n^2
∑=n.x
i =1
Deriving the formula 3
From pure maths, we know that the
formula for this series is:
n.x= ½n²(n²+1)
We then divide both sides by n, to get:
x= ½n(n²+1)
This formula only works for magic squares, which contain
integers (i.e. whole numbers, no decimals)
Create your own magic square
To create your own magic square, follow the link below:
My own magic square
Some useful websites
www.mathforum.org/alejandre/magic.square/adler/adler/whatsquare.html
http://digilander.libero.it/ice00/magic/general/MagicSquare.html
http://www.mrexcel.com/tip069.shtml
http://www.MarkFarrar.co.uk/msqhst01.htm
Further research
Aside from magic squares, there are also a number of
magic shapes you could go onto study
•Magic cubes
•Magic stars
•Magic circles
•Magic word squares
The list goes on!!!