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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.

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Magic squares l.jpg

Magic Squares!!!

By John Burton


What are magic squares l.jpg
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 l.jpg
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

8

9

6

7

12

4

15

14

1


D rers square l.jpg
Dürers Square

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

Here is the year


Some more magic squares l.jpg
Some More Magic Squares

8

1

6

The total for each row, column etc. is 15

3

5

7

4

9

2

The total for each row, column here is 50


How do magic squares work l.jpg
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 l.jpg
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 l.jpg
Constructing magic squares 2

B1

B2

B3

B4

B5

B6

B7


Constructing magic squares 3 l.jpg
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 d rer magic square l.jpg
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

7

4

1

As you can see, these numbers lie on the diagonal lines


Constructing the d rer magic square 2 l.jpg
Constructing the Dürer magic square 2

Next, starting at the bottom left, count backwards from 16, putting the numbers in the blank spaces.

16

13

3

2

5

10

11

8

6

7

9

12

4

1

15

14


Constructing the d rer magic square 3 l.jpg
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

8

9

6

7

12

4

15

14

1


The magic formula l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
Create your own magic square

To create your own magic square, follow the link below:

My own magic square


Some useful websites l.jpg
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 l.jpg
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!!!



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