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# In this investigation, I will explore how magic squares work.

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Introduction

Magic Squares investigation

Scarlett Li

3x3 Magic Square

In this investigation, I will explore how magic squares work. A magic square is a grid which is filled with an arrangement of numbers starting from 1 to n2.  The sum of the numbers in each row, column and diagonal are the same and this is called the ‘Magic Total’.  When doing a 3x3x3 magic square, the magic number is always three times the middle number and it is always three times bigger because it is a 3x3 grid. There are only eight ways which you can add up different numbers to your magic total:

A 3x3 magic square only has eight different solutions and

Middle

6

7

8

9

10

11

12

13

14

 n+3 n-4 n+1 n-2 n n+2 n-1 n+4 n-3

This is the algebraic way of creating a 3x3 magic square

 18 11 16 13 15 17 14 19 12

Here are some more examples with different magic square sequences:

Magic sequence numbers: 11, 12, 13, 14, 15, 16, 17, 18, 19

Magic Total: 135

Magic Number: 45

 7 0 5 2 4 6 3 8 1

Middle Number of square: 15

Magic sequence numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8

Magic Total: 36

Magic Number: 12

 5 -2 3 0 2 4 1 6 -1

Conclusion

It is possible to arrange numbers with some non consecutive numbers:

 3 10 2 14 15 1 13 5 12

Magic set of numbers: 1, 2, 3, 5, 10, 12, 13, 14

Magic Total: 18

Magic Number: 6

Middle Number of square: 2

4x4 Magic Square

I will try to explore how the 4x4 magic square works here and I will try to find the different patterns here.

 1 15 14 4 12 6 7 9 8 10 11 5 13 3 2 16

The formula to work out the magic number is: 4x(42+1)

2

The four corners of the square add up to 34

The four center cells add up to 34

15 and 14 in the top row add up to 34

12 and 8 in the left column add up to 34

The numbers from each row, column and diagonal add up to 34

The magic sequence numbers here are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

When you add all the numbers together it gives you a magic total of 136 and when you divide 136 by 4, it gives you the magic number of 34.

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