1. Magic Squares
Laith Abdulhamid
December 4, 2014

2. Introduction
This project is about “Magic Squares”. The point of this project is to state that we can explain using Algebra. This project is sort of
like evidence. In this project, the “magic squares” will each have an equation with a number of variables that will either sum up or
subtract into a certain number, and each of these variables will represent a number. Each of row and column is supposed to end
up with the same number.

3. Begin the Math
a = 5
b = 3
c = 2
5 + 2 = 7 5 + 3 - 2 = 6 5 - 3 = 2
5 - 3 - 2 = 0 5 = 5 5 + 3 + 2 = 10
5 + 3 = 8 5 - 3 + 2 = 4 5 - 2 = 3
=15
=15
= 15
15 15 15
15
15

4. Proving Algebra to Explain
Row #1 Diagonal #1
a + c + a + b - c + a - b = 15 a + c + 2a - c = 15
Row #2 Diagonal #2
a - b - c + 2a + b + c = 15 a + b + 2a - b = 15
Row #3
a + b + a - b + c + a - c = 15
Column #1
a + c + a - b - c + a + b = 15
Column #2
a + b - c + 2a - b + c = 15
Column #3
a - b + a + b + c + a - c =15
a = 5
b = 3
c = 2

5. 4 x 4 Magic squares
e 3 + 3 - 2 = 4 5 = 5 5 + 5 + 3 + 3 = 16 3 + 3 + 3 = 9
5 + 5 + 2 + 2 = 14 5 + 3 + 3 = 11 2 = 2 5 + 2 = 7
3 - 2 = 1 5 + 3 = 8 5 + 5 + 3 = 13 5 + 5 + 2 = 12
5 + 5 + 5 = 15 5 + 5 = 10 3 = 3 3 + 3 = 6
34
34
34
34 34 34 34
34
34
34

6. Prove ( for the 4x4)
Source: http://www2.potsdam.edu/parksjm/responsesPuzzle96.htm
Row #1 Column #4
2b - c + 3a + 5b = 34 3b + a + c + 2a + c + 2b = 34
Row #2 Diagonal #1
2a + 2c + a + 2b + c + a + c = 34 4a + b + c +3b = 34
Row #3 Diagonal #2
b - c + a + b + 2a + b + 2a + c = 34 2b - c + a + 2b +2a + 3b = 34
Row #4
3b + a + c + 2a + c + 2b = 34
Column #1
2b - c + 2a + 2c + b - c + 3a = 34
Column #2
2a + 2b + a + b + 2a = 34
Column #3
2a + 2b + c + 2a + 2b = 34

7. How I Used Algebra to Explain
The way I used algebra to explain, was by using variables. These variables had values and each
letter (variable) had a different value. a represented 5, b represented 3 and c represented 2.
I chose the number 5, 3 and 2 because those numbers would add up to 15 and 34 when they
were put together in a certain order. for example, in the 3x3 square row 3, it all added up to 15
and the columns and diagonal rows were still compatible and still could add up to 15.