**What is the Math behind Magic Squares?**

A magic square is basically just an arithmetic sequence arranged in a special way. What do we know about an arithmetic sequence? An arithmetic sequence is a sequence of numbers each with a common difference. Looking at it a different way, it is a sequence of number by which each number in the sequence is found by adding a specific number to the first number in the sequence. For Example: The sequence

**1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11..**..
is a sequence that starts with 1 and each successive number in the sequence is found by adding 1 to the previous number in the sequence . Now let’s say we want to know the sum of the first 9 digits in this sequence. We could just add them all up one at a time, but that generally takes more time. And with larger sequences it would take a really long time. So we will set up a formula.

Looking at a arithmetic sequence algebraically we need to set up some variables. We will denote the starting value with,

**a,**and the common difference with**x**and each term is**n**so algebraically an arithmetic sequence looks like this**A, (A + X),( A + 2X), (A +3X), (A +4X), ….. (A + Nx),**
Where (A + nX) is the n

^{th }term^{ }in the sequence.**s**is the_{n }**sum of the ne terms**
When we wish to know the sum of n terms of the sequence what we are saying algebraically is

**s**

_{n}= a + (a + x) +...+ (a + 2x) + …+ (a + (n-2) x) + (a + (n-1) x)
And in order to reduce this to a simple formula we can work with we will take the above sequence and reverse it and add it to itself.

**S**

_{n}= a + (a + x) +...+ (a + 2x) + …+ (a + (n-2) x) + (a + (n-1) x)

__+ S___{n}= (a + (n-1) x) + (a + (n-2) x) +…+ (a + 2x) +… + (a + x) + a

**2 S**

_{n}= (2a + (n-1) x) + (2a + (n-1) x) + …+(2a + (n-1) x) +…+(2a + (n-1) x)
Since we are working with sequences of n terms grouping the last part is as simple as taking

**(2a + (n-1) x),**and multiplying it by n

**2 S**

_{n}= n (2a + (n-1) x)
Then when we divide by 2 we have

**S**_{n}=__n (2a + (n-1) x)__**2**

The resulting equation is close to what we will be using for magic squares.

The only difference is that with

The only difference is that with

**S**_{n}**=**__n (2a + (n-1) x)__is used with linear sequences that can be graphed out on a**2**

number line. In order to modify this for working with magic squares we simply square the n Inside the parenthesis ()

which changes our equation to

which changes our equation to

**S**_{n}=__n (2a + (__**We do this**__n__^{2}-1) x)**2**

because (

**n-1)**is used to represent last term and in magic squares the number of terms is the dimension squared so the last term would be**(n**^{2}-1).
## No comments:

## Post a Comment