A **spiral diagonal** is formed when you take a number, `1`, and increase that number as you move to the right and below. This forms a spiral.

The following slides show how a number spiral is formed for a 4x4 matrix.

The sum of the diagonals in a 3x3 number spiral is $25$. 

In the number spirals diagonal problem, the sum of diagonals for nxn needs to be found.

If one looks closely, one can find a pattern. 
Consider the following matrix.

The values in each diagonal correspond to:

$a_{n} = (9, 25, 49, 81 ...)$
$b_{n} = (7, 21, 43, 73 ...)$
$c_{n} = (5, 17, 37, 65 ...)$
$d_{n} = (3, 13, 31, 57 ...)$

These correspond to the following sequence:

$a_{n} = 4n^2+4n+1$
$b_{n} = 4n^2+2n+1$ 
$c_{n} = 4n^2+1$ 
$d_{n} = 4n^2-2n+1$

The sum of all the terms includes the one in the center; therefore, it becomes:

>**Sum(n) = $1+ \sum_{k=1}^n (a_i + b_i + c_i + d_i)$ 
Sum(n) = $1+\sum_{k=1}^n (16i^2 + 4i +4)$ 
Sum(n) = $1+\sum_{k=1}^n 16i^2 + \sum_{k=1}^n 4i + \sum_{k=1}^n4$**

Using summation rules we can reduce it to: 
>**Sum(n) = $\frac{3+2n(8n^2 + 15n +13)}{3}$**

This simple formula can then be used to calculate the sum of the diagonal for the matrix of any dimension.

>**Remember**: Here, `n` equals 1/2 the number of elements in a diagonal, excluding the center term. 
<centre>**n =** **$\frac{dim-1}{2}$** 
 Where `dim` is the dimension of the matrix.

# Code

The following code shows how to find the sum of the diagonal in a number spiral for an nxn matrix.

#include <iostream>
using namespace std;
// declare function sum
int sum(int dim)
{
 int n = (dim - 1) /2;//find n 
 // use the formula defined above
 int x = (3 + 2 * n * ( 8 * n * n + 15 * n +13)) /3;

 return x;
}
int main() {
 // call the function and print the value
 int diagonal = sum(5);
 cout << diagonal << endl;
 return 0;
}


# declare function sum
def sum(dim):
  n = (dim - 1) /2;#find n 
  # use the formula defined above
  x = (3 + 2 * n * ( 8 * n * n + 15 * n +13)) /3
  return x

def main():
  # call the function and print the value
  diagonal = sum(5)
  print (diagonal)

main()

Python

How to solve the 'number spiral diagonals' problem

Identify patterns in diagonals of a number spiral matrix, use derived formula to calculate sum efficiently.