# Project Euler

## Number spiral diagonals

### Problem 28

Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:

**21** 22 23 24 **25**

20 **7** 8 **9** 10

19 6 **1** 2 11

18 **5** 4 **3** 12

**17** 16 15 14 **13**

It can be verified that the sum of the numbers on the diagonals is 101.

What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed in the same way?

Like Problem 1 and Problem 6, this problem can be solve using summation when examine further. Let look at the 3×3 spiral, we can see that the sum is 1+3+5+7+9, or we can rewrite this as 1+2*(3+9), where 3 and 9 is the minimum and maximum value of the square ring with 3, respectively. For 5×5 spiral, we can write the sum as 1+2*[(3+9) + (13+25)]. If you look further, you will see a pattern such that for ixi spiral, the sum is . To represent this as a summation, the summation need to adjust to only include odd number by having i = 2j+1 and using new upper [(1001 – 1)/2 = 500] and lower [(3-1) / 2 = 1] limits:

by separating and constant terms and pull out the multipliers. We can use the summation formula to obtain the correct answers.