All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
2 diagonal sums (Posted on 2016-04-26) Difficulty: 3 of 5
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 main diagonals in a N by N spiral formed in the same way?
Source: Project Euler

No Solution Yet Submitted by Ady TZIDON    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution Comment 1 of 1
I see you are totaling the numbers on that X-shaped area, rather than summing the two totals of the diagonals (i.e., the 1 at the center counts only once).

The members of each successive shell follow a second degree polynomial function; therefore, the totals follow a third degree polynomial

         73                          81
         
             43 44 45 46 47 48  49
             42 21 22 23 24 25  26
               
             41 20  7  8  9 10  27
               
             40 19  6  1  2 11  28
               
             39 18  5  4  3 12  29
               
             38 17 16 15 14 13  30
                
             37 36 35 34 33 32  31
             
          65                        57
             
 For a 3 x 3 the total is 25
 For a 5 x 5 the total is 101
 For a 7 x 7 the total is 261
 
 t = a*n^3 + b*n^2 + c*n + d
 
 25 = 27a + 9b + 3c + d
 101 = 125a + 25b + 5c + d
 261 = 343a + 49b + 7c + d
 537 = 729a + 81b + 9c + d
 
 Using
 
 https://www.symbolab.com/solver/system-of-equations-calculator
 
 to solve, we get:
 
 t = 2*n^3 / 3 + n^2 / 2 + 4*n/3 - 3/2

This checks out when evaluated.

  Posted by Charlie on 2016-04-26 10:09:11
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information