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 2 diagonal sums (Posted on 2016-04-26)
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

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 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

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