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Project Euler Solutions

Project Euler 28 Solution

Project Euler 28 Solution

Project Euler 28: Find the sum of both diagonals in a square spiral


Project Euler 28 Problem Description

Project Euler 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 both diagonals is 101.

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

Analysis

The hackerrank Project Euler 28 version of this problem extends the size of the square from 1001 to any odd n, where 1 ≤ n ≤ 1018, so any iterative approach will exceed their time limit. A closed form of the summation will have to be determined.

Using a loop to count corners

Looking at the “corners” which form the two principal diagonals produce a simple series: (3, 5, 7, 9), (13, 17, 21, 25), (31, 37, 43, 49), …
This can be quantified as (n2 – 3n + 3, n2 – 2n + 2, n2 – n + 1, n2) and summing them together yields 4n2 – 6n + 6; the sum of the corners for each odd-length square:


3 + 5 + 7 + 9 = 24, 13 + 17 + 21 + 25 = 76, 31 + 37 + 43 + 49 = 160, …

sum, d, size = 1, 1, 1001
 
for n in range(2, size, 2):
  sum += 4*d + 10*n
  d += n*4;
 
print "Answer to PE28 = ",sum

Improved approach: Closed form summation

These sums of corners form the sum of the diagonals, after adding 1 for the center, for every i by i spiral where 3 ≤ in, for odd n and can be expressed as:

 1+\sum\limits_{i=1}^{s} 4(2i+1)^2-6(2i+1)+6, s=\frac{n-1}{2}

and simplified to:

 1+\sum\limits_{i=1}^{s} 16i^2+4i+4

Finally, we can further simplify this summation into a closed form equation by using the algebra of summation notation (Wikipedia or Project Euler Problem 6):

 1+16\cdot\sum\limits_{i=1}^{s}i^2 + 4\cdot\sum\limits_{i=1}^{s}i + \sum\limits_{i=1}^{s}4

 \frac{16s(s + 1)(2s + 1)}{6} + \frac{4s(s + 1)}{2} + 4s + 1
 \frac{16s^3 + 30s^2 + 26s +3}{3}

Or, factored if you prefer:

 (\frac{2s}{3}) (8s^2 + 15s + 13) + 1

Let’s test this equation with the example in the problem statement, n (odd side length ≥ 3) = 5: s = \frac{5-1}{2} = 2

 \frac{16\cdot2^3 + 30\cdot2^2 + 26\cdot2 +3}{3}= 101

HackerRank Project Euler 28 runs up to 10,000 trials with odd values for N up to 1018,
requiring the need for the closed form summation to successfully pass the test cases in under a second.

Project Euler 28
This program and method
solves all test cases for
Project Euler 28 on HackerRank

Project Euler 28 Solution

Runs < 0.001 seconds in Python 2.7.
download arrowUse this link to get the Project Euler 28 Solution Python 2.7 source.

Afterthoughts

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