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problem58.py
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problem58.py
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#!/usr/bin/env python
# -*- coding: utf-8 -*-
"""
Project Euler Problem 58:
Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.
37 36 35 34 33 32 31
38 17 16 15 14 13 30
39 18 5 4 3 12 29
40 19 6 1 2 11 28
41 20 7 8 9 10 27
42 21 22 23 24 25 26
43 44 45 46 47 48 49
It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that
8 out of the 13 numbers lying along both diagonals are prime; that is, a ratio of 8/13 ≈ 62%.
If one complete new layer is wrapped around the spiral above, a square spiral with side length 9 will be formed. If this
process is continued, what is the side length of the square spiral for which the ratio of primes along both diagonals
first falls below 10%?
"""
# The numbers on the diagonals are formed by adding a certain amount, starting at 2, for 4 numbers, then increasing that
# amount by 2. The diagonals are thus: (1), (3, 5, 7, 9), (13, 17, 21, 25), (31, 37, 43, 49), ...
# Using this, simply add side lengths and count primes until primes/total < 0.1
# Runs in ~30 seconds
from math import sqrt
def is_prime(n):
for divisor in range(2, int(sqrt(n)) + 1):
if n % divisor == 0:
return False
return True
primes = 0
total = 1 # including 1
side_length = 1
currently_adding = 0
last = 1
while primes / total >= 0.1 or primes == 0:
if side_length % 100 == 1:
print(side_length, primes / total, primes, total, last)
side_length += 2
currently_adding += 2
new_last = last + currently_adding * 4
for diagonal in range(last + currently_adding, new_last + 1, currently_adding):
if is_prime(diagonal):
primes += 1
total += 4
last = new_last
print(side_length)