mirror of
https://github.com/correl/euler.git
synced 2024-11-30 11:09:52 +00:00
49 lines
1.6 KiB
Python
49 lines
1.6 KiB
Python
|
"""Investigate the number of primes that lie on the diagonals of the spiral grid.
|
||
|
|
|
||
|
|
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%?
|
||
|
|
"""
|
||
|
|
|
||
|
|
from e007 import is_prime
|
||
|
|
|
||
|
|
def spiral_corners_generator():
|
||
|
|
n = 1
|
||
|
|
size = 1
|
||
|
|
while True:
|
||
|
|
corners = []
|
||
|
|
if size == 1:
|
||
|
|
yield [1]
|
||
|
|
else:
|
||
|
|
for i in range(4):
|
||
|
|
n = n + (size - 1)
|
||
|
|
corners.append(n)
|
||
|
|
yield corners
|
||
|
|
size = size + 2
|
||
|
|
|
||
|
|
def main():
|
||
|
|
size = 1
|
||
|
|
diagonals = 0
|
||
|
|
primes = 0
|
||
|
|
for corners in spiral_corners_generator():
|
||
|
|
diagonals = diagonals + len(corners)
|
||
|
|
primes = primes + len([c for c in corners if is_prime(c)])
|
||
|
|
pct = primes / float(diagonals) * 100
|
||
|
|
if size > 7 and pct < 10.0:
|
||
|
|
break
|
||
|
|
size = size + 2
|
||
|
|
|
||
|
|
print 'Side Length: {0}, Percentage: {1}'.format(size, pct)
|
||
|
|
|
||
|
|
if __name__ == '__main__':
|
||
|
|
main()
|