## Question

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 $\frac{8}{13} \approx 62\%$.

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%?

isPrime :: Int -> Bool
isPrime n | n < 1 = False
| otherwise = not $or [n rem x == 0 | x <- [2..floor$ sqrt $fromIntegral n]] primeCorners :: Int -> Int primeCorners n = sum [1 | x <- [1..3], isPrime$ n^2 - x*(n - 1)]

expansion :: [(Int, Int)]
expansion = scanl (\(p, t) x -> (p + primeCorners x, t + 4)) (0, 1) [3,5..]

main :: IO ()
main = print $3 + 2 * (length$ takeWhile id $tail [10*p >= t | (p, t) <- expansion])$ ghc -O2 -o spiral-primes spiral-primes.hs
real   0m15.097s
user   0m14.996s
sys    0m0.012s