66 lines
2.2 KiB
Haskell
66 lines
2.2 KiB
Haskell
-- Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.
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--
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-- 37 36 35 34 33 32 31
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-- 38 17 16 15 14 13 30
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-- 39 18 5 4 3 12 29
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-- 40 19 6 1 2 11 28
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-- 41 20 7 8 9 10 27
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-- 42 21 22 23 24 25 26
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-- 43 44 45 46 47 48 49
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--
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-- 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%.
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--
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-- 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%?
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--
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-- Reflexion:
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-- if n is the side length of the square spiral. The numbers at each 'coin' are
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--
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-- 1 -> 1
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-- 2 -> 3 5 7 9
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-- 3 -> 13 17 21 25
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-- ...
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--
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-- The rule is
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-- (last_max_number + n.(size-1) | n in [1..4])
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-- examples:
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-- last_max_number=1
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-- size = 3
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-- 1+2 = 3, 1+2*2=5, 1+3*2=7, 1+4*2=9
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--
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-- last_max_number=9
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-- size=5
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-- 9+4=13, 9+2*4=17, 9+3*4=21, 9+4*4=25
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import Prime
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next_four_numbers :: (Num a) => a -> a -> [a]
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next_four_numbers n size = map (\p -> n + p*(size-1)) [1,2,3,4]
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nextState (prev,size) = (next_four_numbers (last prev) size, size+2)
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spiral = concat $ map fst $ iterate nextState ([1],3)
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diag_primes = map is_prime spiral
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ratio_func (x:xs) (p,q) = (p,q):(ratio_func xs b)
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where b = if x then (p+1,q+1) else (p,q+1)
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ratios = ratio_func diag_primes (0,0)
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takeNth n p (x:xs) = if n == p
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then x:takeNth n 1 xs
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else takeNth n (p+1) xs
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-- takeWhile (>0.1) (map fst $ takeNth 4 4 ratios)
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main = do
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-- putStrLn $ show $ firsts_spiral_numbers n
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putStrLn $ show $ take (4*n+1) spiral
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putStrLn $ show $ take (4*n+1) diag_primes
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putStrLn $ show $ take 10 $ takeNth 4 4 spiral
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-- putStrLn $ show $ tmp
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putStrLn $ show $ res
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putStrLn $ show $ ((res-1)/2) + 1 + 2
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where
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n=3
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rat=0.1
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tmp = takeWhile (\(p,q) -> (q < 2) || (p/q > rat) ) $ takeNth 4 3 ratios
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(_,res) = last $ tmp
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