Euler problems/51 to 60

Problem 51

Find the smallest prime which, by changing the same part of the number, can form eight different primes.

Solution:

problem_51 = undefined

Problem 52

Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits in some order.

Solution:

problem_52 = head [n | n <- [1..],
                   digits (2*n) == digits (3*n),
                   digits (3*n) == digits (4*n),
                   digits (4*n) == digits (5*n),
                   digits (5*n) == digits (6*n)]
    where digits = sort . show

Problem 53

How many values of C(n,r), for 1 ≤ n ≤ 100, exceed one-million?

Solution:

problem_53 = length [n | n <- [1..100], r <- [1..n], n `choose` r > 10^6]
    where n `choose` r
           | r > n || r < 0 = 0
           | otherwise      = foldl (\z j -> z*(n-j+1) `div` j) n [2..r]

Problem 54

How many hands did player one win in the poker games?

Solution:

problem_54 = undefined

Problem 55

How many Lychrel numbers are there below ten-thousand?

Solution:

problem_55 = length $ filter isLychrel [1..9999]
    where isLychrel n = all notPalindrome (take 50 (tail (iterate revadd n)))
          notPalindrome s = (show s) /= reverse (show s)
          revadd n = n + rev n
              where rev n = read (reverse (show n))

Problem 56

Considering natural numbers of the form, a^b, finding the maximum digital sum.

Solution:

problem_56 = maximum [dsum (a^b) | a <- [1..99], b <-[1..99]]
    where dsum 0 = 0
          dsum n = let ( d, m ) = n `divMod` 10 in m + ( dsum d )

Problem 57

Investigate the expansion of the continued fraction for the square root of two.

Solution:

problem_57 = length $ filter topHeavy $ take 1000 convergents 
    where topHeavy r = numDigits (numerator r) > numDigits (denominator r)
          numDigits = length . show
          convergents = iterate next (3%2)
          next r = 1 + 1/(1+r)

Problem 58

Investigate the number of primes that lie on the diagonals of the spiral grid.

Solution:

base :: (Integral a) => [a]
base = base' 2
    where
        base' n = n:n:n:n:(base' $ n + 2)

pascal = scanl (+) 1 base

ratios :: [Integer] -> [Double]
ratios (x:xs) = 1.0 : ratios' 0 1 xs
    where
        ratios' n d (w:x:y:z:xs) = ((fromInteger num)/(fromInteger den)) : (ratios' num den xs)
            where
                num = (p w + p x + p y + p z + n)
                den = (d + 4)
                p n = case isPrime n of
                        True -> 1
                        False -> 0

problem_58 = fst $ head $ dropWhile (\(_,a) -> a > 0.1) $ zip [1,3..] (ratios pascal)

Problem 59

Using a brute force attack, can you decrypt the cipher using XOR encryption?

Solution:

problem_59 = undefined

Problem 60

Find a set of five primes for which any two primes concatenate to produce another prime.

Solution:

problem_60 = undefined