Euler problems/41 to 50

Problem 41

What is the largest n-digit pandigital prime that exists?

Solution:

problem_41 = head [p | n <- init (tails "987654321"),
                   p <- perms n, isPrime (read p)]
    where perms [] = [[]]
          perms xs = [x:ps | x <- xs, ps <- perms (delete x xs)]
          isPrime n = n > 1 && smallestDivisor n == n
          smallestDivisor n = findDivisor n (2:[3,5..])
          findDivisor n (testDivisor:rest)
              | n `mod` testDivisor == 0      = testDivisor
              | testDivisor*testDivisor >= n  = n
              | otherwise                     = findDivisor n rest

Problem 42

How many triangle words can you make using the list of common English words?

Solution:

problem_42 = undefined

Problem 43

Find the sum of all pandigital numbers with an unusual sub-string divisibility property.

Solution:

problem_43 = undefined

Problem 44

Find the smallest pair of pentagonal numbers whose sum and difference is pentagonal.

Solution:

problem_44 = undefined

Problem 45

After 40755, what is the next triangle number that is also pentagonal and hexagonal?

Solution:

problem_45 =  head . dropWhile (<= 40755) $ match tries (match pents hexes)
    where match (x:xs) (y:ys)
              | x < y  = match xs (y:ys)
              | y < x  = match (x:xs) ys
              | otherwise = x : match xs ys
          tries = [n*(n+1) `div` 2   | n <- [1..]]
          pents = [n*(3*n-1) `div` 2 | n <- [1..]]
          hexes = [n*(2*n-1)         | n <- [1..]]

Problem 46

What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?

Solution:

This solution is inspired by exercise 3.70 in Structure and Interpretation of Computer Programs, (2nd ed.).

problem_46 = head $ oddComposites `orderedDiff` gbSums

oddComposites = filter ((>1) . length . primeFactors) [3,5..]

gbSums = map gbWeight $ weightedPairs gbWeight primes [2*n*n | n <- [1..]]
gbWeight (a,b) = a + b

weightedPairs w (x:xs) (y:ys) =
    (x,y) : mergeWeighted w (map ((,)x) ys) (weightedPairs w xs (y:ys))

mergeWeighted w (x:xs)  (y:ys)
    | w x <= w y  = x : mergeWeighted w xs (y:ys)
    | otherwise   = y : mergeWeighted w (x:xs) ys

x `orderedDiff` [] = x
[] `orderedDiff` y = []
(x:xs) `orderedDiff` (y:ys)
    | x < y     = x : xs `orderedDiff` (y:ys)
    | x > y     = (x:xs) `orderedDiff` ys
    | otherwise = xs `orderedDiff` ys

Problem 47

Find the first four consecutive integers to have four distinct primes factors.

Solution:

problem_47 = undefined

Problem 48

Find the last ten digits of 1¹ + 2² + ... + 1000¹⁰⁰⁰.

Solution: If the problem were more computationally intensive, modular exponentiation might be appropriate. With this problem size the naive approach is sufficient.

problem_48 = sum [n^n | n <- [1..1000]] `mod` 10^10

Problem 49

Find arithmetic sequences, made of prime terms, whose four digits are permutations of each other.

Solution:

problem_49 = undefined

Problem 50

Which prime, below one-million, can be written as the sum of the most consecutive primes?

Solution:

problem_50 = undefined