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:
score :: String -> Int
score = sum . map ((subtract 64) . ord . toUpper)
istrig :: Int -> Bool
istrig n = istrig' n trigs
istrig' :: Int -> [Int] -> Bool
istrig' n (t:ts) | n == t = True
| otherwise = if t < n && head ts > n then False else istrig' n ts
trigs = map (\n -> n*(n+1) `div` 2) [1..]
--get ws from the Euler site
ws = ["A","ABILITY" ... "YOURSELF","YOUTH"]
problem_42 = length $ filter id $ map (istrig . score) ws
Problem 43
Find the sum of all pandigital numbers with an unusual sub-string divisibility property.
Solution:
import Data.List (inits, tails)
perms :: [a] -> [[a]]
perms [] = [[]]
perms (x:xs) = [ p ++ [x] ++ s | xs' <- perms xs
, (p, s) <- zip (inits xs') (tails xs') ]
check :: String -> Bool
check n = all (\x -> (read $ fst x) `mod` snd x == 0) $ zip (map (take 3) $ tail $ tails n) [2,3,5,7,11,13,17]
problem_43 :: Integer
problem_43 = foldr (\x y -> read x + y) 0 $ filter check $ perms "0123456789"
Problem 44
Find the smallest pair of pentagonal numbers whose sum and difference is pentagonal.
Solution:
combine xs = combine' [] xs
where
combine' acc (x:xs) = map (\n -> (n, x)) acc ++ combine' (x:acc) xs
problem_44 = d $ head $ filter f $ combine [p n| n <- [1..]]
where
f (a,b) = t (abs $ b-a) && t (a+b)
d (a,b) = abs (a-b)
p n = n*(3*n-1) `div` 2
t n = p (fromInteger(round((1+sqrt(24*fromInteger(n)+1))/6))) == n
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:
import Data.List (group)
factor_lengths :: [(Integer,Int)]
factor_lengths = [(n, length $ group $ primeFactors n)| n <- [2..]]
problem_47 :: Integer
problem_47 = f factor_lengths
where
f (a:b:c:d:xs)
| 4 == snd a && snd a == snd b && snd b == snd c && snd c == snd d = fst a
| otherwise = f (b:c:d:xs)
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:
I'm new to haskell, improve here :-)
I tidied up your solution a bit, mostly by using and composing library functions where possible...makes it faster on my system. Jim Burton 10:02, 9 July 2007 (UTC)
import Data.List
isprime :: (Integral a) => a -> Bool
isprime n = isprime2 2
where isprime2 x | x < n = if n `mod` x == 0 then False else isprime2 (x+1)
| otherwise = True
-- 'each' works like this: each (4,1234) => [1,2,3,4]
each :: (Int, Int) -> [Int]
each = unfoldr (\(o,y) -> let x = 10 ^ (o-1)
(d,m) = y `divMod` x in
if o == 0 then Nothing else Just (d,(o-1,m)))
ispermut :: Int -> Int -> Bool
ispermut = let f = (sort . each . (,) 4) in (. f) . (==) . f
isin :: (Eq a) => a -> [[a]] -> Bool
isin = any . elem
problem_49_1 :: [Int] -> [[Int]] -> [[Int]]
problem_49_1 [] res = res
problem_49_1 (pr:prims) res = problem_49_1 prims res'
where res' = if pr `isin` res then res else res ++ [pr:(filter (ispermut pr) (pr:prims))]
problem_49 :: [[Int]]
problem_49 = problem_49_1 [n | n <- [1000..9999], isprime n] []
Problem 50
Which prime, below one-million, can be written as the sum of the most consecutive primes?
Solution: (prime and isPrime not included)
findPrimeSum ps | isPrime sumps = Just sumps
| otherwise = findPrimeSum (tail ps) `mplus` findPrimeSum (init ps)
where sumps = sum ps
problem_50 = findPrimeSum $ take 546 primes