# Euler problems/41 to 50

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## Revision as of 03:10, 6 January 2008

## Contents |

## 1 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

## 2 Problem 42

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

Solution:

import Data.Char score = sum . map ((subtract 64) . ord . toUpper) istrig n = istrig' n trigs 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..] problem_42 ws= length $ filter id $ map (istrig . score) ws main=do f<-readFile "words.txt" let words=tail$("":)$read $"["++f++"]" print $problem_42 words

## 3 Problem 43

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

Solution:

import Data.List (inits, tails) perms [] = [[]] perms (x:xs) = [ p ++ [x] ++ s | xs' <- perms xs , (p, s) <- zip (inits xs') (tails xs') ] 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 = foldr (\x y -> read x + y) 0 $ filter check $ perms "0123456789"

## 4 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

## 5 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..]]

## 6 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

## 7 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)

## 8 Problem 48

Find the last ten digits of 1^{1} + 2^{2} + ... + 1000^{1000}.

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

mulMod :: Integral a => a -> a -> a -> a mulMod a b c= (b * c) `rem` a squareMod :: Integral a => a -> a -> a squareMod a b = (b * b) `rem` a pow' :: (Num a, Integral b) => (a -> a -> a) -> (a -> a) -> a -> b -> a pow' _ _ _ 0 = 1 pow' mul sq x' n' = f x' n' 1 where f x n y | n == 1 = x `mul` y | r == 0 = f x2 q y | otherwise = f x2 q (x `mul` y) where (q,r) = quotRem n 2 x2 = sq x powMod :: Integral a => a -> a -> a -> a powMod m = pow' (mulMod m) (squareMod m) problem_48 = flip mod limit$sum [powMod limit n n | n <- [1..1000]] where limit=10^10

## 9 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))] p49a :: [[Int]] p49a = problem_49_1 [n | n <- [1000..9999], isprime n] [] unAdd []=[] unAdd (x:xs)=[x-y|y<-xs]++(unAdd xs) takeEqv []=[] takeEqv (x:xs)=[x|y<-xs,x-y==0]++(takeEqv xs) div2un []=[] div2un (x:xs)=[div (x-y) 2|y<-xs]++(div2un xs) eqvList x y =[a|a<-x,b<-y,a==b] problem_49 =[y| x<-p49a, let y=sort$nub x, length(y)>=4, let z=unAdd y, length(z)/=length(nub z), (eqvList (div2un y) (takeEqv z))/=[] ]

## 10 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