# Euler problems/41 to 50

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(added solution for 49 (a bit ugly though)) |
Jim Burton (Talk | contribs) m (Made some changes to problem 49) |

## Revision as of 10:02, 9 July 2007

## 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:

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

## 3 Problem 43

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

Solution:

problem_43 = undefined

## 4 Problem 44

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

Solution:

problem_44 = undefined

## 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:

problem_47 = undefined

## 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.

problem_48 = sum [n^n | n <- [1..1000]] `mod` 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 library functions where possible...makes it slightly 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 (1234,4) => [1,2,3,4] each :: (Int, Int) -> [Int] each = unfoldr (\(y,o) -> let x = 10 ^ (o-1) (d,m) = y `divMod` x in if o == 0 then Nothing else Just (d,(m,o-1))) ispermut :: Int -> Int -> Bool ispermut x y = sort (each (x,4)) == sort (each (y,4)) 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] []

## 10 Problem 50

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

Solution:

problem_50 = undefined