Euler problems/41 to 50
m (→[http://projecteuler.net/index.php?section=problems&id=48 Problem 48])
(→[http://projecteuler.net/index.php?section=problems&id=41 Problem 41]: a solution)
Revision as of 01:25, 11 April 2007
What is the largest n-digit pandigital prime that exists?
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
How many triangle words can you make using the list of common English words?
problem_42 = undefined
Find the sum of all pandigital numbers with an unusual sub-string divisibility property.
problem_43 = undefined
Find the smallest pair of pentagonal numbers whose sum and difference is pentagonal.
problem_44 = undefined
After 40755, what is the next triangle number that is also pentagonal and hexagonal?
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..]]
What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?
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
Find the first four consecutive integers to have four distinct primes factors.
problem_47 = undefined
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
Find arithmetic sequences, made of prime terms, whose four digits are permutations of each other.
problem_49 = undefined
10 Problem 50
Which prime, below one-million, can be written as the sum of the most consecutive primes?
problem_50 = undefined