Difference between revisions of "Euler problems/41 to 50"
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== [http://projecteuler.net/index.php?section=problems&id=41 Problem 41] == |
== [http://projecteuler.net/index.php?section=problems&id=41 Problem 41] == |
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What is the largest n-digit pandigital prime that exists? |
What is the largest n-digit pandigital prime that exists? |
Revision as of 20:56, 23 June 2007
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:
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