Difference between revisions of "Euler problems/41 to 50"
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+ | == [http://projecteuler.net/index.php?section=problems&id=41 Problem 41] == |
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− | Do them on your own! |
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+ | What is the largest n-digit pandigital prime that exists? |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | import Data.List |
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+ | isprime a = isprimehelper a primes |
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+ | isprimehelper a (p:ps) |
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+ | | a == 1 = False |
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+ | | p*p > a = True |
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+ | | a `mod` p == 0 = False |
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+ | | otherwise = isprimehelper a ps |
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+ | primes = 2 : filter isprime [3,5..] |
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+ | problem_41 = |
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+ | head.filter isprime.filter fun $ [7654321,7654320..] |
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+ | where |
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+ | fun =(=="1234567").sort.show |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=42 Problem 42] == |
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+ | How many triangle words can you make using the list of common English words? |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | import Data.Char |
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+ | trilist = takeWhile (<300) (scanl1 (+) [1..]) |
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+ | wordscore xs = sum $ map (subtract 64 . ord) xs |
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+ | problem_42 megalist= |
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+ | length [ wordscore a | |
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+ | a <- megalist, |
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+ | elem (wordscore a) trilist |
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+ | ] |
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+ | main=do |
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+ | f<-readFile "words.txt" |
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+ | let words=read $"["++f++"]" |
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+ | print $problem_42 words |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=43 Problem 43] == |
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+ | Find the sum of all pandigital numbers with an unusual sub-string divisibility property. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | import Data.List |
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+ | l2n :: (Integral a) => [a] -> a |
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+ | l2n = foldl' (\a b -> 10*a+b) 0 |
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+ | |||
+ | swap (a,b) = (b,a) |
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+ | |||
+ | explode :: (Integral a) => a -> [a] |
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+ | explode = |
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+ | unfoldr (\a -> if a==0 then Nothing else Just $ swap $ quotRem a 10) |
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+ | problem_43 = sum . map l2n . map (\s -> head ([0..9] \\ s):s) |
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+ | . filter (elem 0) . genSeq [] $ [17,13,11,7,5,3,2] |
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+ | |||
+ | mults mi ma n = takeWhile (< ma) $ dropWhile (<mi) $ iterate (+n) n |
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+ | |||
+ | sequ xs ys = tail xs == init ys |
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+ | |||
+ | addZ n xs = replicate (n - length xs) 0 ++ xs |
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+ | |||
+ | genSeq [] (x:xs) = genSeq |
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+ | (filter (not . doub) |
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+ | $ map (addZ 3 . reverse . explode) $ mults 9 1000 x) |
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+ | xs |
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+ | genSeq ys (x:xs) = |
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+ | genSeq (do |
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+ | m <- mults 9 1000 x |
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+ | let s = addZ 3 . reverse . explode $ m |
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+ | y <- filter (sequ s . take 3) $ filter (not . elem (head s)) ys |
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+ | return (head s:y) |
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+ | ) xs |
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+ | genSeq ys [] = ys |
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+ | |||
+ | doub xs = nub xs /= xs |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=44 Problem 44] == |
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+ | Find the smallest pair of pentagonal numbers whose sum and difference is pentagonal. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | import Data.Set |
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+ | problem_44 = |
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+ | head solutions |
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+ | where |
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+ | solutions = |
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+ | [a-b | |
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+ | a <- penta, |
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+ | b <- takeWhile (<a) penta, |
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+ | isPenta (a-b), |
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+ | isPenta (b+a) |
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+ | ] |
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+ | isPenta = (`member` fromList penta) |
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+ | penta = [(n * (3*n-1)) `div` 2 | n <- [1..5000]] |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=45 Problem 45] == |
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+ | After 40755, what is the next triangle number that is also pentagonal and hexagonal? |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | isPent n = |
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+ | (af == 0) && ai `mod` 6 == 5 |
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+ | where |
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+ | (ai, af) = properFraction $ sqrt $ 1 + 24 * (fromInteger n) |
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+ | |||
+ | problem_45 = head [x | x <- scanl (+) 1 [5,9..], x > 40755, isPent x] |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=46 Problem 46] == |
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+ | What is the smallest odd composite that cannot be written as the sum of a prime and twice a square? |
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+ | |||
+ | Solution: |
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+ | |||
+ | This solution is inspired by exercise 3.70 in ''Structure and Interpretation of Computer Programs'', (2nd ed.). |
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+ | |||
+ | millerRabinPrimality on the [[Prime_numbers]] page |
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+ | |||
+ | <haskell> |
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+ | import Data.List |
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+ | isPrime x |
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+ | |x==3=True |
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+ | |otherwise=millerRabinPrimality x 2 |
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+ | problem_46 = |
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+ | find (\x -> not (isPrime x) && check x) [3,5..] |
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+ | where |
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+ | check x = |
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+ | not $ any isPrime $takeWhile (>0) $ map (\y -> x - 2 * y * y) [1..] |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=47 Problem 47] == |
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+ | Find the first four consecutive integers to have four distinct primes factors. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | import Data.List |
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+ | problem_47 = find (all ((==4).snd)) . map (take 4) . tails |
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+ | . zip [1..] . map (length . factors) $ [1..] |
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+ | fstfac x = [(head a ,length a)|a<-group$primeFactors x] |
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+ | fac [(x,y)]=[x^a|a<-[0..y]] |
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+ | fac (x:xs)=[a*b|a<-fac [x],b<-fac xs] |
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+ | factors x=fac$fstfac x |
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+ | primes = 2 : filter ((==1) . length . primeFactors) [3,5..] |
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+ | |||
+ | primeFactors n = factor n primes |
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+ | where |
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+ | factor _ [] = [] |
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+ | factor m (p:ps) | p*p > m = [m] |
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+ | | m `mod` p == 0 = p : [m `div` p] |
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+ | | otherwise = factor m ps |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=48 Problem 48] == |
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+ | Find the last ten digits of 1<sup>1</sup> + 2<sup>2</sup> + ... + 1000<sup>1000</sup>. |
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+ | |||
+ | Solution: |
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+ | If the problem were more computationally intensive, [http://en.wikipedia.org/wiki/Modular_exponentiation modular exponentiation] might be appropriate. With this problem size the naive approach is sufficient. |
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+ | |||
+ | powMod on the [[Prime_numbers]] page |
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+ | |||
+ | <haskell> |
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+ | problem_48 = flip mod limit$sum [powMod limit n n | n <- [1..1000]] |
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+ | where |
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+ | limit=10^10 |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=49 Problem 49] == |
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+ | Find arithmetic sequences, made of prime terms, whose four digits are permutations of each other. |
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+ | |||
+ | Solution: |
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+ | millerRabinPrimality on the [[Prime_numbers]] page |
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+ | |||
+ | <haskell> |
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+ | import Control.Monad |
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+ | import Data.List |
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+ | isPrime x |
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+ | |x==3=True |
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+ | |otherwise=millerRabinPrimality x 2 |
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+ | |||
+ | primes4 = takeWhile (<10000) $ dropWhile (<1000) primes |
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+ | |||
+ | problem_49 = do |
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+ | a <- primes4 |
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+ | b <- dropWhile (<= a) primes4 |
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+ | guard ((sort $ show a) == (sort $ show b)) |
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+ | let c = 2 * b - a |
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+ | guard (c < 10000) |
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+ | guard ((sort $ show a) == (sort $ show c)) |
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+ | guard $ isPrime c |
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+ | return (a, b, c) |
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+ | |||
+ | primes = 2 : filter (\x -> isPrime x ) [3..] |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=50 Problem 50] == |
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+ | Which prime, below one-million, can be written as the sum of the most consecutive primes? |
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+ | |||
+ | Solution: |
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+ | (prime and isPrime not included) |
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+ | |||
+ | <haskell> |
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+ | import Control.Monad |
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+ | findPrimeSum ps |
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+ | | isPrime sumps = Just sumps |
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+ | | otherwise = findPrimeSum (tail ps) `mplus` findPrimeSum (init ps) |
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+ | where |
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+ | sumps = sum ps |
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+ | |||
+ | problem_50 = findPrimeSum $ take 546 primes |
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+ | </haskell> |
Revision as of 04:57, 30 January 2008
Problem 41
What is the largest n-digit pandigital prime that exists?
Solution:
import Data.List
isprime a = isprimehelper a primes
isprimehelper a (p:ps)
| a == 1 = False
| p*p > a = True
| a `mod` p == 0 = False
| otherwise = isprimehelper a ps
primes = 2 : filter isprime [3,5..]
problem_41 =
head.filter isprime.filter fun $ [7654321,7654320..]
where
fun =(=="1234567").sort.show
Problem 42
How many triangle words can you make using the list of common English words?
Solution:
import Data.Char
trilist = takeWhile (<300) (scanl1 (+) [1..])
wordscore xs = sum $ map (subtract 64 . ord) xs
problem_42 megalist=
length [ wordscore a |
a <- megalist,
elem (wordscore a) trilist
]
main=do
f<-readFile "words.txt"
let words=read $"["++f++"]"
print $problem_42 words
Problem 43
Find the sum of all pandigital numbers with an unusual sub-string divisibility property.
Solution:
import Data.List
l2n :: (Integral a) => [a] -> a
l2n = foldl' (\a b -> 10*a+b) 0
swap (a,b) = (b,a)
explode :: (Integral a) => a -> [a]
explode =
unfoldr (\a -> if a==0 then Nothing else Just $ swap $ quotRem a 10)
problem_43 = sum . map l2n . map (\s -> head ([0..9] \\ s):s)
. filter (elem 0) . genSeq [] $ [17,13,11,7,5,3,2]
mults mi ma n = takeWhile (< ma) $ dropWhile (<mi) $ iterate (+n) n
sequ xs ys = tail xs == init ys
addZ n xs = replicate (n - length xs) 0 ++ xs
genSeq [] (x:xs) = genSeq
(filter (not . doub)
$ map (addZ 3 . reverse . explode) $ mults 9 1000 x)
xs
genSeq ys (x:xs) =
genSeq (do
m <- mults 9 1000 x
let s = addZ 3 . reverse . explode $ m
y <- filter (sequ s . take 3) $ filter (not . elem (head s)) ys
return (head s:y)
) xs
genSeq ys [] = ys
doub xs = nub xs /= xs
Problem 44
Find the smallest pair of pentagonal numbers whose sum and difference is pentagonal.
Solution:
import Data.Set
problem_44 =
head solutions
where
solutions =
[a-b |
a <- penta,
b <- takeWhile (<a) penta,
isPenta (a-b),
isPenta (b+a)
]
isPenta = (`member` fromList penta)
penta = [(n * (3*n-1)) `div` 2 | n <- [1..5000]]
Problem 45
After 40755, what is the next triangle number that is also pentagonal and hexagonal?
Solution:
isPent n =
(af == 0) && ai `mod` 6 == 5
where
(ai, af) = properFraction $ sqrt $ 1 + 24 * (fromInteger n)
problem_45 = head [x | x <- scanl (+) 1 [5,9..], x > 40755, isPent x]
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.).
millerRabinPrimality on the Prime_numbers page
import Data.List
isPrime x
|x==3=True
|otherwise=millerRabinPrimality x 2
problem_46 =
find (\x -> not (isPrime x) && check x) [3,5..]
where
check x =
not $ any isPrime $takeWhile (>0) $ map (\y -> x - 2 * y * y) [1..]
Problem 47
Find the first four consecutive integers to have four distinct primes factors.
Solution:
import Data.List
problem_47 = find (all ((==4).snd)) . map (take 4) . tails
. zip [1..] . map (length . factors) $ [1..]
fstfac x = [(head a ,length a)|a<-group$primeFactors x]
fac [(x,y)]=[x^a|a<-[0..y]]
fac (x:xs)=[a*b|a<-fac [x],b<-fac xs]
factors x=fac$fstfac x
primes = 2 : filter ((==1) . length . primeFactors) [3,5..]
primeFactors n = factor n primes
where
factor _ [] = []
factor m (p:ps) | p*p > m = [m]
| m `mod` p == 0 = p : [m `div` p]
| otherwise = factor m ps
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.
powMod on the Prime_numbers page
problem_48 = flip mod limit$sum [powMod limit n n | n <- [1..1000]]
where
limit=10^10
Problem 49
Find arithmetic sequences, made of prime terms, whose four digits are permutations of each other.
Solution: millerRabinPrimality on the Prime_numbers page
import Control.Monad
import Data.List
isPrime x
|x==3=True
|otherwise=millerRabinPrimality x 2
primes4 = takeWhile (<10000) $ dropWhile (<1000) primes
problem_49 = do
a <- primes4
b <- dropWhile (<= a) primes4
guard ((sort $ show a) == (sort $ show b))
let c = 2 * b - a
guard (c < 10000)
guard ((sort $ show a) == (sort $ show c))
guard $ isPrime c
return (a, b, c)
primes = 2 : filter (\x -> isPrime x ) [3..]
Problem 50
Which prime, below one-million, can be written as the sum of the most consecutive primes?
Solution: (prime and isPrime not included)
import Control.Monad
findPrimeSum ps
| isPrime sumps = Just sumps
| otherwise = findPrimeSum (tail ps) `mplus` findPrimeSum (init ps)
where
sumps = sum ps
problem_50 = findPrimeSum $ take 546 primes