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Euler problems/41 to 50

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Do them on your own!
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== [http://projecteuler.net/index.php?section=problems&id=41 Problem 41] ==
 +
What is the largest n-digit pandigital prime that exists?
 +
 
 +
Solution:
 +
<haskell>
 +
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
 +
</haskell>
 +
 
 +
== [http://projecteuler.net/index.php?section=problems&id=42 Problem 42] ==
 +
How many triangle words can you make using the list of common English words?
 +
 
 +
Solution:
 +
<haskell>
 +
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
 +
</haskell>
 +
 
 +
== [http://projecteuler.net/index.php?section=problems&id=43 Problem 43] ==
 +
Find the sum of all pandigital numbers with an unusual sub-string divisibility property.
 +
 
 +
Solution:
 +
<haskell>
 +
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
 +
</haskell>
 +
 
 +
== [http://projecteuler.net/index.php?section=problems&id=44 Problem 44] ==
 +
Find the smallest pair of pentagonal numbers whose sum and difference is pentagonal.
 +
 
 +
Solution:
 +
<haskell>
 +
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]]
 +
</haskell>
 +
 
 +
== [http://projecteuler.net/index.php?section=problems&id=45 Problem 45] ==
 +
After 40755, what is the next triangle number that is also pentagonal and hexagonal?
 +
 
 +
Solution:
 +
<haskell>
 +
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]
 +
</haskell>
 +
 
 +
== [http://projecteuler.net/index.php?section=problems&id=46 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
 +
 
 +
<haskell>
 +
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..]
 +
</haskell>
 +
 
 +
== [http://projecteuler.net/index.php?section=problems&id=47 Problem 47] ==
 +
Find the first four consecutive integers to have four distinct primes factors.
 +
 
 +
Solution:
 +
<haskell>
 +
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
 +
</haskell>
 +
 
 +
== [http://projecteuler.net/index.php?section=problems&id=48 Problem 48] ==
 +
Find the last ten digits of 1<sup>1</sup> + 2<sup>2</sup> + ... + 1000<sup>1000</sup>.
 +
 
 +
Solution:
 +
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.
 +
 
 +
powMod  on the [[Prime_numbers]] page
 +
 
 +
<haskell>
 +
problem_48 = flip mod limit$sum [powMod limit n n | n <- [1..1000]]
 +
    where
 +
    limit=10^10
 +
</haskell>
 +
 
 +
== [http://projecteuler.net/index.php?section=problems&id=49 Problem 49] ==
 +
Find arithmetic sequences, made of prime terms, whose four digits are permutations of each other.
 +
 
 +
Solution:
 +
millerRabinPrimality on the [[Prime_numbers]] page
 +
 
 +
<haskell>
 +
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..]
 +
</haskell>
 +
 
 +
== [http://projecteuler.net/index.php?section=problems&id=50 Problem 50] ==
 +
Which prime, below one-million, can be written as the sum of the most consecutive primes?
 +
 
 +
Solution:
 +
(prime and isPrime not included)
 +
 
 +
<haskell>
 +
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
 +
</haskell>

Revision as of 04:57, 30 January 2008

Contents

1 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

2 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

3 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

4 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]]

5 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]

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.).

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..]

7 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

8 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

9 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..]

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)

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