# Euler problems/21 to 30

### From HaskellWiki

m (Corrected the links to the Euler project) |
(→Problem 24: Added another method for Problem 24) |
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− | [[Category:Programming exercise spoilers]] |
+ | == [http://projecteuler.net/index.php?section=problems&id=21 Problem 21] == |

− | == [http://projecteuler.net/index.php?section=view&id=21 Problem 21] == |
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Evaluate the sum of all amicable pairs under 10000. |
Evaluate the sum of all amicable pairs under 10000. |
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− | Solution: |
+ | Solution: |

+ | (http://www.research.att.com/~njas/sequences/A063990) |
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+ | |||

This is a little slow because of the naive method used to compute the divisors. |
This is a little slow because of the naive method used to compute the divisors. |
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<haskell> |
<haskell> |
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Line 12: | Line 12: | ||

</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section=view&id=22 Problem 22] == |
+ | Here is an alternative using a faster way of computing the sum of divisors. |

+ | <haskell> |
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+ | problem_21_v2 = sum [n | n <- [2..9999], let m = d n, |
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+ | m > 1, m < 10000, n == d m, d m /= d (d m)] |
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+ | d n = product [(p * product g - 1) `div` (p - 1) | |
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+ | g <- group $ primeFactors n, let p = head g |
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+ | ] - n |
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+ | primeFactors = pf primes |
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+ | where |
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+ | pf ps@(p:ps') n |
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+ | | p * p > n = [n] |
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+ | | r == 0 = p : pf ps q |
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+ | | otherwise = pf ps' n |
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+ | where (q, r) = n `divMod` p |
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+ | primes = 2 : filter (null . tail . primeFactors) [3,5..] |
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+ | </haskell> |
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+ | |||

+ | == [http://projecteuler.net/index.php?section=problems&id=22 Problem 22] == |
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What is the total of all the name scores in the file of first names? |
What is the total of all the name scores in the file of first names? |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | -- apply to a list of names |
+ | import Data.List |

− | problem_22 :: [String] -> Int |
+ | import Data.Char |

− | problem_22 = sum . zipWith (*) [ 1 .. ] . map score |
+ | problem_22 = |

− | where score = sum . map ( subtract 64 . ord ) |
+ | do input <- readFile "names.txt" |

+ | let names = sort $ read$"["++ input++"]" |
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+ | let scores = zipWith score names [1..] |
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+ | print . sum $ scores |
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+ | where score w i = (i *) . sum . map (\c -> ord c - ord 'A' + 1) $ w |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section=view&id=23 Problem 23] == |
+ | == [http://projecteuler.net/index.php?section=problems&id=23 Problem 23] == |

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers. |
Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers. |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | problem_23 = undefined |
+ | --http://www.research.att.com/~njas/sequences/A048242 |

+ | import Data.Array |
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+ | n = 28124 |
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+ | abundant n = eulerTotient n - n > n |
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+ | abunds_array = listArray (1,n) $ map abundant [1..n] |
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+ | abunds = filter (abunds_array !) [1..n] |
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+ | |||

+ | rests x = map (x-) $ takeWhile (<= x `div` 2) abunds |
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+ | isSum = any (abunds_array !) . rests |
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+ | |||

+ | problem_23 = print . sum . filter (not . isSum) $ [1..n] |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section=view&id=24 Problem 24] == |
+ | == [http://projecteuler.net/index.php?section=problems&id=24 Problem 24] == |

What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9? |
What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9? |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | perms [] = [[]] |
+ | import Data.List |

− | perms xs = do |
+ | |

− | x <- xs |
+ | fac 0 = 1 |

− | map ( x: ) ( perms . delete x $ xs ) |
+ | fac n = n * fac (n - 1) |

+ | perms [] _= [] |
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+ | perms xs n= x : perms (delete x xs) (mod n m) |
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+ | where m = fac $ length xs - 1 |
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+ | y = div n m |
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+ | x = xs!!y |
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+ | |||

+ | problem_24 = perms "0123456789" 999999 |
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+ | </haskell> |
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− | problem_24 = ( perms "0123456789" ) !! 999999 |
+ | Or, using Data.List.permutations, |

+ | <haskell> |
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+ | import Data.List |
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+ | problem_24 = (!! 999999) . sort $ permutations ['0'..'9'] |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section=view&id=25 Problem 25] == |
+ | Casey Hawthorne |

+ | |||

+ | For Project Euler #24 you don't need to generate all the lexicographic permutations by Knuth's method or any other. |
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+ | |||

+ | You're only looking for the millionth lexicographic permutation of "0123456789" |
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+ | |||

+ | <haskell> |
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+ | |||

+ | -- Plan of attack. |
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+ | |||

+ | -- The "x"s are different numbers |
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+ | -- 0xxxxxxxxx represents 9! = 362880 permutations/numbers |
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+ | -- 1xxxxxxxxx represents 9! = 362880 permutations/numbers |
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+ | -- 2xxxxxxxxx represents 9! = 362880 permutations/numbers |
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+ | |||

+ | |||

+ | -- 20xxxxxxxx represents 8! = 40320 |
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+ | -- 21xxxxxxxx represents 8! = 40320 |
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+ | |||

+ | -- 23xxxxxxxx represents 8! = 40320 |
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+ | -- 24xxxxxxxx represents 8! = 40320 |
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+ | -- 25xxxxxxxx represents 8! = 40320 |
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+ | -- 26xxxxxxxx represents 8! = 40320 |
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+ | -- 27xxxxxxxx represents 8! = 40320 |
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+ | |||

+ | |||

+ | module Euler where |
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+ | |||

+ | import Data.List |
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+ | |||

+ | factorial n = product [1..n] |
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+ | |||

+ | -- lexOrder "0123456789" 1000000 "" |
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+ | |||

+ | lexOrder digits left s |
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+ | | len == 0 = s ++ digits |
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+ | | quot > 0 && rem == 0 = lexOrder (digits\\(show (digits!!(quot-1)))) rem (s ++ [(digits!!(quot-1))]) |
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+ | | quot == 0 && rem == 0 = lexOrder (digits\\(show (digits!!len))) rem (s ++ [(digits!!len)]) |
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+ | | rem == 0 = lexOrder (digits\\(show (digits!!(quot+1)))) rem (s ++ [(digits!!(quot+1))]) |
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+ | | otherwise = lexOrder (digits\\(show (digits!!(quot)))) rem (s ++ [(digits!!(quot))]) |
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+ | where |
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+ | len = (length digits) - 1 |
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+ | (quot,rem) = quotRem left (factorial len) |
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+ | |||

+ | </haskell> |
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+ | |||

+ | == [http://projecteuler.net/index.php?section=problems&id=25 Problem 25] == |
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What is the first term in the Fibonacci sequence to contain 1000 digits? |
What is the first term in the Fibonacci sequence to contain 1000 digits? |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | valid ( i, n ) = length ( show n ) == 1000 |
+ | fibs = 0:1:(zipWith (+) fibs (tail fibs)) |

+ | t = 10^999 |
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+ | |||

+ | problem_25 = length w |
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+ | where |
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+ | w = takeWhile (< t) fibs |
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+ | </haskell> |
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+ | |||

+ | |||

+ | Casey Hawthorne |
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+ | |||

+ | I believe you mean the following: |
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+ | |||

+ | <haskell> |
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+ | |||

+ | fibs = 0:1:(zipWith (+) fibs (tail fibs)) |
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− | problem_25 = fst . head . filter valid . zip [ 1 .. ] $ fibs |
+ | last (takeWhile (<10^1000) fibs) |

− | where fibs = 1 : 1 : 2 : zipWith (+) fibs ( tail fibs ) |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section=view&id=26 Problem 26] == |
+ | == [http://projecteuler.net/index.php?section=problems&id=26 Problem 26] == |

Find the value of d < 1000 for which 1/d contains the longest recurring cycle. |
Find the value of d < 1000 for which 1/d contains the longest recurring cycle. |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | problem_26 = fst $ maximumBy (\a b -> snd a `compare` snd b) |
+ | problem_26 = fst $ maximumBy (comparing snd) |

[(n,recurringCycle n) | n <- [1..999]] |
[(n,recurringCycle n) | n <- [1..999]] |
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where recurringCycle d = remainders d 10 [] |
where recurringCycle d = remainders d 10 [] |
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remainders d 0 rs = 0 |
remainders d 0 rs = 0 |
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remainders d r rs = let r' = r `mod` d |
remainders d r rs = let r' = r `mod` d |
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− | in case findIndex (== r') rs of |
+ | in case elemIndex r' rs of |

Just i -> i + 1 |
Just i -> i + 1 |
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Nothing -> remainders d (10*r') (r':rs) |
Nothing -> remainders d (10*r') (r':rs) |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section=view&id=27 Problem 27] == |
+ | == [http://projecteuler.net/index.php?section=problems&id=27 Problem 27] == |

Find a quadratic formula that produces the maximum number of primes for consecutive values of n. |
Find a quadratic formula that produces the maximum number of primes for consecutive values of n. |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | problem_27 = undefined |
+ | problem_27 = -(2*a-1)*(a^2-a+41) |

+ | where n = 1000 |
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+ | m = head $ filter (\x->x^2-x+41>n) [1..] |
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+ | a = m-1 |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section=view&id=28 Problem 28] == |
+ | == [http://projecteuler.net/index.php?section=problems&id=28 Problem 28] == |

What is the sum of both diagonals in a 1001 by 1001 spiral? |
What is the sum of both diagonals in a 1001 by 1001 spiral? |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | corners :: Int -> (Int, Int, Int, Int) |
+ | problem_28 = sum (map (\n -> 4*(n-2)^2+10*(n-1)) [3,5..1001]) + 1 |

− | corners i = (n*n, 1+(n*(2*m)), 2+(n*(2*m-1)), 3+(n*(2*m-2))) |
+ | </haskell> |

− | where m = (i-1) `div` 2 |
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− | n = 2*m+1 |
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− | sumcorners :: Int -> Int |
+ | Alternatively, one can use the fact that the distance between the diagonal numbers increases by 2 in every concentric square. Each square contains four gaps, so the following <hask>scanl</hask> does the trick: |

− | sumcorners i = a+b+c+d where (a, b, c, d) = corners i |
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− | sumdiags :: Int -> Int |
+ | <haskell> |

− | sumdiags i | even i = error "not a spiral" |
+ | euler28 n = sum $ scanl (+) 0 |

− | | i == 3 = s + 1 |
+ | (1:(concatMap (replicate 4) [2,4..(n-1)])) |

− | | otherwise = s + sumdiags (i-2) |
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− | where s = sumcorners i |
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− | |||

− | problem_28 = sumdiags 1001 |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section=view&id=29 Problem 29] == |
+ | == [http://projecteuler.net/index.php?section=problems&id=29 Problem 29] == |

How many distinct terms are in the sequence generated by a<sup>b</sup> for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100? |
How many distinct terms are in the sequence generated by a<sup>b</sup> for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100? |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | problem_29 = length . group . sort $ [a^b | a <- [2..100], b <- [2..100]] |
+ | import Control.Monad |

+ | problem_29 = length . group . sort $ liftM2 (^) [2..100] [2..100] |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section=view&id=30 Problem 30] == |
+ | We can also solve it in a more naive way, without using Monads, like this: |

+ | <haskell> |
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+ | import List |
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+ | problem_29 = length $ nub pr29_help |
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+ | where pr29_help = [z | y <- [2..100], |
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+ | z <- lift y] |
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+ | lift y = map (\x -> x^y) [2..100] |
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+ | </haskell> |
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+ | |||

+ | Simpler: |
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+ | |||

+ | <haskell> |
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+ | import List |
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+ | problem_29 = length $ nub [x^y | x <- [2..100], y <- [2..100]] |
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+ | </haskell> |
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+ | |||

+ | Instead of using lists, the Set data structure can be used for a significant speed increase: |
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+ | |||

+ | <haskell> |
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+ | import Set |
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+ | problem_29 = size $ fromList [x^y | x <- [2..100], y <- [2..100]] |
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+ | </haskell> |
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+ | |||

+ | == [http://projecteuler.net/index.php?section=problems&id=30 Problem 30] == |
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Find the sum of all the numbers that can be written as the sum of fifth powers of their digits. |
Find the sum of all the numbers that can be written as the sum of fifth powers of their digits. |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | problem_30 = undefined |
+ | import Data.Char (digitToInt) |

− | </haskell> |
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+ | limit :: Integer |
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+ | limit = snd $ head $ dropWhile (\(a,b) -> a > b) $ zip (map (9^5*) [1..]) (map (10^) [1..]) |
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− | [[Category:Tutorials]] |
+ | fifth :: Integer -> Integer |

− | [[Category:Code]] |
+ | fifth = sum . map ((^5) . toInteger . digitToInt) . show |

+ | |||

+ | problem_30 :: Integer |
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+ | problem_30 = sum $ filter (\n -> n == fifth n) [2..limit] |
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+ | </haskell> |

## Latest revision as of 03:52, 14 November 2011

## Contents |

## [edit] 1 Problem 21

Evaluate the sum of all amicable pairs under 10000.

Solution: (http://www.research.att.com/~njas/sequences/A063990)

This is a little slow because of the naive method used to compute the divisors.

problem_21 = sum [m+n | m <- [2..9999], let n = divisorsSum ! m, amicable m n] where amicable m n = m < n && n < 10000 && divisorsSum ! n == m divisorsSum = array (1,9999) [(i, sum (divisors i)) | i <- [1..9999]] divisors n = [j | j <- [1..n `div` 2], n `mod` j == 0]

Here is an alternative using a faster way of computing the sum of divisors.

problem_21_v2 = sum [n | n <- [2..9999], let m = d n, m > 1, m < 10000, n == d m, d m /= d (d m)] d n = product [(p * product g - 1) `div` (p - 1) | g <- group $ primeFactors n, let p = head g ] - n primeFactors = pf primes where pf ps@(p:ps') n | p * p > n = [n] | r == 0 = p : pf ps q | otherwise = pf ps' n where (q, r) = n `divMod` p primes = 2 : filter (null . tail . primeFactors) [3,5..]

## [edit] 2 Problem 22

What is the total of all the name scores in the file of first names?

Solution:

import Data.List import Data.Char problem_22 = do input <- readFile "names.txt" let names = sort $ read$"["++ input++"]" let scores = zipWith score names [1..] print . sum $ scores where score w i = (i *) . sum . map (\c -> ord c - ord 'A' + 1) $ w

## [edit] 3 Problem 23

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

Solution:

--http://www.research.att.com/~njas/sequences/A048242 import Data.Array n = 28124 abundant n = eulerTotient n - n > n abunds_array = listArray (1,n) $ map abundant [1..n] abunds = filter (abunds_array !) [1..n] rests x = map (x-) $ takeWhile (<= x `div` 2) abunds isSum = any (abunds_array !) . rests problem_23 = print . sum . filter (not . isSum) $ [1..n]

## [edit] 4 Problem 24

What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?

Solution:

import Data.List fac 0 = 1 fac n = n * fac (n - 1) perms [] _= [] perms xs n= x : perms (delete x xs) (mod n m) where m = fac $ length xs - 1 y = div n m x = xs!!y problem_24 = perms "0123456789" 999999

Or, using Data.List.permutations,

import Data.List problem_24 = (!! 999999) . sort $ permutations ['0'..'9']

Casey Hawthorne

For Project Euler #24 you don't need to generate all the lexicographic permutations by Knuth's method or any other.

You're only looking for the millionth lexicographic permutation of "0123456789"

-- Plan of attack. -- The "x"s are different numbers -- 0xxxxxxxxx represents 9! = 362880 permutations/numbers -- 1xxxxxxxxx represents 9! = 362880 permutations/numbers -- 2xxxxxxxxx represents 9! = 362880 permutations/numbers -- 20xxxxxxxx represents 8! = 40320 -- 21xxxxxxxx represents 8! = 40320 -- 23xxxxxxxx represents 8! = 40320 -- 24xxxxxxxx represents 8! = 40320 -- 25xxxxxxxx represents 8! = 40320 -- 26xxxxxxxx represents 8! = 40320 -- 27xxxxxxxx represents 8! = 40320 module Euler where import Data.List factorial n = product [1..n] -- lexOrder "0123456789" 1000000 "" lexOrder digits left s | len == 0 = s ++ digits | quot > 0 && rem == 0 = lexOrder (digits\\(show (digits!!(quot-1)))) rem (s ++ [(digits!!(quot-1))]) | quot == 0 && rem == 0 = lexOrder (digits\\(show (digits!!len))) rem (s ++ [(digits!!len)]) | rem == 0 = lexOrder (digits\\(show (digits!!(quot+1)))) rem (s ++ [(digits!!(quot+1))]) | otherwise = lexOrder (digits\\(show (digits!!(quot)))) rem (s ++ [(digits!!(quot))]) where len = (length digits) - 1 (quot,rem) = quotRem left (factorial len)

## [edit] 5 Problem 25

What is the first term in the Fibonacci sequence to contain 1000 digits?

Solution:

fibs = 0:1:(zipWith (+) fibs (tail fibs)) t = 10^999 problem_25 = length w where w = takeWhile (< t) fibs

Casey Hawthorne

I believe you mean the following:

fibs = 0:1:(zipWith (+) fibs (tail fibs)) last (takeWhile (<10^1000) fibs)

## [edit] 6 Problem 26

Find the value of d < 1000 for which 1/d contains the longest recurring cycle.

Solution:

problem_26 = fst $ maximumBy (comparing snd) [(n,recurringCycle n) | n <- [1..999]] where recurringCycle d = remainders d 10 [] remainders d 0 rs = 0 remainders d r rs = let r' = r `mod` d in case elemIndex r' rs of Just i -> i + 1 Nothing -> remainders d (10*r') (r':rs)

## [edit] 7 Problem 27

Find a quadratic formula that produces the maximum number of primes for consecutive values of n.

Solution:

problem_27 = -(2*a-1)*(a^2-a+41) where n = 1000 m = head $ filter (\x->x^2-x+41>n) [1..] a = m-1

## [edit] 8 Problem 28

What is the sum of both diagonals in a 1001 by 1001 spiral?

Solution:

problem_28 = sum (map (\n -> 4*(n-2)^2+10*(n-1)) [3,5..1001]) + 1

euler28 n = sum $ scanl (+) 0 (1:(concatMap (replicate 4) [2,4..(n-1)]))

## [edit] 9 Problem 29

How many distinct terms are in the sequence generated by a^{b} for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?

Solution:

import Control.Monad problem_29 = length . group . sort $ liftM2 (^) [2..100] [2..100]

We can also solve it in a more naive way, without using Monads, like this:

import List problem_29 = length $ nub pr29_help where pr29_help = [z | y <- [2..100], z <- lift y] lift y = map (\x -> x^y) [2..100]

Simpler:

import List problem_29 = length $ nub [x^y | x <- [2..100], y <- [2..100]]

Instead of using lists, the Set data structure can be used for a significant speed increase:

import Set problem_29 = size $ fromList [x^y | x <- [2..100], y <- [2..100]]

## [edit] 10 Problem 30

Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.

Solution:

import Data.Char (digitToInt) limit :: Integer limit = snd $ head $ dropWhile (\(a,b) -> a > b) $ zip (map (9^5*) [1..]) (map (10^) [1..]) fifth :: Integer -> Integer fifth = sum . map ((^5) . toInteger . digitToInt) . show problem_30 :: Integer problem_30 = sum $ filter (\n -> n == fifth n) [2..limit]