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Euler problems/21 to 30

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Do them on your own!
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== [http://projecteuler.net/index.php?section=problems&id=21 Problem 21] ==
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Evaluate the sum of all amicable pairs under 10000.
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Solution:
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<haskell>
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problem_21 =
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sum [n |
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n <- [2..9999],
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let m = eulerTotient n,
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m > 1,
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m < 10000,
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n == eulerTotient m
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]
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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?
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Solution:
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<haskell>
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import Data.List
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import Data.Char
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problem_22 = do
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input <- readFile "names.txt"
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let names = sort $ read$"["++ input++"]"
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let scores = zipWith score names [1..]
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print $ show $ sum $ scores
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where
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score w i = (i *) $ sum $ map (\c -> ord c - ord 'A' + 1) w
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</haskell>
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== [http://projecteuler.net/index.php?section=problems&id=23 Problem 23] ==
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Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.
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Solution:
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<haskell>
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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 = putStrLn $ show $ foldl1 (+) $ filter (not . isSum) [1..n]
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</haskell>
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== [http://projecteuler.net/index.php?section=problems&id=24 Problem 24] ==
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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:
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<haskell>
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import Data.List
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fac 0 = 1
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fac n = n * fac (n - 1)
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perms [] _= []
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perms xs n=
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x:( perms ( delete x $ xs ) (mod n m))
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where
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m=fac$(length(xs) -1)
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y=div n m
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x = xs!!y
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problem_24 = perms "0123456789" 999999
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</haskell>
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== [http://projecteuler.net/index.php?section=problems&id=25 Problem 25] ==
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What is the first term in the Fibonacci sequence to contain 1000 digits?
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Solution:
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<haskell>
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import Data.List
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fib x
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|x==0=0
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|x==1=1
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|x==2=1
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|odd x=(fib (d+1))^2+(fib d)^2
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|otherwise=(fib (d+1))^2-(fib (d-1))^2
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where
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d=div x 2
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phi=(1+sqrt 5)/2
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dig x=floor( (fromInteger x-1) * log 10 /log phi)
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problem_25 =
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head[a|a<-[dig num..],(>=limit)$fib a]
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where
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num=1000
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limit=10^(num-1)
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</haskell>
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== [http://projecteuler.net/index.php?section=problems&id=26 Problem 26] ==
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Find the value of d < 1000 for which 1/d contains the longest recurring cycle.
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Solution:
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<haskell>
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next n d = (n `mod` d):next (10*n`mod`d) d
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idigs n = tail $ take (1+n) $ next 1 n
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pos x = map fst . filter ((==x) . snd) . zip [1..]
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periods n = let d = idigs n in pos (head d) (tail d)
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problem_26 =
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snd$maximum [(m,a)|
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a<-[800..1000] ,
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let k=periods a,
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not$null k,
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let m=head k
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]
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</haskell>
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== [http://projecteuler.net/index.php?section=problems&id=27 Problem 27] ==
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Find a quadratic formula that produces the maximum number of primes for consecutive values of n.
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Solution:
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<haskell>
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eulerCoefficients n
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= [((len, a*b), (a, b))
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| b <- takeWhile (<n) primes, a <- [-b+1..n-1],
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let len = length $ takeWhile (isPrime . (\x -> x^2 + a*x + b)) [0..],
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if b == 2 then even a else odd a, len > 39]
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problem_27 = snd . fst . maximum . eulerCoefficients $ 1000
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</haskell>
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== [http://projecteuler.net/index.php?section=problems&id=28 Problem 28] ==
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What is the sum of both diagonals in a 1001 by 1001 spiral?
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Solution:
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<haskell>
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problem_28 = sum (map (\n -> 4*(n-2)^2+10*(n-1)) [3,5..1001]) + 1
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</haskell>
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== [http://projecteuler.net/index.php?section=problems&id=29 Problem 29] ==
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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:
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<haskell>
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import Control.Monad
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problem_29 = length . group . sort $ liftM2 (^) [2..100] [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.
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Solution:
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<haskell>
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import Data.Array
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import Data.Char
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p = listArray (0,9) $ map (^5) [0..9]
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upperLimit = 295277
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candidates =
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[ n |
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n <- [10..upperLimit],
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(sum $ digits n) `mod` 10 == last(digits n),
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powersum n == n
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]
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where
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digits n = map digitToInt $ show n
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powersum n = sum $ map (p!) $ digits n
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problem_30 = sum candidates
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</haskell>

Revision as of 04:55, 30 January 2008

Contents

1 Problem 21

Evaluate the sum of all amicable pairs under 10000.

Solution:

problem_21 = 
    sum [n |
    n <- [2..9999],
    let m = eulerTotient  n,
    m > 1,
    m < 10000,
    n ==  eulerTotient  m
    ]

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 $ show $ sum $ scores
    where
    score w i = (i *) $ sum $ map (\c -> ord c - ord 'A' + 1) w

3 Problem 23

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

Solution:

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 = putStrLn $ show $ foldl1 (+) $ filter (not . isSum) [1..n]

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

5 Problem 25

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

Solution:

import Data.List
fib x
    |x==0=0
    |x==1=1
    |x==2=1
    |odd x=(fib (d+1))^2+(fib d)^2
    |otherwise=(fib (d+1))^2-(fib (d-1))^2
    where
    d=div x 2
 
phi=(1+sqrt 5)/2
dig x=floor( (fromInteger x-1) * log 10 /log phi)
problem_25 = 
    head[a|a<-[dig num..],(>=limit)$fib a]
    where
    num=1000
    limit=10^(num-1)

6 Problem 26

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

Solution:

next n d = (n `mod` d):next (10*n`mod`d) d
 
idigs n = tail $ take (1+n) $ next 1 n
 
pos x = map fst . filter ((==x) . snd) . zip [1..]
 
periods n = let d = idigs n in pos (head d) (tail d)
 
problem_26 = 
    snd$maximum [(m,a)|
    a<-[800..1000] ,
    let k=periods a,
    not$null k,
    let m=head k
    ]

7 Problem 27

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

Solution:

eulerCoefficients n 
  = [((len, a*b), (a, b)) 
      | b <- takeWhile (<n) primes, a <- [-b+1..n-1],
        let len = length $ takeWhile (isPrime . (\x -> x^2 + a*x + b)) [0..],
        if b == 2 then even a else odd a, len > 39]
 
problem_27 = snd . fst . maximum . eulerCoefficients $ 1000

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

9 Problem 29

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

Solution:

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

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.Array
import Data.Char
 
p = listArray (0,9) $ map (^5) [0..9]
 
upperLimit = 295277
 
candidates = 
    [ n |
    n <- [10..upperLimit],
    (sum $ digits n) `mod` 10 == last(digits n),
    powersum n == n
    ]
    where 
    digits n = map digitToInt $ show n
    powersum n = sum $ map (p!) $ digits n
 
problem_30 = sum candidates