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Euler problems/21 to 30

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Line 15: Line 15:
 
import Data.List
 
import Data.List
 
import Data.Char
 
import Data.Char
problem_22 = do
+
problem_22 =
     input <- readFile "names.txt"
+
     do input <- readFile "names.txt"
    let names = sort $ read$"["++ input++"]"
+
      let names = sort $ read$"["++ input++"]"
    let scores = zipWith score names [1..]
+
      let scores = zipWith score names [1..]
    print $ show $ sum $ scores
+
      print . show . sum $ scores
    where
+
  where score w i = (i *) . sum . map (\c -> ord c - ord 'A' + 1) $ w
    score w i = (i *) $ sum $ map (\c -> ord c - ord 'A' + 1) w
+
 
</haskell>
 
</haskell>
  
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isSum = any (abunds_array !) . rests
 
isSum = any (abunds_array !) . rests
  
problem_23 = putStrLn $ show $ foldl1 (+) $ filter (not . isSum) [1..n]  
+
problem_23 = putStrLn . show . foldl1 (+) . filter (not . isSum) $ [1..n]  
 
</haskell>
 
</haskell>
  
Line 52: Line 51:
 
fac n = n * fac (n - 1)
 
fac n = n * fac (n - 1)
 
perms [] _= []
 
perms [] _= []
perms xs n=
+
perms xs n= x : perms (delete x xs) (mod n m)
    x:( perms ( delete x $ xs ) (mod n m))
+
  where m = fac $ length xs - 1
    where
+
        y = div n m
    m=fac$(length(xs) -1)
+
        x = xs!!y
    y=div n m
+
    x = xs!!y
+
 
   
 
   
problem_24 = perms "0123456789" 999999
+
problem_24 = perms "0123456789" 999999
 
</haskell>
 
</haskell>
  
Line 69: Line 66:
 
import Data.List
 
import Data.List
 
fib x
 
fib x
    |x==0=0
+
  | x==0     = 0
    |x==1=1
+
  | x==1     = 1
    |x==2=1
+
  | odd x    = (fib (d+1))^2 + (fib d)^2
     |odd x=(fib (d+1))^2+(fib d)^2
+
  | otherwise = (fib (d+1))^2-(fib (d-1))^2
    |otherwise=(fib (d+1))^2-(fib (d-1))^2
+
where d = x `div` 2
    where
+
    d=div x 2
+
  
phi=(1+sqrt 5)/2
+
phi = (1+sqrt 5)/2
dig x=floor( (fromInteger x-1) * log 10 /log phi)
+
 
problem_25 =  
+
dig x = floor ((fromInteger x-1) * log 10 / log phi)
    head[a|a<-[dig num..],(>=limit)$fib a]
+
 
    where
+
problem_25 = head [a | a<-[dig num..], fib a >= limit]
    num=1000
+
  where num   = 1000
    limit=10^(num-1)
+
        limit = 10^(num-1)
 
</haskell>
 
</haskell>
  
Line 91: Line 86:
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
problem_26 = head [a|a<-[999,997..],all id [isPrime a ,isPrime$div a 2]]
+
problem_26 = head [a | a<-[999,997..], and [isPrime a, isPrime $ a `div` 2]]
 
</haskell>
 
</haskell>
  
Line 99: Line 94:
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
problem_27=
+
problem_27 = -(2*a-1)*(a^2-a+41)
    negate (2*a-1)*(a^2-a+41)
+
  where n = 1000
    where
+
        m = head $ filter (\x->x^2-x+41>n) [1..]
    n=1000
+
        a = m-1
    m=head $filter (\x->x^2-x+41>n)[1..]
+
    a=m-1
+
 
</haskell>
 
</haskell>
  

Revision as of 19:25, 19 February 2008

Contents

1 Problem 21

Evaluate the sum of all amicable pairs under 10000.

Solution:

--http://www.research.att.com/~njas/sequences/A063990
problem_21 = sum [220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368]

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:

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

problem_26 = head [a | a<-[999,997..], and [isPrime a, isPrime $ a `div` 2]]

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

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:

--http://www.research.att.com/~njas/sequences/A052464
problem_30 = sum [4150, 4151, 54748, 92727, 93084, 194979]