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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>
Line 39: Line 38:
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]
Retrieved from "http://www.haskell.org/haskellwiki/Euler_problems/21_to_30"