Difference between revisions of "Euler problems/21 to 30"

== [http://projecteuler.net/index.php?section=problems&id=21 Problem 21] ==

Evaluate the sum of all amicable pairs under 10000.

Solution:

<haskell>

problem_21 =

sum [n |

n <- [2..9999],

let m = eulerTotient n,

m > 1,

m < 10000,

n == eulerTotient m

]

</haskell>

== [http://projecteuler.net/index.php?section=problems&id=22 Problem 22] ==

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

Solution:

<haskell>

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

</haskell>

== [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.

Solution:

<haskell>

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]

</haskell>

== [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?

Solution:

<haskell>

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

</haskell>

== [http://projecteuler.net/index.php?section=problems&id=25 Problem 25] ==

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

Solution:

<haskell>

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)

</haskell>

== [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.

Solution:

<haskell>

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

]

</haskell>

== [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.

Solution:

<haskell>

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

</haskell>

== [http://projecteuler.net/index.php?section=problems&id=28 Problem 28] ==

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

Solution:

<haskell>

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

</haskell>

== [http://projecteuler.net/index.php?section=problems&id=29 Problem 29] ==

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

Solution:

<haskell>

import Control.Monad

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

</haskell>

== [http://projecteuler.net/index.php?section=problems&id=30 Problem 30] ==

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

Solution:

<haskell>

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

</haskell>