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Euler problems/21 to 30

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== [http://projecteuler.net/index.php?section=problems&id=21 Problem 21] ==
+
Do them on your own!
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<sup>b</sup> 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>
 

Revision as of 21:42, 29 January 2008

Do them on your own!