Difference between revisions of "Euler problems/21 to 30"
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<haskell> |
<haskell> |
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+ | problem_26 = head [a|a<-[999,997..],all id [isPrime a ,isPrime$div a 2]] |
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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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</haskell> |
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Solution: |
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<haskell> |
<haskell> |
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+ | problem_27= |
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− | eulerCoefficients n |
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− | + | negate (2*a-1)*(a^2-a+41) |
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− | | b <- takeWhile (<n) primes, a <- [-b+1..n-1], |
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+ | n=1000 |
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− | let len = length $ takeWhile (isPrime . (\x -> x^2 + a*x + b)) [0..], |
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+ | m=head $filter (\x->x^2-x+41>n)[1..] |
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− | if b == 2 then even a else odd a, len > 39] |
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+ | a=m-1 |
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− | problem_27 = snd . fst . maximum . eulerCoefficients $ 1000 |
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</haskell> |
</haskell> |
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Revision as of 01:33, 17 February 2008
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
]
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
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]
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$(length(xs) -1)
y=div n m
x = xs!!y
problem_24 = perms "0123456789" 999999
Problem 25
What is the first term in the Fibonacci sequence to contain 1000 digits?
Solution:
import Data.List
fib x
|x==0=0
|x==1=1
|x==2=1
|odd x=(fib (d+1))^2+(fib d)^2
|otherwise=(fib (d+1))^2-(fib (d-1))^2
where
d=div x 2
phi=(1+sqrt 5)/2
dig x=floor( (fromInteger x-1) * log 10 /log phi)
problem_25 =
head[a|a<-[dig num..],(>=limit)$fib a]
where
num=1000
limit=10^(num-1)
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..],all id [isPrime a ,isPrime$div a 2]]
Problem 27
Find a quadratic formula that produces the maximum number of primes for consecutive values of n.
Solution:
problem_27=
negate (2*a-1)*(a^2-a+41)
where
n=1000
m=head $filter (\x->x^2-x+41>n)[1..]
a=m-1
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
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]
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