Difference between revisions of "Euler problems/21 to 30"
(Removing category tags. See Talk:Euler_problems) 

Line 82:  Line 82:  
Solution: 
Solution: 

<haskell> 
<haskell> 

−  perms [] = [ 
+  perms [] _= [] 
−  perms xs = do 
+  perms xs n= do 
−  +  let m=fac$(length(xs) 1) 

−  +  let y=div n m 

+  let x = xs!!y 

+  x:( perms ( delete x $ xs ) (mod n m)) 

−  problem_24 = 
+  problem_24 = perms "0123456789" 999999 
</haskell> 
</haskell> 

Revision as of 08:46, 1 December 2007
Contents
Problem 21
Evaluate the sum of all amicable pairs under 10000.
Solution: This is a little slow because of the naive method used to compute the divisors.
problem_21 = sum [m+n  m < [2..9999], let n = divisorsSum ! m, amicable m n]
where amicable m n = m < n && n < 10000 && divisorsSum ! n == m
divisorsSum = array (1,9999)
[(i, sum (divisors i))  i < [1..9999]]
divisors n = [j  j < [1..n `div` 2], n `mod` j == 0]
Here is an alternative using a faster way of computing the sum of divisors.
problem_21_v2 = sum [n  n < [2..9999], let m = d n,
m > 1, m < 10000, n == d m]
d n = product [(p * product g  1) `div` (p  1) 
g < group $ primeFactors n, let p = head g
]  n
primeFactors = pf primes
where
pf ps@(p:ps') n
 p * p > n = [n]
 r == 0 = p : pf ps q
 otherwise = pf ps' n
where (q, r) = n `divMod` p
primes = 2 : filter (null . tail . primeFactors) [3,5..]
Problem 22
What is the total of all the name scores in the file of first names?
Solution:
 apply to a list of names
problem_22 :: [String] > Int
problem_22 = sum . zipWith (*) [ 1 .. ] . map score
where score = sum . map ( subtract 64 . ord )
Problem 23
Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.
Solution:
import Data.Set hiding (filter, map)
import Data.List (scanl, group)
problem_23 :: Integer
problem_23 = sum [1..28123]  (fold (+) 0 $ abundant_sums $ abundant 28123)
abundant_sums :: [Integer] > Set Integer
abundant_sums [] = empty
abundant_sums l@(x:xs) = union (fromList [x + a  a < takeWhile (\y > y <= 28123  x) l]) (abundant_sums xs)
abundant :: Integer > [Integer]
abundant n = [a  a < [1..n], (sum $ factors a)  a > a]
primes :: [Integer]
primes = 2 : filter ((==1) . length . primeFactors) [3,5..]
primeFactors :: Integer > [Integer]
primeFactors n = factor n primes
where
factor _ [] = []
factor m (p:ps)  p*p > m = [m]
 m `mod` p == 0 = p : factor (m `div` p) (p:ps)
 otherwise = factor m ps
factors :: Integer > [Integer]
factors = perms . map (tail . scanl (*) 1) . group . primeFactors
where
perms :: (Integral a) => [[a]] > [a]
perms [] = [1]
perms (x:xs) = perms xs ++ concatMap (\z > map (*z) $ perms xs) x
Problem 24
What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?
Solution:
perms [] _= []
perms xs n= do
let m=fac$(length(xs) 1)
let y=div n m
let x = xs!!y
x:( perms ( delete x $ xs ) (mod n m))
problem_24 = perms "0123456789" 999999
Problem 25
What is the first term in the Fibonacci sequence to contain 1000 digits?
Solution:
valid ( i, n ) = length ( show n ) == 1000
problem_25 = fst . head . filter valid . zip [ 1 .. ] $ fibs
where fibs = 1 : 1 : 2 : zipWith (+) fibs ( tail fibs )
Problem 26
Find the value of d < 1000 for which 1/d contains the longest recurring cycle.
Solution:
problem_26 = fst $ maximumBy (\a b > snd a `compare` snd b)
[(n,recurringCycle n)  n < [1..999]]
where recurringCycle d = remainders d 10 []
remainders d 0 rs = 0
remainders d r rs = let r' = r `mod` d
in case findIndex (== r') rs of
Just i > i + 1
Nothing > remainders d (10*r') (r':rs)
Problem 27
Find a quadratic formula that produces the maximum number of primes for consecutive values of n.
Solution:
The following is written in literate Haskell:
> import Data.List
To be sure we get the maximum type checking of the compiler,
we switch off the default type
> default ()
Generate a list of primes.
It works by filtering out numbers that are
divisable by a previously found prime
> primes :: [Int]
> primes = sieve (2 : [3, 5..])
> where
> sieve (p:xs) = p : sieve (filter (\x > x `mod` p > 0) xs)
> isPrime :: Int > Bool
> isPrime x = x `elem` (takeWhile (<= x) primes)
The lists of values we are going to try for a and b;
b must be a prime, as n² + an + b is equal to b when n = 0
> testRangeA :: [Int]
> testRangeA = [1000 .. 1000]
> testRangeB :: [Int]
> testRangeB = takeWhile (< 1000) primes
The search
> bestCoefficients :: (Int, Int, Int)
> bestCoefficients =
> maximumBy (\(x, _, _) (y, _, _) > compare x y) $
> [f a b  a < testRangeA, b < testRangeB]
> where
Generate a list of results of the quadratic formula
(only the contiguous primes)
wrap the result in a triple, together with a and b
> f :: Int > Int > (Int, Int, Int)
> f a b = ( length $ contiguousPrimes a b
> , a
> , b
> )
> contiguousPrimes :: Int > Int > [Int]
> contiguousPrimes a b = takeWhile isPrime (map (quadratic a b) [0..])
The quadratic formula
> quadratic :: Int > Int > Int > Int
> quadratic a b n = n * n + a * n + b
> problem_27 =
> do
> let (l, a, b) = bestCoefficients
>
> putStrLn $ ""
> putStrLn $ "Problem Euler 27"
> putStrLn $ ""
> putStrLn $ "The best quadratic formula found is:"
> putStrLn $ " n * n + " ++ show a ++ " * n + " ++ show b
> putStrLn $ ""
> putStrLn $ "The number of primes is: " ++ (show l)
> putStrLn $ ""
> putStrLn $ "The primes are:"
> print $ take l $ contiguousPrimes a b
> putStrLn $ ""
Problem 28
What is the sum of both diagonals in a 1001 by 1001 spiral?
Solution:
corners :: Int > (Int, Int, Int, Int)
corners i = (n*n, 1+(n*(2*m)), 2+(n*(2*m1)), 3+(n*(2*m2)))
where m = (i1) `div` 2
n = 2*m+1
sumcorners :: Int > Int
sumcorners i = a+b+c+d where (a, b, c, d) = corners i
sumdiags :: Int > Int
sumdiags i  even i = error "not a spiral"
 i == 3 = s + 1
 otherwise = s + sumdiags (i2)
where s = sumcorners i
problem_28 = sumdiags 1001
You can note that from 1 to 3 there's (+2), and such too for 5, 7 and 9, it then goes up to (+4) 4 times, and so on, adding 2 to the number to add for each level of the spiral. You can so avoid all need for multiplications and just do additions with the following code :
problem_28 = sum . scanl (+) 1 . concatMap (replicate 4) $ [2,4..1000]
Problem 29
How many distinct terms are in the sequence generated by a^{b} for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?
Solution:
problem_29 = length . group . sort $ [a^b  a < [2..100], b < [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.Char (ord)
limit :: Integer
limit = snd $ head $ dropWhile (\(a,b) > a > b) $ zip (map (9^5*) [1..]) (map (10^) [1..])
fifth :: Integer > Integer
fifth n = foldr (\a b > (toInteger(ord a)  48)^5 + b) 0 $ show n
problem_30 :: Integer
problem_30 = sum $ filter (\n > n == fifth n) [2..limit]