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Euler problems/21 to 30

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m (Corrected the links to the Euler project)
(Added a solution to problem 27)
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Solution:
 
Solution:
  +
  +
The following is written in [http://haskell.org/haskellwiki/Literate_programming#Haskell_and_literate_programming literate Haskell]:
 
<haskell>
 
<haskell>
problem_27 = undefined
+
> 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; code from the Haskell Café mailing list, slightly modified
  +
It works by filtering out numbers that are divisable by a previously found prime
  +
  +
> -- primes :: [Int]
  +
> -- primes = 2 : [x | x <- [3, 5..], all (\p -> x `mod` p > 0) (factorsToTry x)]
  +
> -- where
  +
> -- factorsToTry x = takeWhile (\p -> p * p <= x) primes
  +
  +
> 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 =
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> maximumBy (\(x, _, _) (y, _, _) -> compare x y) $
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> [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 $ ""
  +
  +
 
</haskell>
 
</haskell>
   

Revision as of 12:58, 25 July 2007

Contents

1 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]

2 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 )

3 Problem 23

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

Solution:

problem_23 = undefined

4 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 = do
    x <- xs
    map ( x: ) ( perms . delete x $ xs )
 
problem_24 = ( perms "0123456789" ) !! 999999

5 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 )

6 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)

7 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; code from the Haskell Café mailing list, slightly modified
It works by filtering out numbers that are divisable by a previously found prime
 
> -- primes :: [Int]
> -- primes = 2 : [x | x <- [3, 5..], all (\p -> x `mod` p > 0) (factorsToTry x)]
> --   where
> --     factorsToTry x = takeWhile (\p -> p * p <= x) primes
 
> 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+ 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 $ ""

8 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*m-1)), 3+(n*(2*m-2))) 
    where m = (i-1) `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 (i-2) 
           where s = sumcorners i
 
problem_28 = sumdiags 1001

9 Problem 29

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

Solution:

problem_29 = length . group . sort $ [a^b | a <- [2..100], b <- [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:

problem_30 = undefined