Euler problems/1 to 10

Problem 1

Add all the natural numbers below 1000 that are multiples of 3 or 5.

Solution:

problem_1 = sum [ x |
    x <- [1..999],
    (x `mod` 3 == 0) ||  (x `mod` 5 == 0)
    ]

problem_1_v2 = sum $ filter (\x -> ( x `mod` 3 == 0 || x `mod` 5 == 0 ) ) [1..999]

---

sum1to n = n * (n+1) `div` 2

problem_1_v3 = sumStep 3 999 + sumStep 5 999 - sumStep 15 999
    where sumStep s n = s * sum1to (n `div` s)

Problem 2

Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed one million.

Solution:

problem_2 = sum [ x |
    x <- takeWhile (<= 1000000) fibs,
    x `mod` 2 == 0
    ]
    where
    fibs = 1 : 1 : zipWith (+) fibs (tail fibs)

The following two solutions use the fact that the even-valued terms in the Fibonacci sequence themselves form a Fibonacci-like sequence that satisfies evenFib 0 = 0, evenFib 1 = 2, evenFib (n+2) = evenFib n + 4 * evenFib (n+1).

problem_2_v2 = sumEvenFibs $ numEvenFibsLessThan 1000000
sumEvenFibs n = (evenFib n + evenFib (n+1) - 2) `div` 4
evenFib n = round $ (2 + sqrt 5) ** (fromIntegral n) / sqrt 5
numEvenFibsLessThan n =
  floor $ (log (fromIntegral n - 0.5) + 0.5*log 5) / log (2 + sqrt 5)

The first two solutions work because 10^6 is small. The following solution also works for much larger numbers (up to at least 10^1000000 on my computer):

problem_2_v3 = sumEvenFibsLessThan 1000000
sumEvenFibsLessThan n = (a + b - 1) `div` 2
  where
    n2 = n `div` 2
    (a, b) = foldr f (0,1) $ takeWhile ((<= n2) . fst) $ iterate times2E (1, 4)
    f x y | fst z <= n2 = z
          | otherwise   = y
      where z = x `addE` y
addE (a, b) (c, d) = let ac = a*c in (a*d + b*c - 4*ac, ac + b*d)
times2E (a, b) = addE (a, b) (a, b)

Problem 3

Find the largest prime factor of 317584931803.

Solution:

primes = 2 : filter ((==1) . length . primeFactors) [3,5..]
primeFactors n = factor n primes
    where factor n (p:ps) | p*p > n        = [n]
                          | n `mod` p == 0 = p : factor (n `div` p) (p:ps)
                          | otherwise      = factor n ps

problem_3 = last (primeFactors 317584931803)

This can be improved by using null . tail instead of (== 1) . length.

Problem 4

Find the largest palindrome made from the product of two 3-digit numbers.

Solution:

problem_4 = foldr max 0 [ x |
    y <- [100..999],
    z <- [100..999], 
    let x = y * z, 
    let s = show x,
    s == reverse s
    ]

An alternative to avoid evaluating twice the same pair of numbers:

problem_4' = foldr1 max [ x |
    y <- [100..999],
    z <- [y..999],
    let x = y * z,
    let s = show x,
    s == reverse s
    ]

Problem 5

What is the smallest number divisible by each of the numbers 1 to 20?

Solution:

problem_5 = head [ x |
    x <- [2520,5040..],
    all (\y -> x `mod` y == 0) [1..20]
    ]

An alternative solution that takes advantage of the Prelude to avoid use of the generate and test idiom:

problem_5' = foldr1 lcm [1..20]

Problem 6

What is the difference between the sum of the squares and the square of the sums?

Solution:

problem_6 = sum [ x^2 | x <- [1..100]] - (sum [1..100])^2

Problem 7

Find the 10001st prime.

Solution:

primes = 2 : filter ((==1) . length . primeFactors) [3,5..]
primeFactors n = factor n primes
    where factor n (p:ps) | p*p > n        = [n]
                          | n `mod` p == 0 = p : factor (n `div` p) (p:ps)
                          | otherwise      = factor n ps
problem_7 = head $ drop 10000 primes

As above, this can be improved by using null . tail instead of (== 1) . length.

Here is an alternative that uses a sieve of Eratosthenes:

primes' = 2 : 3 : sieve (tail primes') [5,7..]
  where
    sieve (p:ps) x = let (h, _:t) = span (p*p <) x
                     in h ++ sieve ps (filter (\q -> q `mod` p /= 0) t
problem_7_v2 = primes' !! 10000

Problem 8

Discover the largest product of five consecutive digits in the 1000-digit number.

Solution:

import Data.Char
groupsOf _ [] = []
groupsOf n xs = take n xs : groupsOf n ( tail xs )
 
problem_8 x= maximum . map product . groupsOf 5 $ x
main=do
    t<-readFile "p8.log" 
    let digits=map digitToInt $foldl (++) ""$lines t
    print $problem_8 digits

Problem 9

There is only one Pythagorean triplet, {a, b, c}, for which a + b + c = 1000. Find the product abc.

Solution:

problem_9 = head [a*b*c | a <- [1..500], b <- [a..500], let c = 1000-a-b, a^2 + b^2 == c^2]

Another solution using Pythagorean Triplets generation:

triplets l =  [[a,b,c]|
    m <- [2..limit],
    n <- [1..(m-1)], 
    let a = m^2 - n^2, 
    let b = 2*m*n, 
    let c = m^2 + n^2,
    a+b+c==l
    ]
    where limit = floor $ sqrt $ fromIntegral l
problem_9 = product $ head $ triplets 1000

Problem 10

Calculate the sum of all the primes below one million.

Solution:

problem_10 = sum (takeWhile (< 1000000) primes)