# Euler problems/1 to 10

### From HaskellWiki

(Removing category tags. See Talk:Euler_problems) |
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Solution: | Solution: | ||

<haskell> | <haskell> | ||

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

+ | x <- [1..999], | ||

+ | (x `mod` 3 == 0) || (x `mod` 5 == 0) | ||

+ | ] | ||

</haskell> | </haskell> | ||

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Solution: | Solution: | ||

<haskell> | <haskell> | ||

− | problem_2 = sum [ x | x <- takeWhile (<= 1000000) fibs, x `mod` 2 == 0] | + | problem_2 = sum [ x | |

− | + | x <- takeWhile (<= 1000000) fibs, | |

+ | x `mod` 2 == 0 | ||

+ | ] | ||

+ | where | ||

+ | fibs = 1 : 1 : zipWith (+) fibs (tail fibs) | ||

</haskell> | </haskell> | ||

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Solution: | Solution: | ||

<haskell> | <haskell> | ||

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

+ | y <- [100..999], | ||

+ | z <- [100..999], | ||

+ | let x = y * z, | ||

+ | let s = show x, | ||

+ | s == reverse s | ||

+ | ] | ||

</haskell> | </haskell> | ||

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

<haskell> | <haskell> | ||

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

+ | y <- [100..999], | ||

+ | z <- [y..999], | ||

+ | let x = y * z, | ||

+ | let s = show x, | ||

+ | s == reverse s | ||

+ | ] | ||

</haskell> | </haskell> | ||

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Solution: | Solution: | ||

<haskell> | <haskell> | ||

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

+ | x <- [2520,5040..], | ||

+ | all (\y -> x `mod` y == 0) [1..20] | ||

+ | ] | ||

</haskell> | </haskell> | ||

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

Line 140: | Line 162: | ||

Solution: | Solution: | ||

<haskell> | <haskell> | ||

− | + | import Data.Char | |

− | + | ||

− | + | ||

groupsOf _ [] = [] | groupsOf _ [] = [] | ||

groupsOf n xs = take n xs : groupsOf n ( tail xs ) | groupsOf n xs = take n xs : groupsOf n ( tail xs ) | ||

− | + | ||

− | problem_8 = maximum . map product . groupsOf 5 $ digits | + | 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 | ||

</haskell> | </haskell> | ||

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Another solution using Pythagorean Triplets generation: | Another solution using Pythagorean Triplets generation: | ||

<haskell> | <haskell> | ||

− | + | triplets l = [[a,b,c]| | |

− | triplets l = [ | + | 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 | where limit = floor $ sqrt $ fromIntegral l | ||

− | + | problem_9 = product $ head $ triplets 1000 | |

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | problem_9 = | + | |

− | + | ||

− | + | ||

</haskell> | </haskell> | ||

## Revision as of 10:57, 5 January 2008

## Contents |

## 1 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)

## 2 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

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)

## 3 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

instead of

## 4 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 ]

## 5 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]

## 6 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

## 7 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

instead of

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

## 8 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

## 9 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

## 10 Problem 10

Calculate the sum of all the primes below one million.

Solution:

problem_10 = sum (takeWhile (< 1000000) primes)