Project Euler欧拉计划

1

If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.

Find the sum of all the multiples of 3 or 5 below 1000.

answer = sum $ takeWhile (< 1000) [ n | n <- [ 1 .. 999 ],
                                        mod n 3 == 0 || mod n 5 == 0
                                  ]

2

Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.

answer = sum $ filter even $ takeWhile (<= 4000000) $ map fst $
                                                               iterate (\ (a,b) -> (b, a+b))
                                                                       (1, 2)

3

The prime factors of 13195 are 5, 7, 13 and 29.

What is the largest prime factor of the number 600851475143 ?

answer = maximum $ fact 600851475143
    where
        -- Get the factors of N, not sorted
        fact n = factors n 2
            where
                -- Get the factors of N
                -- p <= (the next factor of n),  p <- Z+
                factors n p | n == 1     = [1]
                            | n  > 1     = if    mod n p == 0              -- if    p is a factor
                                           then  p:(factors (div n p) p)
                                           else  factors n (succ p)        -- else  try next

4

A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99.

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

answer = maximum $ [ x*y | x <- [ 100 .. 999 ],
                           y <- [ 100 .. 999 ],
                           isPalindrome $ show(x*y)
                   ]
                   where
                       isPalindrome l = l == reverse(l)

5

2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.

What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

foldr lcm 1 [1..20]

6

The sum of the squares of the first ten natural numbers is,

1^2 + 2^2 + ... + 10^2 = 385

The square of the sum of the first ten natural numbers is,

(1 + 2 + ... + 10)^2 = 552 = 3025

Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 − 385 = 2640.

Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.

ans = s2 - s1
   where s1 = sum $ map (\ a -> a^2 ) [ 1 .. 100 ]
         s2 = (sum [ 1 .. 100 ])^2

出乎意料地快。

8

The four adjacent digits in the 1000-digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832.

73167176531330624919225119674426574742355349194934

96983520312774506326239578318016984801869478851843

85861560789112949495459501737958331952853208805511

12540698747158523863050715693290963295227443043557

66896648950445244523161731856403098711121722383113

62229893423380308135336276614282806444486645238749

30358907296290491560440772390713810515859307960866

70172427121883998797908792274921901699720888093776

65727333001053367881220235421809751254540594752243

52584907711670556013604839586446706324415722155397

53697817977846174064955149290862569321978468622482

83972241375657056057490261407972968652414535100474

82166370484403199890008895243450658541227588666881

16427171479924442928230863465674813919123162824586

17866458359124566529476545682848912883142607690042

24219022671055626321111109370544217506941658960408

07198403850962455444362981230987879927244284909188

84580156166097919133875499200524063689912560717606

05886116467109405077541002256983155200055935729725

71636269561882670428252483600823257530420752963450

Find the thirteen adjacent digits in the 1000-digit number that have the greatest product. What is the value of this product?

import Data.Char
ans = maximum productLst
 where productLst = getProductLst number
        where number = "7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319528532088055111254069874715852386305071569329096329522744304355766896648950445244523161731856403098711121722383113622298934233803081353362766142828064444866452387493035890729629049156044077239071381051585930796086670172427121883998797908792274921901699720888093776657273330010533678812202354218097512545405947522435258490771167055601360483958644670632441572215539753697817977846174064955149290862569321978468622482839722413756570560574902614079729686524145351004748216637048440319989000889524345065854122758866688116427171479924442928230863465674813919123162824586178664583591245665294765456828489128831426076900422421902267105562632111110937054421750694165896040807198403850962455444362981230987879927244284909188845801561660979191338754992005240636899125607176060588611646710940507754100225698315520005593572972571636269561882670428252483600823257530420752963450"
              getProductLst l | len == 13   = [ productOf1to13 ]
                              | otherwise   = productOf1to13:getProductLst(tail l)
               where len = length l
                     productOf1to13 = product from1to13
                      where from1to13 = map (\ a -> read (a:[]) :: Integer) $ take 13 l
--                                                                 ^^^^^^^
--                                           Necessary. Or you will get some negative numbers.

9

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,

a^2 + b^2 = c^2

For example, 3^2 + 4^2 = 9 + 16 = 25 = 5^2.

There exists exactly one Pythagorean triplet for which a + b + c = 1000.

Find the product abc.

ans = product' $ head $ [ (a, b, c) | c <- [ 1 .. 1000 ],
                                      b <- [ 1 .. c ],
                                      a <- [ 1 .. b ],
                                      a + b + c == 1000,
                                      a^2 + b^2 == c^2
                        ]
 where product' (x, y, z) = x * y * z