# Difference between revisions of "Euler problems/1 to 10"

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− | Do them on your own! |
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+ | == [http://projecteuler.net/index.php?section=problems&id=1 Problem 1] == |
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+ | Add all the natural numbers below 1000 that are multiples of 3 or 5. |
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+ | |||

+ | Two solutions using <hask>sum</hask>: |
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+ | <haskell> |
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+ | import Data.List (union) |
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+ | problem_1' = sum (union [3,6..999] [5,10..999]) |
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+ | |||

+ | problem_1 = sum [x | x <- [1..999], x `mod` 3 == 0 || x `mod` 5 == 0] |
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+ | </haskell> |
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+ | |||

+ | Another solution which uses algebraic relationships: |
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+ | |||

+ | <haskell> |
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+ | problem_1 = sumStep 3 999 + sumStep 5 999 - sumStep 15 999 |
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+ | where |
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+ | sumStep s n = s * sumOnetoN (n `div` s) |
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+ | sumOnetoN n = n * (n+1) `div` 2 |
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+ | </haskell> |
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+ | |||

+ | == [http://projecteuler.net/index.php?section=problems&id=2 Problem 2] == |
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+ | Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed one million. |
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+ | |||

+ | Solution: |
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+ | <haskell> |
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+ | problem_2 = sum [ x | x <- takeWhile (<= 1000000) fibs, even x] |
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+ | where |
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+ | fibs = 1 : 1 : zipWith (+) fibs (tail fibs) |
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+ | </haskell> |
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+ | |||

+ | The following two solutions use the fact that the even-valued terms in |
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+ | the Fibonacci sequence themselves form a Fibonacci-like sequence |
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+ | that satisfies |
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+ | <hask>evenFib 0 = 0, evenFib 1 = 2, evenFib (n+2) = evenFib n + 4 * evenFib (n+1)</hask>. |
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+ | <haskell> |
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+ | problem_2 = sumEvenFibs $ numEvenFibsLessThan 1000000 |
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+ | where |
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+ | sumEvenFibs n = (evenFib n + evenFib (n+1) - 2) `div` 4 |
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+ | evenFib n = round $ (2 + sqrt 5) ** (fromIntegral n) / sqrt 5 |
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+ | numEvenFibsLessThan n = |
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+ | floor $ (log (fromIntegral n - 0.5) + 0.5*log 5) / log (2 + sqrt 5) |
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+ | </haskell> |
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+ | |||

+ | The first two solutions work because 10^6 is small. |
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+ | The following solution also works for much larger numbers |
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+ | (up to at least 10^1000000 on my computer): |
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+ | <haskell> |
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+ | problem_2 = sumEvenFibsLessThan 1000000 |
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+ | |||

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

+ | times2E (a, b) = addE (a, b) (a, b) |
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+ | |||

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

+ | |||

+ | Another elegant, quick solution, based on some background mathematics as in comments: |
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+ | |||

+ | <haskell> |
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+ | -- Every third term is even, and every third term beautifully follows: |
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+ | -- fib n = 4*fib n-3 + fib n-6 |
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+ | evenFibs = 2 : 8 : zipWith (+) (map (4*) (tail evenFibs)) evenFibs |
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+ | |||

+ | -- So, evenFibs are: e(n) = 4*e(n-1) + e(n-2) |
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+ | -- [there4]:4e(n) = e(n+1) - e(n-1) |
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+ | -- 4e(n-1) = e(n) - e(n-2) |
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+ | -- 4e(n-2) = e(n-1) - e(n-3) |
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+ | -- ... |
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+ | -- 4e(3) = e(4) - e(2) |
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+ | -- 4e(2) = e(3) - e(1) |
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+ | -- 4e(1) = e(2) - e(0) |
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+ | -- ------------------------------- |
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+ | -- Total: 4([sum] e(k) - e(0)) = e(n+1) + e(n) - e(1) - e(0) |
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+ | -- => [sum] e(k) = (e(n+1) + e(n) - e(1) + 3e(0))/4 = 1089154 for |
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+ | -- first 10 terms |
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+ | |||

+ | sumEvenFibsBelow :: Int -> Int |
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+ | sumEvenFibsBelow n = ((last $ take (x+1) evenFibs) + |
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+ | (last $ take x evenFibs) - |
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+ | 8 + 6) `div` 4 |
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+ | where x = length (takeWhile (<= n) evenFibs) |
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+ | |||

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

+ | == [http://projecteuler.net/index.php?section=problems&id=3 Problem 3] == |
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+ | Find the largest prime factor of 600851475143. |
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+ | |||

+ | Solution: |
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+ | <haskell> |
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+ | primes = 2 : filter (null . tail . primeFactors) [3,5..] |
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+ | |||

+ | primeFactors n = factor n primes |
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+ | where |
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+ | factor n (p:ps) |
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+ | | p*p > n = [n] |
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+ | | n `mod` p == 0 = p : factor (n `div` p) (p:ps) |
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+ | | otherwise = factor n ps |
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+ | |||

+ | problem_3 = last (primeFactors 600851475143) |
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+ | </haskell> |
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+ | |||

+ | == [http://projecteuler.net/index.php?section=problems&id=4 Problem 4] == |
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+ | Find the largest palindrome made from the product of two 3-digit numbers. |
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+ | |||

+ | Solution: |
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+ | <haskell> |
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+ | problem_4 = |
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+ | maximum [x | y<-[100..999], z<-[y..999], let x=y*z, let s=show x, s==reverse s] |
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+ | </haskell> |
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+ | |||

+ | == [http://projecteuler.net/index.php?section=problems&id=5 Problem 5] == |
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+ | What is the smallest number divisible by each of the numbers 1 to 20? |
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+ | |||

+ | Solution: |
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+ | <haskell> |
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+ | problem_5 = foldr1 lcm [1..20] |
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+ | </haskell> |
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+ | |||

+ | Another solution: <code>16*9*5*7*11*13*17*19</code>. Product of maximal powers of primes in the range. |
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+ | |||

+ | == [http://projecteuler.net/index.php?section=problems&id=6 Problem 6] == |
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+ | What is the difference between the sum of the squares and the square of the sums? |
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+ | |||

+ | Solution: |
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+ | <!-- |
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+ | <haskell> |
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+ | fun n = a - b |
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+ | where |
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+ | a=(n^2 * (n+1)^2) `div` 4 |
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+ | b=(n * (n+1) * (2*n+1)) `div` 6 |
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+ | |||

+ | problem_6 = fun 100 |
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+ | </haskell> |
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+ | --> |
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+ | <!-- Might just be me, but I find this a LOT easier to read. Perhaps not as good mathematically, but it runs in an instant, even for n = 25000. |
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+ | <haskell> |
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+ | fun n = a - b |
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+ | where |
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+ | a = (sum [1..n])^2 |
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+ | b = sum (map (^2) [1..n]) |
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+ | |||

+ | problem_6 = fun 100 |
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+ | </haskell> |
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+ | --> |
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+ | <!-- I just made it a oneliner... --> |
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+ | <haskell> |
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+ | problem_6 = (sum [1..100])^2 - sum (map (^2) [1..100]) |
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+ | </haskell> |
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+ | |||

+ | == [http://projecteuler.net/index.php?section=problems&id=7 Problem 7] == |
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+ | Find the 10001st prime. |
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+ | |||

+ | Solution: |
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+ | <haskell> |
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+ | --primes in problem_3 |
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+ | problem_7 = primes !! 10000 |
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+ | </haskell> |
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+ | == [http://projecteuler.net/index.php?section=problems&id=8 Problem 8] == |
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+ | Discover the largest product of thirteen consecutive digits in the 1000-digit number. |
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+ | |||

+ | Solution: |
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+ | <!-- |
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+ | <haskell> |
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+ | import Data.Char |
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+ | groupsOf _ [] = [] -- incorrect, overall: last |
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+ | groupsOf n xs = -- subsequences will be shorter than n!! |
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+ | take n xs : groupsOf n ( tail xs ) |
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+ | |||

+ | problem_8 x = maximum . map product . groupsOf 5 $ x |
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+ | main = do t <- readFile "p8.log" |
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+ | let digits = map digitToInt $concat $ lines t |
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+ | print $ problem_8 digits |
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+ | </haskell> |
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+ | --> |
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+ | <haskell> |
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+ | import Data.Char |
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+ | import Data.List |
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+ | |||

+ | euler_8 = do |
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+ | str <- readFile "number.txt" |
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+ | print . maximum . map product |
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+ | . foldr (zipWith (:)) (repeat []) |
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+ | . take 13 . tails . map (fromIntegral . digitToInt) |
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+ | . concat . lines $ str |
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+ | </haskell> |
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+ | |||

+ | == [http://projecteuler.net/index.php?section=problems&id=9 Problem 9] == |
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+ | There is only one Pythagorean triplet, {''a'', ''b'', ''c''}, for which ''a'' + ''b'' + ''c'' = 1000. Find the product ''abc''. |
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+ | |||

+ | Solution: |
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+ | <haskell> |
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+ | triplets l = [[a,b,c] | m <- [2..limit], |
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+ | n <- [1..(m-1)], |
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+ | let a = m^2 - n^2, |
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+ | let b = 2*m*n, |
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+ | let c = m^2 + n^2, |
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+ | a+b+c==l] |
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+ | where limit = floor . sqrt . fromIntegral $ l |
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+ | |||

+ | problem_9 = product . head . triplets $ 1000 |
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+ | </haskell> |
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+ | |||

+ | == [http://projecteuler.net/index.php?section=problems&id=10 Problem 10] == |
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+ | Calculate the sum of all the primes below one million. |
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+ | |||

+ | Solution: |
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+ | <haskell> |
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+ | --primes in problem_3 |
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+ | problem_10 = sum (takeWhile (< 1000000) primes) |
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+ | </haskell> |

## Latest revision as of 02:31, 8 May 2016

## Contents

## Problem 1

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

Two solutions using `sum`

:

```
import Data.List (union)
problem_1' = sum (union [3,6..999] [5,10..999])
problem_1 = sum [x | x <- [1..999], x `mod` 3 == 0 || x `mod` 5 == 0]
```

Another solution which uses algebraic relationships:

```
problem_1 = sumStep 3 999 + sumStep 5 999 - sumStep 15 999
where
sumStep s n = s * sumOnetoN (n `div` s)
sumOnetoN n = n * (n+1) `div` 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, even x]
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 = sumEvenFibs $ numEvenFibsLessThan 1000000
where
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 = 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) = (a*d + b*c - 4*ac, ac + b*d)
where ac=a*c
times2E (a, b) = addE (a, b) (a, b)
```

Another elegant, quick solution, based on some background mathematics as in comments:

```
-- Every third term is even, and every third term beautifully follows:
-- fib n = 4*fib n-3 + fib n-6
evenFibs = 2 : 8 : zipWith (+) (map (4*) (tail evenFibs)) evenFibs
-- So, evenFibs are: e(n) = 4*e(n-1) + e(n-2)
-- [there4]:4e(n) = e(n+1) - e(n-1)
-- 4e(n-1) = e(n) - e(n-2)
-- 4e(n-2) = e(n-1) - e(n-3)
-- ...
-- 4e(3) = e(4) - e(2)
-- 4e(2) = e(3) - e(1)
-- 4e(1) = e(2) - e(0)
-- -------------------------------
-- Total: 4([sum] e(k) - e(0)) = e(n+1) + e(n) - e(1) - e(0)
-- => [sum] e(k) = (e(n+1) + e(n) - e(1) + 3e(0))/4 = 1089154 for
-- first 10 terms
sumEvenFibsBelow :: Int -> Int
sumEvenFibsBelow n = ((last $ take (x+1) evenFibs) +
(last $ take x evenFibs) -
8 + 6) `div` 4
where x = length (takeWhile (<= n) evenFibs)
```

## Problem 3

Find the largest prime factor of 600851475143.

Solution:

```
primes = 2 : filter (null . tail . 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 600851475143)
```

## Problem 4

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

Solution:

```
problem_4 =
maximum [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 = foldr1 lcm [1..20]
```

Another solution: `16*9*5*7*11*13*17*19`

. Product of maximal powers of primes in the range.

## Problem 6

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

Solution:

```
problem_6 = (sum [1..100])^2 - sum (map (^2) [1..100])
```

## Problem 7

Find the 10001st prime.

Solution:

```
--primes in problem_3
problem_7 = primes !! 10000
```

## Problem 8

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

Solution:

```
import Data.Char
import Data.List
euler_8 = do
str <- readFile "number.txt"
print . maximum . map product
. foldr (zipWith (:)) (repeat [])
. take 13 . tails . map (fromIntegral . digitToInt)
. concat . lines $ str
```

## Problem 9

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

Solution:

```
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:

```
--primes in problem_3
problem_10 = sum (takeWhile (< 1000000) primes)
```