# Euler problems/11 to 20

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− | == [http://projecteuler.net/index.php?section=problems&id=11 Problem 11] == |
+ | Do them on your own! |

− | What is the greatest product of four numbers on the same straight line in the [http://projecteuler.net/index.php?section=problems&id=11 20 by 20 grid]? |
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− | |||

− | Solution: |
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− | using Array and Arrows, for fun : |
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− | <haskell> |
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− | import Control.Arrow |
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− | import Data.Array |
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− | |||

− | input :: String -> Array (Int,Int) Int |
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− | input = listArray ((1,1),(20,20)) . map read . words |
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− | |||

− | senses = [(+1) *** id,(+1) *** (+1), id *** (+1), (+1) *** (\n -> n - 1)] |
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− | |||

− | inArray a i = inRange (bounds a) i |
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− | |||

− | prods :: Array (Int, Int) Int -> [Int] |
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− | prods a = [product xs | |
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− | i <- range $ bounds a |
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− | , s <- senses |
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− | , let is = take 4 $ iterate s i |
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− | , all (inArray a) is |
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− | , let xs = map (a!) is |
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− | ] |
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− | main = getContents >>= print . maximum . prods . input |
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− | </haskell> |
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− | |||

− | == [http://projecteuler.net/index.php?section=problems&id=12 Problem 12] == |
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− | What is the first triangle number to have over five-hundred divisors? |
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− | Solution: |
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− | <haskell> |
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− | --primeFactors in problem_3 |
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− | problem_12 = |
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− | head $ filter ((> 500) . nDivisors) triangleNumbers |
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− | where |
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− | triangleNumbers = scanl1 (+) [1..] |
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− | nDivisors n = |
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− | product $ map ((+1) . length) (group (primeFactors n)) |
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− | </haskell> |
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− | |||

− | == [http://projecteuler.net/index.php?section=problems&id=13 Problem 13] == |
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− | Find the first ten digits of the sum of one-hundred 50-digit numbers. |
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− | Solution: |
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− | <haskell> |
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− | sToInt =(+0).read |
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− | main=do |
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− | a<-readFile "p13.log" |
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− | let b=map sToInt $lines a |
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− | let c=take 10 $ show $ sum b |
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− | print c |
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− | </haskell> |
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− | |||

− | == [http://projecteuler.net/index.php?section=problems&id=14 Problem 14] == |
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− | Find the longest sequence using a starting number under one million. |
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− | Solution: |
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− | Faster solution, using an Array to memoize length of sequences : |
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− | <haskell> |
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− | import Data.Array |
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− | import Data.List |
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− | |||

− | syrs n = |
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− | a |
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− | where |
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− | a = listArray (1,n) $ 0:[1 + syr n x | x <- [2..n]] |
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− | syr n x = |
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− | if x' <= n then a ! x' else 1 + syr n x' |
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− | where |
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− | x' = if even x then x `div` 2 else 3 * x + 1 |
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− | |||

− | main = |
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− | print $ foldl' maxBySnd (0,0) $ assocs $ syrs 1000000 |
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− | where |
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− | maxBySnd x@(_,a) y@(_,b) = if a > b then x else y |
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− | </haskell> |
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− | |||

− | == [http://projecteuler.net/index.php?section=problems&id=15 Problem 15] == |
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− | Starting in the top left corner in a 20 by 20 grid, how many routes are there to the bottom right corner? |
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− | Solution: |
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− | Here is a bit of explanation, and a few more solutions: |
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− | Each route has exactly 40 steps, with 20 of them horizontal and 20 of |
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− | them vertical. We need to count how many different ways there are of |
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− | choosing which steps are horizontal and which are vertical. So we have: |
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− | |||

− | <haskell> |
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− | problem_15 = |
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− | product [21..40] `div` product [2..20] |
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− | </haskell> |
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− | |||

− | The first solution calculates this using the clever trick of contructing |
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− | [http://en.wikipedia.org/wiki/Pascal's_triangle Pascal's triangle] |
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− | along its diagonals. |
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− | Here is another solution that constructs Pascal's triangle in the usual way, |
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− | row by row: |
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− | |||

− | <haskell> |
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− | problem_15_v2 = |
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− | iterate (\r -> zipWith (+) (0:r) (r++[0])) [1] !! 40 !! 20 |
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− | </haskell> |
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− | |||

− | == [http://projecteuler.net/index.php?section=problems&id=16 Problem 16] == |
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− | What is the sum of the digits of the number 2<sup>1000</sup>? |
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− | |||

− | Solution: |
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− | <haskell> |
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− | import Data.Char |
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− | problem_16 = sum k |
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− | where |
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− | s=show $2^1000 |
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− | k=map digitToInt s |
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− | </haskell> |
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− | |||

− | == [http://projecteuler.net/index.php?section=problems&id=17 Problem 17] == |
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− | How many letters would be needed to write all the numbers in words from 1 to 1000? |
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− | Solution: |
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− | <haskell> |
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− | import Char |
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− | |||

− | one = ["one","two","three","four","five","six","seven","eight", |
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− | "nine","ten","eleven","twelve","thirteen","fourteen","fifteen", |
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− | "sixteen","seventeen","eighteen", "nineteen"] |
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− | ty = ["twenty","thirty","forty","fifty","sixty","seventy","eighty","ninety"] |
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− | |||

− | decompose x |
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− | | x == 0 = [] |
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− | | x < 20 = one !! (x-1) |
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− | | x >= 20 && x < 100 = |
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− | ty !! (firstDigit (x) - 2) ++ decompose ( x - firstDigit (x) * 10) |
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− | | x < 1000 && x `mod` 100 ==0 = |
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− | one !! (firstDigit (x)-1) ++ "hundred" |
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− | | x > 100 && x <= 999 = |
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− | one !! (firstDigit (x)-1) ++ "hundredand" ++decompose ( x - firstDigit (x) * 100) |
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− | | x == 1000 = "onethousand" |
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− | |||

− | where |
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− | firstDigit x = digitToInt$head (show x) |
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− | |||

− | problem_17 = |
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− | length$concat (map decompose [1..1000]) |
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− | </haskell> |
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− | |||

− | == [http://projecteuler.net/index.php?section=problems&id=18 Problem 18] == |
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− | Find the maximum sum travelling from the top of the triangle to the base. |
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− | Solution: |
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− | <haskell> |
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− | problem_18 = |
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− | head $ foldr1 g tri |
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− | where |
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− | f x y z = x + max y z |
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− | g xs ys = zipWith3 f xs ys $ tail ys |
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− | tri = [ |
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− | [75], |
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− | [95,64], |
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− | [17,47,82], |
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− | [18,35,87,10], |
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− | [20,04,82,47,65], |
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− | [19,01,23,75,03,34], |
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− | [88,02,77,73,07,63,67], |
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− | [99,65,04,28,06,16,70,92], |
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− | [41,41,26,56,83,40,80,70,33], |
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− | [41,48,72,33,47,32,37,16,94,29], |
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− | [53,71,44,65,25,43,91,52,97,51,14], |
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− | [70,11,33,28,77,73,17,78,39,68,17,57], |
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− | [91,71,52,38,17,14,91,43,58,50,27,29,48], |
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− | [63,66,04,68,89,53,67,30,73,16,69,87,40,31], |
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− | [04,62,98,27,23,09,70,98,73,93,38,53,60,04,23]] |
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− | </haskell> |
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− | |||

− | == [http://projecteuler.net/index.php?section=problems&id=19 Problem 19] == |
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− | You are given the following information, but you may prefer to do some research for yourself. |
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− | * 1 Jan 1900 was a Monday. |
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− | * Thirty days has September, |
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− | * April, June and November. |
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− | * All the rest have thirty-one, |
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− | * Saving February alone, |
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− | Which has twenty-eight, rain or shine. |
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− | And on leap years, twenty-nine. |
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− | * A leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400. |
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− | How many Sundays fell on the first of the month during the twentieth century |
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− | (1 Jan 1901 to 31 Dec 2000)? |
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− | Solution: |
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− | <haskell> |
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− | problem_19 = |
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− | length $ filter (== sunday) $ drop 12 $ take 1212 since1900 |
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− | since1900 = |
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− | scanl nextMonth monday $ concat $ |
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− | replicate 4 nonLeap ++ cycle (leap : replicate 3 nonLeap) |
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− | nonLeap = |
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− | [31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31] |
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− | leap = |
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− | 31 : 29 : drop 2 nonLeap |
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− | nextMonth x y = |
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− | (x + y) `mod` 7 |
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− | sunday = 0 |
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− | monday = 1 |
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− | </haskell> |
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− | |||

− | Here is an alternative that is simpler, but it is cheating a bit: |
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− | <haskell> |
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− | import Data.Time.Calendar |
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− | import Data.Time.Calendar.WeekDate |
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− | |||

− | problem_19_v2 = |
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− | length [() | |
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− | y <- [1901..2000], |
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− | m <- [1..12], |
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− | let (_, _, d) = toWeekDate $ fromGregorian y m 1, |
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− | d == 7 |
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− | ] |
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− | </haskell> |
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− | |||

− | == [http://projecteuler.net/index.php?section=problems&id=20 Problem 20] == |
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− | Find the sum of digits in 100! |
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− | Solution: |
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− | <haskell> |
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− | numPrime x p=takeWhile(>0) [div x (p^a)|a<-[1..]] |
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− | fastFactorial n= |
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− | product[a^x| |
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− | a<-takeWhile(<n) primes, |
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− | let x=sum$numPrime n a |
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− | ] |
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− | digits n |
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− | |n<10=[n] |
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− | |otherwise= y:digits x |
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− | where |
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− | (x,y)=divMod n 10 |
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− | problem_20= sum $ digits $fastFactorial 100 |
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− | </haskell> |

## Revision as of 21:39, 29 January 2008

Do them on your own!