Difference between revisions of "Euler problems/11 to 20"
Jim Burton (talk  contribs) (Added solution for problem 14) 
(→Problem 14: streamline code) 

(41 intermediate revisions by 17 users not shown)  
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−  [[Category:Programming exercise spoilers]] 

== [http://projecteuler.net/index.php?section=problems&id=11 Problem 11] == 
== [http://projecteuler.net/index.php?section=problems&id=11 Problem 11] == 

−  What is the greatest product of four numbers on the same straight line in the [http://projecteuler.net/index.php?section= 
+  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]? 
Solution: 
Solution: 

+  using Array and Arrows, for fun : 

<haskell> 
<haskell> 

−  problem_11 = undefined 

+  import Control.Arrow 

+  import Data.Array 

+  
+  input :: String > Array (Int,Int) Int 

+  input = listArray ((1,1),(20,20)) . map read . words 

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

+  
+  inArray a i = inRange (bounds a) i 

+  
+  prods :: Array (Int, Int) Int > [Int] 

+  prods a = [product xs  i < range $ bounds a, 

+  s < senses, 

+  let is = take 4 $ iterate s i, 

+  all (inArray a) is, 

+  let xs = map (a!) is] 

+  main = print . maximum . prods . input =<< getContents 

</haskell> 
</haskell> 

Line 13:  Line 29:  
Solution: 
Solution: 

<haskell> 
<haskell> 

+  primeFactors in problem_3 

problem_12 = head $ filter ((> 500) . nDivisors) triangleNumbers 
problem_12 = head $ filter ((> 500) . nDivisors) triangleNumbers 

−  where triangleNumbers = scanl1 (+) [1..] 

+  where nDivisors n = product $ map ((+1) . length) (group (primeFactors n)) 

−  nDivisors n = product $ map ((+1) . length) (group (primeFactors n)) 

+  triangleNumbers = scanl1 (+) [1..] 

−  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 

</haskell> 
</haskell> 

Line 28:  Line 40:  
Solution: 
Solution: 

<haskell> 
<haskell> 

−  nums = ...  put the numbers in a list 

+  
−  problem_13 = take 10 . show . sum $ nums 

+  main = do xs < fmap (map read . lines) (readFile "p13.log") 

+  print . take 10 . show . sum $ xs 

</haskell> 
</haskell> 

Line 36:  Line 49:  
Solution: 
Solution: 

+  <haskell> 

+  import Data.List 

+  
+  problem_14 = j 1000000 where 

+  f :: Int > Integer > Int 

+  f k 1 = k 

+  f k n = f (k+1) $ if even n then div n 2 else 3*n + 1 

+  g x y = if snd x < snd y then y else x 

+  h x n = g x (n, f 1 n) 

+  j n = fst $ foldl' h (1,1) [2..n1] 

+  </haskell> 

+  
+  Faster solution, using unboxed types and parallel computation: 

<haskell> 
<haskell> 

−  p14s :: Integer > [Integer] 

+  import Control.Parallel 

−  p14s n = n : p14s' n 

+  import Data.Word 

−  where p14s' n = if n' == 1 then [1] else n' : p14s' n' 

+  
−  where n' = if even n then n `div` 2 else (3*n)+1 

+  collatzLen :: Int > Word32 > Int 

+  collatzLen c 1 = c 

+  collatzLen c n = collatzLen (c+1) $ if n `mod` 2 == 0 then n `div` 2 else 3*n+1 

+  
+  pmax x n = x `max` (collatzLen 1 n, n) 

+  
+  solve xs = foldl pmax (1,1) xs 

+  
+  main = print soln 

+  where 

+  s1 = solve [2..500000] 

+  s2 = solve [500001..1000000] 

+  soln = s2 `par` (s1 `pseq` max s1 s2) 

+  </haskell> 

+  
+  Even faster solution, using an Array to memoize length of sequences : 

+  <haskell> 

+  import Data.Array 

+  import Data.List 

+  import Data.Ord (comparing) 

+  
+  syrs n = 

+  a 

+  where 

+  a = listArray (1,n) $ 0 : map syr [2..n] 

+  syr x = 

+  if y <= n then 1 + a ! y else 1 + syr y 

+  where 

+  y = if even x then x `div` 2 else 3 * x + 1 

+  
+  main = 

+  print . maximumBy (comparing snd) . assocs . syrs $ 1000000 

+  </haskell> 

+  
+  <! 

+  This is a trivial solution without any memoization, right? 

+  
+  Using a list to memoize the lengths 

+  
+  <haskell> 

+  import Data.List 

+  
+   computes the sequence for a given n 

+  l n = n:unfoldr f n where 

+  f 1 = Nothing  we're done here 

+   for reasons of speed we do div and mod in one go 

+  f n = let (d,m)=divMod n 2 in case m of 

+  0 > Just (d,d)  n was even 

+  otherwise > let k = 3*n+1 in Just (k,k)  n was odd 

+  
+  
+  answer = foldl1' f $  computes the maximum of a list of tuples 

+   save the length of the sequence and the number generating it in a tuple 

+  [(length $! l x, x)  x < [1..1000000]] where 

+  f (a,c) (b,d)  one tuple is greater than other if the first component (=sequencelength) is greater 

+   a > b = (a,c) 

+   otherwise = (b,d) 

−  problem_14 = fst $ head $ qsort [(x, length $ p14s x)  x < [1 .. 999999]] 

+  main = print answer 

−  where qsort [] = [] 

−  qsort (x:xs) = qsort left ++ [x] ++ qsort right 

−  where left = filter ((>(snd x)) . snd) xs 

−  right = filter ((<=(snd x)) . snd) xs 

</haskell> 
</haskell> 

+  > 

== [http://projecteuler.net/index.php?section=problems&id=15 Problem 15] == 
== [http://projecteuler.net/index.php?section=problems&id=15 Problem 15] == 

Line 53:  Line 132:  
Solution: 
Solution: 

+  A direct computation: 

+  <haskell> 

+  problem_15 = iterate (scanl1 (+)) (repeat 1) !! 20 !! 20 

+  </haskell> 

+  
+  Thinking about it as a problem in combinatorics: 

+  
+  Each route has exactly 40 steps, with 20 of them horizontal and 20 of 

+  them vertical. We need to count how many different ways there are of 

+  choosing which steps are horizontal and which are vertical. So we have: 

+  
<haskell> 
<haskell> 

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

Line 62:  Line 152:  
Solution: 
Solution: 

<haskell> 
<haskell> 

−  dsum 0 = 0 

+  import Data.Char 

−  dsum n = let ( d, m ) = n `divMod` 10 in m + ( dsum d ) 

+  problem_16 = sum k 

−  
+  where s = show (2^1000) 

−  problem_16 = dsum ( 2^1000 ) 

+  k = map digitToInt s 

</haskell> 
</haskell> 

Line 73:  Line 163:  
Solution: 
Solution: 

<haskell> 
<haskell> 

−   not a very concise or beautiful solution, but food for improvements :) 

+  import Char 

−  names = concat $ 

+  one = ["one","two","three","four","five","six","seven","eight", 

−  [zip [(0, n)  n < [0..19]] 

+  "nine","ten","eleven","twelve","thirteen","fourteen","fifteen", 

−  ["", "One", "Two", "Three", "Four", "Five", "Six", "Seven", "Eight" 

+  "sixteen","seventeen","eighteen", "nineteen"] 

−  +  ty = ["twenty","thirty","forty","fifty","sixty","seventy","eighty","ninety"] 

−  ,"Sixteen", "Seventeen", "Eighteen", "Nineteen"] 

−  ,zip [(1, n)  n < [0..9]] 

−  ["", "Ten", "Twenty", "Thirty", "Fourty", "Fifty", "Sixty", "Seventy" 

−  ,"Eighty", "Ninety"] 

−  ,[((2,0), "")] 

−  ,[((2, n), look (0,n) ++ " Hundred and")  n < [1..9]] 

−  ,[((3,0), "")] 

−  ,[((3, n), look (0,n) ++ " Thousand")  n < [1..9]]] 

−  look n = fromJust . lookup n $ names 

+  decompose x 

+   x == 0 = [] 

+   x < 20 = one !! (x1) 

+   x >= 20 && x < 100 = 

+  ty !! (firstDigit (x)  2) ++ decompose ( x  firstDigit (x) * 10) 

+   x < 1000 && x `mod` 100 ==0 = 

+  one !! (firstDigit (x)1) ++ "hundred" 

+   x > 100 && x <= 999 = 

+  one !! (firstDigit (x)1) ++ "hundredand" ++decompose ( x  firstDigit (x) * 100) 

+   x == 1000 = "onethousand" 

−  spell n = unwords $ if last s == "and" then init s else s 

+  where firstDigit x = digitToInt . head . show $ x 

−  where 

−  s = words . unwords $ map look digs' 

−  digs = reverse . zip [0..] . reverse . map digitToInt . show $ n 

−  digs' = case lookup 1 digs of 

−  Just 1 > 

−  let [ten,one] = filter (\(a,_) > a<=1) digs in 

−  (digs \\ [ten,one]) ++ [(0,(snd ten)*10+(snd one))] 

−  otherwise > digs 

−  problem_17 
+  problem_17 = length . concatMap decompose $ [1..1000] 
</haskell> 
</haskell> 

Line 100:  Line 191:  
Solution: 
Solution: 

<haskell> 
<haskell> 

−  problem_18 = 
+  problem_18 = head $ foldr1 g tri 
+  where 

+  f x y z = x + max y z 

+  g xs ys = zipWith3 f xs ys $ tail ys 

+  tri = [ 

+  [75], 

+  [95,64], 

+  [17,47,82], 

+  [18,35,87,10], 

+  [20,04,82,47,65], 

+  [19,01,23,75,03,34], 

+  [88,02,77,73,07,63,67], 

+  [99,65,04,28,06,16,70,92], 

+  [41,41,26,56,83,40,80,70,33], 

+  [41,48,72,33,47,32,37,16,94,29], 

+  [53,71,44,65,25,43,91,52,97,51,14], 

+  [70,11,33,28,77,73,17,78,39,68,17,57], 

+  [91,71,52,38,17,14,91,43,58,50,27,29,48], 

+  [63,66,04,68,89,53,67,30,73,16,69,87,40,31], 

+  [04,62,98,27,23,09,70,98,73,93,38,53,60,04,23]] 

</haskell> 
</haskell> 

== [http://projecteuler.net/index.php?section=problems&id=19 Problem 19] == 
== [http://projecteuler.net/index.php?section=problems&id=19 Problem 19] == 

−  How many Sundays fell on the first of the month during the twentieth century? 

+  You are given the following information, but you may prefer to do some research for yourself. 

+  * 1 Jan 1900 was a Monday. 

+  * Thirty days has September, 

+  * April, June and November. 

+  * All the rest have thirtyone, 

+  * Saving February alone, 

+  Which has twentyeight, rain or shine. 

+  And on leap years, twentynine. 

+  * A leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400. 

+  
+  How many Sundays fell on the first of the month during the twentieth century 

+  (1 Jan 1901 to 31 Dec 2000)? 

Solution: 
Solution: 

<haskell> 
<haskell> 

−  problem_19 = undefined 

+  problem_19 = length . filter (== sunday) . drop 12 . take 1212 $ since1900 

−  </haskell> 

+  since1900 = scanl nextMonth monday . concat $ 

+  replicate 4 nonLeap ++ cycle (leap : replicate 3 nonLeap) 

−  == [http://projecteuler.net/index.php?section=problems&id=20 Problem 20] == 

+  nonLeap = [31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31] 

−  Find the sum of digits in 100! 

−  Solution: 

+  leap = 31 : 29 : drop 2 nonLeap 

−  <haskell> 

+  
−  problem_20 = let fac n = product [1..n] in 

+  nextMonth x y = (x + y) `mod` 7 

−  foldr ((+) . Data.Char.digitToInt) 0 $ show $ fac 100 

+  
+  sunday = 0 

+  monday = 1 

</haskell> 
</haskell> 

−  Alternate solution, summing digits directly, which is faster than the show, digitToInt route. 

+  Here is an alternative that is simpler, but it is cheating a bit: 

<haskell> 
<haskell> 

−  dsum 0 = 0 

+  import Data.Time.Calendar 

−  dsum n = let ( d, m ) = n `divMod` 10 in m + ( dsum d ) 

+  import Data.Time.Calendar.WeekDate 

−  problem_20' = dsum . product $ [ 1 .. 100 ] 

+  problem_19_v2 = length [()  y < [1901..2000], 

+  m < [1..12], 

+  let (_, _, d) = toWeekDate $ fromGregorian y m 1, 

+  d == 7] 

</haskell> 
</haskell> 

−  [[Category:Tutorials]] 

+  == [http://projecteuler.net/index.php?section=problems&id=20 Problem 20] == 

−  [[Category:Code]] 

+  Find the sum of digits in 100! 

+  
+  Solution: 

+  <haskell> 

+  problem_20 = sum $ map Char.digitToInt $ show $ product [1..100] 

+  </haskell> 
Latest revision as of 15:16, 16 September 2015
Contents
Problem 11
What is the greatest product of four numbers on the same straight line in the 20 by 20 grid?
Solution: using Array and Arrows, for fun :
import Control.Arrow
import Data.Array
input :: String > Array (Int,Int) Int
input = listArray ((1,1),(20,20)) . map read . words
senses = [(+1) *** id,(+1) *** (+1), id *** (+1), (+1) *** (\n > n  1)]
inArray a i = inRange (bounds a) i
prods :: Array (Int, Int) Int > [Int]
prods a = [product xs  i < range $ bounds a,
s < senses,
let is = take 4 $ iterate s i,
all (inArray a) is,
let xs = map (a!) is]
main = print . maximum . prods . input =<< getContents
Problem 12
What is the first triangle number to have over fivehundred divisors?
Solution:
primeFactors in problem_3
problem_12 = head $ filter ((> 500) . nDivisors) triangleNumbers
where nDivisors n = product $ map ((+1) . length) (group (primeFactors n))
triangleNumbers = scanl1 (+) [1..]
Problem 13
Find the first ten digits of the sum of onehundred 50digit numbers.
Solution:
main = do xs < fmap (map read . lines) (readFile "p13.log")
print . take 10 . show . sum $ xs
Problem 14
Find the longest sequence using a starting number under one million.
Solution:
import Data.List
problem_14 = j 1000000 where
f :: Int > Integer > Int
f k 1 = k
f k n = f (k+1) $ if even n then div n 2 else 3*n + 1
g x y = if snd x < snd y then y else x
h x n = g x (n, f 1 n)
j n = fst $ foldl' h (1,1) [2..n1]
Faster solution, using unboxed types and parallel computation:
import Control.Parallel
import Data.Word
collatzLen :: Int > Word32 > Int
collatzLen c 1 = c
collatzLen c n = collatzLen (c+1) $ if n `mod` 2 == 0 then n `div` 2 else 3*n+1
pmax x n = x `max` (collatzLen 1 n, n)
solve xs = foldl pmax (1,1) xs
main = print soln
where
s1 = solve [2..500000]
s2 = solve [500001..1000000]
soln = s2 `par` (s1 `pseq` max s1 s2)
Even faster solution, using an Array to memoize length of sequences :
import Data.Array
import Data.List
import Data.Ord (comparing)
syrs n =
a
where
a = listArray (1,n) $ 0 : map syr [2..n]
syr x =
if y <= n then 1 + a ! y else 1 + syr y
where
y = if even x then x `div` 2 else 3 * x + 1
main =
print . maximumBy (comparing snd) . assocs . syrs $ 1000000
Problem 15
Starting in the top left corner in a 20 by 20 grid, how many routes are there to the bottom right corner?
Solution: A direct computation:
problem_15 = iterate (scanl1 (+)) (repeat 1) !! 20 !! 20
Thinking about it as a problem in combinatorics:
Each route has exactly 40 steps, with 20 of them horizontal and 20 of them vertical. We need to count how many different ways there are of choosing which steps are horizontal and which are vertical. So we have:
problem_15 = product [21..40] `div` product [2..20]
Problem 16
What is the sum of the digits of the number 2^{1000}?
Solution:
import Data.Char
problem_16 = sum k
where s = show (2^1000)
k = map digitToInt s
Problem 17
How many letters would be needed to write all the numbers in words from 1 to 1000?
Solution:
import Char
one = ["one","two","three","four","five","six","seven","eight",
"nine","ten","eleven","twelve","thirteen","fourteen","fifteen",
"sixteen","seventeen","eighteen", "nineteen"]
ty = ["twenty","thirty","forty","fifty","sixty","seventy","eighty","ninety"]
decompose x
 x == 0 = []
 x < 20 = one !! (x1)
 x >= 20 && x < 100 =
ty !! (firstDigit (x)  2) ++ decompose ( x  firstDigit (x) * 10)
 x < 1000 && x `mod` 100 ==0 =
one !! (firstDigit (x)1) ++ "hundred"
 x > 100 && x <= 999 =
one !! (firstDigit (x)1) ++ "hundredand" ++decompose ( x  firstDigit (x) * 100)
 x == 1000 = "onethousand"
where firstDigit x = digitToInt . head . show $ x
problem_17 = length . concatMap decompose $ [1..1000]
Problem 18
Find the maximum sum travelling from the top of the triangle to the base.
Solution:
problem_18 = head $ foldr1 g tri
where
f x y z = x + max y z
g xs ys = zipWith3 f xs ys $ tail ys
tri = [
[75],
[95,64],
[17,47,82],
[18,35,87,10],
[20,04,82,47,65],
[19,01,23,75,03,34],
[88,02,77,73,07,63,67],
[99,65,04,28,06,16,70,92],
[41,41,26,56,83,40,80,70,33],
[41,48,72,33,47,32,37,16,94,29],
[53,71,44,65,25,43,91,52,97,51,14],
[70,11,33,28,77,73,17,78,39,68,17,57],
[91,71,52,38,17,14,91,43,58,50,27,29,48],
[63,66,04,68,89,53,67,30,73,16,69,87,40,31],
[04,62,98,27,23,09,70,98,73,93,38,53,60,04,23]]
Problem 19
You are given the following information, but you may prefer to do some research for yourself.
 1 Jan 1900 was a Monday.
 Thirty days has September,
 April, June and November.
 All the rest have thirtyone,
 Saving February alone,
Which has twentyeight, rain or shine. And on leap years, twentynine.
 A leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400.
How many Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000)?
Solution:
problem_19 = length . filter (== sunday) . drop 12 . take 1212 $ since1900
since1900 = scanl nextMonth monday . concat $
replicate 4 nonLeap ++ cycle (leap : replicate 3 nonLeap)
nonLeap = [31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31]
leap = 31 : 29 : drop 2 nonLeap
nextMonth x y = (x + y) `mod` 7
sunday = 0
monday = 1
Here is an alternative that is simpler, but it is cheating a bit:
import Data.Time.Calendar
import Data.Time.Calendar.WeekDate
problem_19_v2 = length [()  y < [1901..2000],
m < [1..12],
let (_, _, d) = toWeekDate $ fromGregorian y m 1,
d == 7]
Problem 20
Find the sum of digits in 100!
Solution:
problem_20 = sum $ map Char.digitToInt $ show $ product [1..100]