Difference between revisions of "Euler problems/31 to 40"
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| + | Do them on your own! |
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| − | == [http://projecteuler.net/index.php?section=problems&id=31 Problem 31] == |
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| − | Investigating combinations of English currency denominations. |
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| − | |||
| − | Solution: |
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| − | |||
| − | This is the naive doubly recursive solution. Speed would be greatly improved by use of [[memoization]], dynamic programming, or the closed form. |
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| − | <haskell> |
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| − | problem_31 = |
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| − | ways [1,2,5,10,20,50,100,200] !!200 |
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| − | where |
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| − | ways [] = 1 : repeat 0 |
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| − | ways (coin:coins) =n |
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| − | where |
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| − | n = zipWith (+) (ways coins) (take coin (repeat 0) ++ n) |
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| − | </haskell> |
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| − | |||
| − | A beautiful solution, making usage of laziness and recursion to implement a dynamic programming scheme, blazingly fast despite actually generating the combinations and not only counting them : |
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| − | <haskell> |
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| − | coins = [1,2,5,10,20,50,100,200] |
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| − | |||
| − | combinations = foldl (\without p -> |
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| − | let (poor,rich) = splitAt p without |
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| − | with = poor ++ |
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| − | zipWith (++) (map (map (p:)) with) |
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| − | rich |
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| − | in with |
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| − | ) ([[]] : repeat []) |
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| − | |||
| − | problem_31 = |
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| − | length $ combinations coins !! 200 |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=32 Problem 32] == |
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| − | Find the sum of all numbers that can be written as pandigital products. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Control.Monad |
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| − | combs 0 xs = [([],xs)] |
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| − | combs n xs = [(y:ys,rest)|y<-xs, (ys,rest)<-combs (n-1) (delete y xs)] |
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| − | |||
| − | l2n :: (Integral a) => [a] -> a |
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| − | l2n = foldl' (\a b -> 10*a+b) 0 |
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| − | |||
| − | swap (a,b) = (b,a) |
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| − | |||
| − | explode :: (Integral a) => a -> [a] |
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| − | explode = |
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| − | unfoldr (\a -> if a==0 then Nothing else Just $ swap $ quotRem a 10) |
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| − | |||
| − | pandigiticals = nub $ do |
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| − | (beg,end) <- combs 5 [1..9] |
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| − | n <- [1,2] |
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| − | let (a,b) = splitAt n beg |
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| − | res = l2n a * l2n b |
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| − | guard $ sort (explode res) == end |
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| − | return res |
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| − | problem_32 = sum pandigiticals |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=33 Problem 33] == |
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| − | Discover all the fractions with an unorthodox cancelling method. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Data.Ratio |
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| − | problem_33 = denominator $product $ rs |
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| − | {- |
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| − | xy/yz = x/z |
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| − | (10x + y)/(10y+z) = x/z |
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| − | 9xz + yz = 10xy |
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| − | -} |
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| − | rs=[(10*x+y)%(10*y+z) | |
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| − | x <- t, |
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| − | y <- t, |
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| − | z <- t, |
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| − | x /= y , |
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| − | (9*x*z) + (y*z) == (10*x*y) |
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| − | ] |
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| − | where |
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| − | t=[1..9] |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=34 Problem 34] == |
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| − | Find the sum of all numbers which are equal to the sum of the factorial of their digits. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Data.Map (fromList ,(!)) |
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| − | digits n |
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| − | {- 123->[3,2,1] |
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| − | -} |
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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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| − | -- 123 ->321 |
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| − | problem_34 = |
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| − | sum[ x | x <- [3..100000], x == facsum x ] |
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| − | where |
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| − | fact n = product [1..n] |
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| − | fac=fromList [(a,fact a)|a<-[0..9]] |
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| − | facsum x= sum [fac!a|a<-digits x] |
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| − | </haskell> |
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| − | |||
| − | Here's another (slighly simpler) way: |
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| − | <haskell> |
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| − | import Data.Char |
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| − | |||
| − | fac n = product [1..n] |
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| − | |||
| − | digits n = map digitToInt $ show n |
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| − | |||
| − | sum_fac n = sum $ map fac $ digits n |
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| − | |||
| − | problem_34_v2 = sum [ x | x <- [3..10^5], x == sum_fac x ] |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=35 Problem 35] == |
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| − | How many circular primes are there below one million? |
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| − | |||
| − | Solution: |
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| − | millerRabinPrimality on the [[Prime_numbers]] page |
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| − | <haskell> |
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| − | isPrime x |
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| − | |x==1=False |
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| − | |x==2=True |
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| − | |x==3=True |
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| − | |otherwise=millerRabinPrimality x 2 |
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| − | permutations n = |
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| − | take l $ map (read . take l) $ |
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| − | tails $ take (2*l -1) $ cycle s |
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| − | where |
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| − | s = show n |
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| − | l = length s |
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| − | circular_primes [] = [] |
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| − | circular_primes (x:xs) |
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| − | | all isPrime p = x : circular_primes xs |
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| − | | otherwise = circular_primes xs |
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| − | where |
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| − | p = permutations x |
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| − | x=[1,3,7,9] |
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| − | dmm=(\x y->x*10+y) |
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| − | x3=[foldl dmm 0 [a,b,c]|a<-x,b<-x,c<-x] |
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| − | x4=[foldl dmm 0 [a,b,c,d]|a<-x,b<-x,c<-x,d<-x] |
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| − | x5=[foldl dmm 0 [a,b,c,d,e]|a<-x,b<-x,c<-x,d<-x,e<-x] |
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| − | x6=[foldl dmm 0 [a,b,c,d,e,f]|a<-x,b<-x,c<-x,d<-x,e<-x,f<-x] |
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| − | problem_35 = |
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| − | (+13)$length $ circular_primes $ [a|a<-foldl (++) [] [x3,x4,x5,x6],isPrime a] |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=36 Problem 36] == |
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| − | Find the sum of all numbers less than one million, which are palindromic in base 10 and base 2. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | isPalin [] = True |
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| − | isPalin [a] = True |
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| − | isPalin (x:xs) = |
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| − | if x == last xs then isPalin $ sansLast xs else False |
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| − | where |
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| − | sansLast xs = reverse $ tail $ reverse xs |
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| − | toBase2 0 = [] |
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| − | toBase2 x = (show $ mod x 2) : toBase2 (div x 2) |
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| − | isbothPalin x = |
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| − | isPalin (show x) && isPalin (toBase2 x) |
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| − | problem_36= |
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| − | sum $ filter isbothPalin $ filter (not.even) [1..1000000] |
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| − | </haskell> |
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| − | |||
| − | Alternate Solution: |
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| − | <haskell> |
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| − | import Numeric |
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| − | import Data.Char |
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| − | |||
| − | isPalindrome x = x == reverse x |
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| − | |||
| − | showBin n = showIntAtBase 2 intToDigit n "" |
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| − | |||
| − | problem_36_v2 = sum [ n | n <- [1,3..10^6-1], |
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| − | isPalindrome (show n) && |
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| − | isPalindrome (showBin n)] |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=37 Problem 37] == |
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| − | Find the sum of all eleven primes that are both truncatable from left to right and right to left. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | -- isPrime in p35 |
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| − | clist n = |
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| − | filter isLeftTruncatable $ if isPrime n then n:ns else [] |
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| − | where |
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| − | ns = concatMap (clist . ((10*n) +)) [1,3,7,9] |
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| − | |||
| − | isLeftTruncatable = |
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| − | all isPrime . map read . init . tail . tails . show |
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| − | problem_37 = |
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| − | sum $ filter (>=10) $ concatMap clist [2,3,5,7] |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=38 Problem 38] == |
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| − | What is the largest 1 to 9 pandigital that can be formed by multiplying a fixed number by 1, 2, 3, ... ? |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Data.List |
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| − | |||
| − | mult n i vs |
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| − | | length (concat vs) >= 9 = concat vs |
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| − | | otherwise = mult n (i+1) (vs ++ [show (n * i)]) |
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| − | |||
| − | problem_38 = |
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| − | maximum $ map read $ filter |
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| − | ((['1'..'9'] ==) .sort) $ |
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| − | [ mult n 1 [] | n <- [2..9999] ] |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=39 Problem 39] == |
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| − | If p is the perimeter of a right angle triangle, {a, b, c}, which value, for p ≤ 1000, has the most solutions? |
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| − | |||
| − | Solution: |
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| − | We use the well known formula to generate primitive Pythagorean triples. All we need are the perimeters, and they have to be scaled to produce all triples in the problem space. |
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| − | <haskell> |
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| − | problem_39 = |
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| − | head $ perims !! indexMax |
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| − | where |
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| − | perims = group $ sort [n*p | p <- pTriples, n <- [1..1000 `div` p]] |
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| − | counts = map length perims |
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| − | Just indexMax = findIndex (== (maximum counts)) $ counts |
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| − | pTriples = |
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| − | [p | |
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| − | n <- [1..floor (sqrt 1000)], |
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| − | m <- [n+1..floor (sqrt 1000)], |
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| − | even n || even m, |
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| − | gcd n 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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| − | let p = a + b + c, |
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| − | p < 1000 |
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| − | ] |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=40 Problem 40] == |
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| − | Finding the nth digit of the fractional part of the irrational number. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | takeLots :: [Int] -> [a] -> [a] |
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| − | takeLots = |
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| − | t 1 |
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| − | where |
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| − | t i [] _ = [] |
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| − | t i jj@(j:js) (x:xs) |
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| − | | i == j = x : t (i+1) js xs |
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| − | | otherwise = t (i+1) jj xs |
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| − | |||
| − | digitos :: [Int] |
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| − | digitos = |
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| − | d [1] |
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| − | where |
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| − | d k = reverse k ++ d (mais k) |
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| − | mais (9:is) = 0 : mais is |
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| − | mais (i:is) = (i+1) : is |
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| − | mais [] = [1] |
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| − | |||
| − | problem_40 = |
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| − | product $ takeLots [10^n | n <- [0..6]] digitos |
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| − | </haskell> |
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| − | |||
| − | Here's how I did it, I think this is much easier to read: |
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| − | |||
| − | <haskell> |
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| − | num = concatMap show [1..] |
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| − | |||
| − | problem_40_v2 = product $ map (\x -> digitToInt (num !! (10^x-1))) [0..6] |
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| − | </haskell> |
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Revision as of 21:43, 29 January 2008
Do them on your own!