Euler problems/71 to 80
From HaskellWiki
(Difference between revisions)
(Alternative solution for Problem 76) |
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fractions :: [Ratio Integer] | fractions :: [Ratio Integer] | ||
| - | fractions = [f | d <- [1..1000000], let n = (d * 3) `div` 7, let f = n%d, f /= 3%7] | + | fractions = |
| + | [f | | ||
| + | d <- [1..1000000], | ||
| + | let n = (d * 3) `div` 7, | ||
| + | let f = n%d, | ||
| + | f /= 3%7 | ||
| + | ] | ||
problem_71 :: Integer | problem_71 :: Integer | ||
| Line 63: | Line 69: | ||
factorDigits :: Array Integer Integer | factorDigits :: Array Integer Integer | ||
| - | factorDigits = array (0,2177281) [(x,n)|x <- [0..2177281], let n = sum $ map (\y -> fact (toInteger $ ord y - 48)) $ show x] | + | factorDigits = |
| + | array (0,2177281) | ||
| + | [(x,n)| | ||
| + | x <- [0..2177281], | ||
| + | let n = sum $ | ||
| + | map (\y -> fact (toInteger $ ord y - 48)) $ | ||
| + | show x | ||
| + | ] | ||
cycle :: Integer -> Integer | cycle :: Integer -> Integer | ||
| Line 93: | Line 106: | ||
problem_74 :: Int | problem_74 :: Int | ||
| - | problem_74 = length $ filter (\(_,b) -> isChainLength 59 b) $ zip ([0..] :: [Integer]) $ take 1000000 $ elems factorDigits | + | problem_74 = |
| + | length $ filter | ||
| + | (\(_,b) -> isChainLength 59 b) $ | ||
| + | zip ([0..] :: [Integer]) $ | ||
| + | take 1000000 $ elems factorDigits | ||
</haskell> | </haskell> | ||
| Line 131: | Line 148: | ||
This is only slightly harder than [[Euler problems/31 to 40#39|problem 39]]. The search condition is simpler but the search space is larger. | This is only slightly harder than [[Euler problems/31 to 40#39|problem 39]]. The search condition is simpler but the search space is larger. | ||
<haskell> | <haskell> | ||
| - | problem_75 = length . filter ((== 1) . length) $ group perims | + | problem_75 = |
| + | length . filter ((== 1) . length) $ group perims | ||
where perims = sort [scale*p | p <- pTriples, scale <- [1..10^6 `div` p]] | where perims = sort [scale*p | p <- pTriples, scale <- [1..10^6 `div` p]] | ||
pTriples = [p | | pTriples = [p | | ||
| Line 152: | Line 170: | ||
Calculated using Euler's pentagonal formula and a list for memoization. | Calculated using Euler's pentagonal formula and a list for memoization. | ||
<haskell> | <haskell> | ||
| - | partitions = 1 : [sum [s * partitions !! p| (s,p) <- zip signs $ parts n]| n <- [1..]] | + | partitions = |
| + | 1 : [sum | ||
| + | [s * partitions !! p| | ||
| + | (s,p) <- zip signs $ parts n | ||
| + | ]| | ||
| + | n <- [1..]] | ||
where | where | ||
signs = cycle [1,1,(-1),(-1)] | signs = cycle [1,1,(-1),(-1)] | ||
| Line 178: | Line 201: | ||
combinations acc 0 _ = [acc] | combinations acc 0 _ = [acc] | ||
combinations acc _ [] = [] | combinations acc _ [] = [] | ||
| - | combinations acc value prim@(x:xs) = combinations (acc ++ [x]) value' prim' ++ combinations acc value xs | + | combinations acc value prim@(x:xs) = |
| + | combinations (acc ++ [x]) value' prim' ++ combinations acc value xs | ||
where | where | ||
value' = value - x | value' = value - x | ||
| Line 199: | Line 223: | ||
partitions :: Array Int Integer | partitions :: Array Int Integer | ||
| - | partitions = array (0,1000000) $ (0,1) : [(n,sum [s * partitions ! p| (s,p) <- zip signs $ parts n])| n <- [1..1000000]] | + | partitions = |
| + | array (0,1000000) $ | ||
| + | (0,1) : | ||
| + | [(n,sum [s * partitions ! p| | ||
| + | (s,p) <- zip signs $ parts n])| | ||
| + | n <- [1..1000000]] | ||
where | where | ||
signs = cycle [1,1,(-1),(-1)] | signs = cycle [1,1,(-1),(-1)] | ||
| Line 207: | Line 236: | ||
problem_78 :: Int | problem_78 :: Int | ||
| - | problem_78 = head $ filter (\x -> (partitions ! x) `mod` 1000000 == 0) [1..] | + | problem_78 = |
| + | head $ filter (\x -> (partitions ! x) `mod` 1000000 == 0) [1..] | ||
</haskell> | </haskell> | ||
| Line 218: | Line 248: | ||
<haskell> | <haskell> | ||
import Data.List | import Data.List | ||
| - | + | ||
problem_79 :: String -> String | problem_79 :: String -> String | ||
| - | problem_79 file = map fst $ sortBy (\(_,a) (_,b) -> compare (length b) (length a)) $ zip digs order | + | problem_79 file = |
| + | map fst $ | ||
| + | sortBy (\(_,a) (_,b) -> | ||
| + | compare (length b) (length a)) $ | ||
| + | zip digs order | ||
where | where | ||
| - | + | nums = lines file | |
| - | + | digs = | |
| - | + | map head $ group $ | |
| - | + | sort $ filter (\c -> c >= '0' && c <= '9') file | |
| + | prec = concatMap (\(x:y:z:_) -> [[x,y],[y,z],[x,z]]) nums | ||
| + | order = | ||
| + | map (\n -> map head $ | ||
| + | group $ sort $ map (\(_:x:_) -> x) $ | ||
| + | filter (\(x:_) -> x == n) prec) digs | ||
| + | main=do | ||
| + | f<-readFile "keylog.txt" | ||
| + | print$problem_79 f | ||
</haskell> | </haskell> | ||
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squareDigs' :: Integer -> Integer -> [Integer] -> [Integer] | squareDigs' :: Integer -> Integer -> [Integer] -> [Integer] | ||
| - | squareDigs' p r (x:xs) = x' : squareDigs' (p*10 + x') r' xs | + | squareDigs' p r (x:xs) = |
| + | x' : squareDigs' (p*10 + x') r' xs | ||
where | where | ||
| - | + | n = 100*r + x | |
| - | + | (x',r') = | |
| - | + | last $ takeWhile | |
| + | (\(_,a) -> a >= 0) $ | ||
| + | scanl (\(_,b) (a',b') -> (a',b-b')) (0,n) rs | ||
| + | rs = [y|y <- zip [1..] [(20*p+1),(20*p+3)..]] | ||
sumDigits n = sum $ take 100 $ squareDigs n | sumDigits n = sum $ take 100 $ squareDigs n | ||
problem_80 :: Integer | problem_80 :: Integer | ||
| - | problem_80 = sum $ map sumDigits [x|x <- [1..100] \\ [n^2|n<-[1..10]]] | + | problem_80 = |
| + | sum $ map sumDigits | ||
| + | [x|x <- [1..100] \\ [n^2|n<-[1..10]]] | ||
</haskell> | </haskell> | ||
Revision as of 04:04, 6 January 2008
Contents |
1 Problem 71
Listing reduced proper fractions in ascending order of size.
Solution:
import Data.Ratio (Ratio, (%), numerator) fractions :: [Ratio Integer] fractions = [f | d <- [1..1000000], let n = (d * 3) `div` 7, let f = n%d, f /= 3%7 ] problem_71 :: Integer problem_71 = numerator $ maximum $ fractions
2 Problem 72
How many elements would be contained in the set of reduced proper fractions for d ≤ 1,000,000?
Solution:
Using the Farey Sequence method, the solution is the sum of phi (n) from 1 to 1000000.
See problem 69 for phi function
problem_72 = sum [phi x|x <- [1..1000000]]
3 Problem 73
How many fractions lie between 1/3 and 1/2 in a sorted set of reduced proper fractions?
Solution:
import Data.Ratio (Ratio, (%), numerator, denominator) median :: Ratio Int -> Ratio Int -> Ratio Int median a b = ((numerator a) + (numerator b)) % ((denominator a) + (denominator b)) count :: Ratio Int -> Ratio Int -> Int count a b | d > 10000 = 1 | otherwise = count a m + count m b where m = median a b d = denominator m problem_73 :: Int problem_73 = (count (1%3) (1%2)) - 1
4 Problem 74
Determine the number of factorial chains that contain exactly sixty non-repeating terms.
Solution:
import Data.Array (Array, array, (!), elems) import Data.Char (ord) import Data.List (foldl1') import Prelude hiding (cycle) fact :: Integer -> Integer fact 0 = 1 fact n = foldl1' (*) [1..n] factorDigits :: Array Integer Integer factorDigits = array (0,2177281) [(x,n)| x <- [0..2177281], let n = sum $ map (\y -> fact (toInteger $ ord y - 48)) $ show x ] cycle :: Integer -> Integer cycle 145 = 1 cycle 169 = 3 cycle 363601 = 3 cycle 1454 = 3 cycle 871 = 2 cycle 45361 = 2 cycle 872 = 2 cycle 45362 = 2 cycle _ = 0 isChainLength :: Integer -> Integer -> Bool isChainLength len n | len < 0 = False | t = isChainLength (len-1) n' | otherwise = (len - c) == 0 where c = cycle n t = c == 0 n' = factorDigits ! n -- | strict version of the maximum function maximum' :: (Ord a) => [a] -> a maximum' [] = undefined maximum' [x] = x maximum' (a:b:xs) = let m = max a b in m `seq` maximum' (m : xs) problem_74 :: Int problem_74 = length $ filter (\(_,b) -> isChainLength 59 b) $ zip ([0..] :: [Integer]) $ take 1000000 $ elems factorDigits
Slightly faster solution :
{-# OPTIONS_GHC -fbang-patterns #-} import Data.List import Data.Array explode 0 = [] explode n = n `mod` 10 : explode (n `quot` 10) count :: (a -> Bool) -> [a] -> Int count pred xs = lgo 0 xs where lgo !c [] = c lgo !c (y:ys) | pred y = lgo (c + 1) ys | otherwise = lgo c ys known = [([1,2,145,40585],1),([871,45361,872,45362],2),([169,363601,1454],3)] mChain = array (1,1000000) $ (concat $ expand known) ++ [(x, n)|x<-[3..1000000] , not $ x `elem` concat (map fst known) , let n = 1 + chain (sumFactDigits x)] where expand [] = [] expand ((xs,len):xxs) = map (flip (,) len) xs : expand xxs chain x | x <= 1000000 = mChain ! x | otherwise = 1 + chain (sumFactDigits x) sumFactDigits = foldl' (\a b -> a + facts !! b) 0 . explode facts = scanl (*) 1 [1..9] problem_74 = count (== 60) $ elems mChain
5 Problem 75
Find the number of different lengths of wire can that can form a right angle triangle in only one way.
Solution: This is only slightly harder than problem 39. The search condition is simpler but the search space is larger.
problem_75 = length . filter ((== 1) . length) $ group perims where perims = sort [scale*p | p <- pTriples, scale <- [1..10^6 `div` p]] pTriples = [p | n <- [1..1000], m <- [n+1..1000], even n || even m, gcd n m == 1, let a = m^2 - n^2, let b = 2*m*n, let c = m^2 + n^2, let p = a + b + c, p <= 10^6]
6 Problem 76
How many different ways can one hundred be written as a sum of at least two positive integers?
Solution:
Calculated using Euler's pentagonal formula and a list for memoization.
partitions = 1 : [sum [s * partitions !! p| (s,p) <- zip signs $ parts n ]| n <- [1..]] where signs = cycle [1,1,(-1),(-1)] suite = map penta $ concat [[n,(-n)]|n <- [1..]] penta n = n*(3*n - 1) `div` 2 parts n = takeWhile (>= 0) [n-x| x <- suite] problem_76 = partitions !! 100 - 1
Here is a simpler solution: For each n, we create the list of the number of partitions of n whose lowest number is i, for i=1..n. We build up the list of these lists for n=0..100.
build x = (map sum (zipWith drop [0..] x) ++ [1]) : x problem_76' = (sum $ head $ iterate build [] !! 100) - 1
7 Problem 77
What is the first value which can be written as the sum of primes in over five thousand different ways?
Solution:
Brute force but still finds the solution in less than one second.
combinations acc 0 _ = [acc] combinations acc _ [] = [] combinations acc value prim@(x:xs) = combinations (acc ++ [x]) value' prim' ++ combinations acc value xs where value' = value - x prim' = dropWhile (>value') prim problem_77 :: Integer problem_77 = head $ filter f [1..] where f n = (length $ combinations [] n $ takeWhile (<n) primes) > 5000
8 Problem 78
Investigating the number of ways in which coins can be separated into piles.
Solution:
Same as problem 76 but using array instead of lists to speedup things.
import Data.Array partitions :: Array Int Integer partitions = array (0,1000000) $ (0,1) : [(n,sum [s * partitions ! p| (s,p) <- zip signs $ parts n])| n <- [1..1000000]] where signs = cycle [1,1,(-1),(-1)] suite = map penta $ concat [[n,(-n)]|n <- [1..]] penta n = n*(3*n - 1) `div` 2 parts n = takeWhile (>= 0) [n-x| x <- suite] problem_78 :: Int problem_78 = head $ filter (\x -> (partitions ! x) `mod` 1000000 == 0) [1..]
9 Problem 79
By analysing a user's login attempts, can you determine the secret numeric passcode?
Solution:
A bit ugly but works fine
import Data.List problem_79 :: String -> String problem_79 file = map fst $ sortBy (\(_,a) (_,b) -> compare (length b) (length a)) $ zip digs order where nums = lines file digs = map head $ group $ sort $ filter (\c -> c >= '0' && c <= '9') file prec = concatMap (\(x:y:z:_) -> [[x,y],[y,z],[x,z]]) nums order = map (\n -> map head $ group $ sort $ map (\(_:x:_) -> x) $ filter (\(x:_) -> x == n) prec) digs main=do f<-readFile "keylog.txt" print$problem_79 f
10 Problem 80
Calculating the digital sum of the decimal digits of irrational square roots.
Solution:
import Data.List ((\\)) hundreds :: Integer -> [Integer] hundreds n = hundreds' [] n where hundreds' acc 0 = acc hundreds' acc n = hundreds' (m : acc) d where (d,m) = divMod n 100 squareDigs :: Integer -> [Integer] squareDigs n = p : squareDigs' p r xs where (x:xs) = hundreds n ++ repeat 0 p = floor $ sqrt $ fromInteger x r = x - (p^2) squareDigs' :: Integer -> Integer -> [Integer] -> [Integer] squareDigs' p r (x:xs) = x' : squareDigs' (p*10 + x') r' xs where n = 100*r + x (x',r') = last $ takeWhile (\(_,a) -> a >= 0) $ scanl (\(_,b) (a',b') -> (a',b-b')) (0,n) rs rs = [y|y <- zip [1..] [(20*p+1),(20*p+3)..]] sumDigits n = sum $ take 100 $ squareDigs n problem_80 :: Integer problem_80 = sum $ map sumDigits [x|x <- [1..100] \\ [n^2|n<-[1..10]]]
