# Euler problems/71 to 80

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(Faster (5X) solution to problem 74) |
(Removing category tags. See Talk:Euler_problems) |

## Revision as of 12:11, 30 September 2007

## 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

## 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

## 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]]]