Euler problems/71 to 80
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
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]]
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
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
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]
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
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
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..]
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
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]]]