Euler problems/71 to 80

Problem 71

Listing reduced proper fractions in ascending order of size.

Solution:

-- http://mathworld.wolfram.com/FareySequence.html 
import Data.Ratio ((%), numerator,denominator)
fareySeq a b
    |da2<=10^6=fareySeq a1 b
    |otherwise=na
    where
    na=numerator a
    nb=numerator b
    da=denominator a
    db=denominator b
    a1=(na+nb)%(da+db)
    da2=denominator a1
problem_71=fareySeq (0%1) (3%7)

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.

groups=1000
eulerTotient n = product (map (\(p,i) -> p^(i-1) * (p-1)) factors)
    where factors = fstfac n
fstfac x = [(head a ,length a)|a<-group$primeFactors x] 
p72 n= sum [eulerTotient x|x <- [groups*n+1..groups*(n+1)]]
problem_72 = sum [p72 x|x <- [0..999]]

Problem 73

How many fractions lie between 1/3 and 1/2 in a sorted set of reduced proper fractions?

Solution:

import Data.Array
twix k = crude k - fd2 - sum [ar!(k `div` m) | m <- [3 .. k `div` 5], odd m]
    where
    fd2 = crude (k `div` 2)
    ar = array (5,k `div` 3) $
          ((5,1):[(j, crude j - sum [ar!(j `div` m) | m <- [2 .. j `div` 5]])
                      | j <- [6 .. k `div` 3]])
    crude j = 
        m*(3*m+r-2) + s
        where
            (m,r) = j `divMod` 6
            s = case r of
                  5 -> 1
                  _ -> 0
 
problem_73 =  twix 10000

Problem 74

Determine the number of factorial chains that contain exactly sixty non-repeating terms.

Solution:

import Data.List
explode 0 = []
explode n = n `mod` 10 : explode (n `quot` 10)
 
chain 2    = 1
chain 1    = 1
chain 145    = 1
chain 40585    = 1
chain 169    = 3
chain 363601 = 3
chain 1454   = 3
chain 871    = 2
chain 45361  = 2
chain 872    = 2
chain 45362  = 2
chain x = 1 + chain (sumFactDigits x)
makeIncreas 1 minnum  = [[a]|a<-[minnum..9]]
makeIncreas digits minnum  = [a:b|a<-[minnum ..9],b<-makeIncreas (digits-1) a]
p74=
    sum[div p6 $countNum a|
    a<-tail$makeIncreas  6 1,
    let k=digitToN a,
    chain k==60
    ]
    where
    p6=facts!! 6
sumFactDigits = foldl' (\a b -> a + facts !! b) 0 . explode
factorial n = if n == 0 then 1 else n * factorial (n - 1)
digitToN = foldl' (\a b -> 10*a + b) 0 .dropWhile (==0)
facts = scanl (*) 1 [1..9]
countNum xs=ys
    where
    ys=product$map (factorial.length)$group xs 
problem_74= length[k|k<-[1..9999],chain k==60]+p74
test = print $ [a|a<-tail$makeIncreas 6 0,let k=digitToN a,chain k==60]

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:

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

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.

counter = foldl (\without p ->
                     let (poor,rich) = splitAt p without
                         with = poor ++ 
                                zipWith (+) with rich
                     in with
                ) (1 : repeat 0)
 
problem_77 =  
    find ((>5000) . (ways !!)) $ [1..]
    where
    ways = counter $ take 100 primes

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
main=do 
    f<-readFile "keylog.txt"
    print$problem_79 f

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]]]