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Euler problems/71 to 80

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== [http://projecteuler.net/index.php?section=view&id=71 Problem 71] ==
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Do them on your own!
Listing reduced proper fractions in ascending order of size.
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Solution:
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<haskell>
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-- http://mathworld.wolfram.com/FareySequence.html
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import Data.Ratio ((%), numerator,denominator)
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fareySeq a b
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    |da2<=10^6=fareySeq a1 b
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    |otherwise=na
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    where
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    na=numerator a
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    nb=numerator b
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    da=denominator a
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    db=denominator b
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    a1=(na+nb)%(da+db)
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    da2=denominator a1
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problem_71=fareySeq (0%1) (3%7)
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</haskell>
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== [http://projecteuler.net/index.php?section=view&id=72 Problem 72] ==
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How many elements would be contained in the set of reduced proper fractions for d ≤ 1,000,000?
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Solution:
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Using the [http://mathworld.wolfram.com/FareySequence.html Farey Sequence] method, the solution is the sum of phi (n) from 1 to 1000000.
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<haskell>
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groups=1000
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eulerTotient n = product (map (\(p,i) -> p^(i-1) * (p-1)) factors)
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    where factors = fstfac n
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fstfac x = [(head a ,length a)|a<-group$primeFactors x]
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p72 n= sum [eulerTotient x|x <- [groups*n+1..groups*(n+1)]]
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problem_72 = sum [p72 x|x <- [0..999]]
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</haskell>
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== [http://projecteuler.net/index.php?section=view&id=73 Problem 73] ==
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How many fractions lie between 1/3 and 1/2 in a sorted set of reduced proper fractions?
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Solution:
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<haskell>
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import Data.Array
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twix k = crude k - fd2 - sum [ar!(k `div` m) | m <- [3 .. k `div` 5], odd m]
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    where
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    fd2 = crude (k `div` 2)
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    ar = array (5,k `div` 3) $
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          ((5,1):[(j, crude j - sum [ar!(j `div` m) | m <- [2 .. j `div` 5]])
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                      | j <- [6 .. k `div` 3]])
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    crude j =
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        m*(3*m+r-2) + s
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        where
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            (m,r) = j `divMod` 6
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            s = case r of
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                  5 -> 1
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                  _ -> 0
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problem_73 =  twix 10000
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</haskell>
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== [http://projecteuler.net/index.php?section=view&id=74 Problem 74] ==
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Determine the number of factorial chains that contain exactly sixty non-repeating terms.
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Solution:
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<haskell>
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import Data.List
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explode 0 = []
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explode n = n `mod` 10 : explode (n `quot` 10)
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chain 2    = 1
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chain 1    = 1
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chain 145    = 1
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chain 40585    = 1
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chain 169    = 3
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chain 363601 = 3
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chain 1454  = 3
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chain 871    = 2
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chain 45361  = 2
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chain 872    = 2
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chain 45362  = 2
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chain x = 1 + chain (sumFactDigits x)
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makeIncreas 1 minnum  = [[a]|a<-[minnum..9]]
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makeIncreas digits minnum  = [a:b|a<-[minnum ..9],b<-makeIncreas (digits-1) a]
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p74=
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    sum[div p6 $countNum a|
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    a<-tail$makeIncreas  6 1,
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    let k=digitToN a,
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    chain k==60
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    ]
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    where
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    p6=facts!! 6
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sumFactDigits = foldl' (\a b -> a + facts !! b) 0 . explode
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factorial n = if n == 0 then 1 else n * factorial (n - 1)
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digitToN = foldl' (\a b -> 10*a + b) 0 .dropWhile (==0)
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facts = scanl (*) 1 [1..9]
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countNum xs=ys
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    where
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    ys=product$map (factorial.length)$group xs
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problem_74= length[k|k<-[1..9999],chain k==60]+p74
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test = print $ [a|a<-tail$makeIncreas 6 0,let k=digitToN a,chain k==60]
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</haskell>
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== [http://projecteuler.net/index.php?section=view&id=75 Problem 75] ==
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Find the number of different lengths of wire can that can form a right angle triangle in only one way.
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Solution:
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<haskell>
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import Data.Array
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triplets =
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    [p |
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    n <- [2..706],
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    m <- [1..n-1],
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    gcd m n == 1,
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    let p = 2 * (n^2 + m*n),
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    odd (m + n),
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    p <= 10^6
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    ]
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hist bnds ns =
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    accumArray (+) 0 bnds [(n, 1) |
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        n <- ns,
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        inRange bnds n
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        ]
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problem_75 =
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    length $ filter (\(_,b) -> b == 1) $ assocs arr
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    where
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    arr = hist (12,10^6) $ concatMap multiples triplets
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    multiples n = takeWhile (<=10^6) [n, 2*n..]
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</haskell>
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== [http://projecteuler.net/index.php?section=view&id=76 Problem 76] ==
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How many different ways can one hundred be written as a sum of at least two positive integers?
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Solution:
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Here is a simpler solution: For each n, we create the list of the number of partitions of n
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whose lowest number is i, for i=1..n. We build up the list of these lists for n=0..100.
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<haskell>
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build x = (map sum (zipWith drop [0..] x) ++ [1]) : x
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problem_76 = (sum $ head $ iterate build [] !! 100) - 1
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</haskell>
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== [http://projecteuler.net/index.php?section=view&id=77 Problem 77] ==
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What is the first value which can be written as the sum of primes in over five thousand different ways?
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Solution:
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Brute force but still finds the solution in less than one second.
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<haskell>
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counter = foldl (\without p ->
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                    let (poor,rich) = splitAt p without
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                        with = poor ++
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                                zipWith (+) with rich
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                    in with
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                ) (1 : repeat 0)
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problem_77 = 
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    find ((>5000) . (ways !!)) $ [1..]
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    where
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    ways = counter $ take 100 primes
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</haskell>
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== [http://projecteuler.net/index.php?section=view&id=78 Problem 78] ==
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Investigating the number of ways in which coins can be separated into piles.
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Solution:
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<haskell>
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import Data.Array
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partitions :: Array Int Integer
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partitions =
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    array (0,1000000) $
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    (0,1) :
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    [(n,sum [s * partitions ! p|
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    (s,p) <- zip signs $ parts n])|
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    n <- [1..1000000]]
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    where
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        signs = cycle [1,1,(-1),(-1)]
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        suite = map penta $ concat [[n,(-n)]|n <- [1..]]
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        penta n = n*(3*n - 1) `div` 2
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        parts n = takeWhile (>= 0) [n-x| x <- suite]
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problem_78 :: Int
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problem_78 =
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    head $ filter (\x -> (partitions ! x) `mod` 1000000 == 0) [1..]
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</haskell>
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== [http://projecteuler.net/index.php?section=view&id=79 Problem 79] ==
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By analysing a user's login attempts, can you determine the secret numeric passcode?
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Solution:
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<haskell>
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import Data.Char (digitToInt, intToDigit)
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import Data.Graph (buildG, topSort)
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import Data.List (intersect)
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p79 file=
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    (+0)$read . intersect graphWalk $ usedDigits
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    where
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    usedDigits = intersect "0123456789" $ file
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    edges = concat . map (edgePair . map digitToInt) . words $ file
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    graphWalk = map intToDigit . topSort . buildG (0, 9) $ edges
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    edgePair [x, y, z] = [(x, y), (y, z)]
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    edgePair _        = undefined
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problem_79 = do
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    f<-readFile  "keylog.txt"
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    print $p79 f
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</haskell>
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== [http://projecteuler.net/index.php?section=view&id=80 Problem 80] ==
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Calculating the digital sum of the decimal digits of irrational square roots.
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Solution:
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<haskell>
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import Data.Char
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problem_80=
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    sum [f x |
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    a <- [1..100],
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    x <- [intSqrt $ a * t],
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    x * x /= a * t
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    ]
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    where
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    t=10^202
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    f = (sum . take 100 . map (flip (-) (ord '0') .ord) . show)
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</haskell>
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Revision as of 21:46, 29 January 2008

Do them on your own!