Difference between revisions of "Euler problems/71 to 80"

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Do them on your own!
  
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== [http://projecteuler.net/index.php?section=view&id=71 Problem 71] ==
  +
Listing reduced proper fractions in ascending order of size.
  +
  +
Solution:
  +
<haskell>
  +
-- http://mathworld.wolfram.com/FareySequence.html
  +
import Data.Ratio ((%), numerator,denominator)
  +
fareySeq a b
  +
|da2<=10^6=fareySeq a1 b
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|otherwise=na
  +
where
  +
na=numerator a
  +
nb=numerator b
  +
da=denominator a
  +
db=denominator b
  +
a1=(na+nb)%(da+db)
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da2=denominator a1
  +
problem_71=fareySeq (0%1) (3%7)
  +
</haskell>
  +
  +
== [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?
  +
  +
Solution:
  +
  +
Using the [http://mathworld.wolfram.com/FareySequence.html Farey Sequence] method, the solution is the sum of phi (n) from 1 to 1000000.
  +
<haskell>
  +
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]]
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</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=view&id=73 Problem 73] ==
  +
How many fractions lie between 1/3 and 1/2 in a sorted set of reduced proper fractions?
  +
  +
Solution:
  +
<haskell>
  +
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) $
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((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
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(m,r) = j `divMod` 6
  +
s = case r of
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5 -> 1
  +
_ -> 0
  +
  +
problem_73 = twix 10000
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=view&id=74 Problem 74] ==
  +
Determine the number of factorial chains that contain exactly sixty non-repeating terms.
  +
  +
Solution:
  +
<haskell>
  +
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)
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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
  +
]
  +
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
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test = print $ [a|a<-tail$makeIncreas 6 0,let k=digitToN a,chain k==60]
  +
</haskell>
  +
== [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.
  +
  +
Solution:
  +
<haskell>
  +
import Data.Array
  +
  +
triplets =
  +
[p |
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n <- [2..706],
  +
m <- [1..n-1],
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gcd m n == 1,
  +
let p = 2 * (n^2 + m*n),
  +
odd (m + n),
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p <= 10^6
  +
]
  +
  +
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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]
  +
  +
problem_75 =
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length $ filter (\(_,b) -> b == 1) $ assocs arr
  +
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..]
  +
</haskell>
  +
  +
== [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?
  +
  +
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.
  +
<haskell>
  +
build x = (map sum (zipWith drop [0..] x) ++ [1]) : x
  +
problem_76 = (sum $ head $ iterate build [] !! 100) - 1
  +
</haskell>
  +
  +
== [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?
  +
  +
Solution:
  +
  +
Brute force but still finds the solution in less than one second.
  +
<haskell>
  +
counter = foldl (\without p ->
  +
let (poor,rich) = splitAt p without
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with = poor ++
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zipWith (+) with rich
  +
in with
  +
) (1 : repeat 0)
  +
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problem_77 =
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find ((>5000) . (ways !!)) $ [1..]
  +
where
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ways = counter $ take 100 primes
  +
</haskell>
  +
  +
== [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.
  +
  +
Solution:
  +
<haskell>
  +
import Data.Array
  +
  +
partitions :: Array Int Integer
  +
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]
  +
  +
problem_78 :: Int
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problem_78 =
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head $ filter (\x -> (partitions ! x) `mod` 1000000 == 0) [1..]
  +
</haskell>
  +
  +
== [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?
  +
  +
Solution:
  +
<haskell>
  +
import Data.Char (digitToInt, intToDigit)
  +
import Data.Graph (buildG, topSort)
  +
import Data.List (intersect)
  +
  +
p79 file=
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(+0)$read . intersect graphWalk $ usedDigits
  +
where
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usedDigits = intersect "0123456789" $ file
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edges = concat . map (edgePair . map digitToInt) . words $ file
  +
graphWalk = map intToDigit . topSort . buildG (0, 9) $ edges
  +
edgePair [x, y, z] = [(x, y), (y, z)]
  +
edgePair _ = undefined
  +
  +
problem_79 = do
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f<-readFile "keylog.txt"
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print $p79 f
  +
</haskell>
  +
  +
== [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.
  +
  +
Solution:
  +
<haskell>
  +
import Data.Char
  +
problem_80=
  +
sum [f x |
  +
a <- [1..100],
  +
x <- [intSqrt $ a * t],
  +
x * x /= a * t
  +
]
  +
where
  +
t=10^202
  +
f = (sum . take 100 . map (flip (-) (ord '0') .ord) . show)
  +
</haskell>

Revision as of 04:59, 30 January 2008

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: 

import Data.Array
 
triplets = 
    [p | 
    n <- [2..706],
    m <- [1..n-1],
    gcd m n == 1, 
    let p = 2 * (n^2 + m*n),
    odd (m + n),
    p <= 10^6
    ]
 
hist bnds ns = 
    accumArray (+) 0 bnds [(n, 1) |
        n <- ns,
        inRange bnds n
        ]
 
problem_75 = 
    length $ filter (\(_,b) -> b == 1) $ assocs arr
    where
    arr = hist (12,10^6) $ concatMap multiples triplets
    multiples n = takeWhile (<=10^6) [n, 2*n..]

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: 

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: 

import Data.Char (digitToInt, intToDigit)
import Data.Graph (buildG, topSort)
import Data.List (intersect)
 
p79 file= 
    (+0)$read . intersect graphWalk $ usedDigits
    where
    usedDigits = intersect "0123456789" $ file
    edges = concat . map (edgePair . map digitToInt) . words $ file
    graphWalk = map intToDigit . topSort . buildG (0, 9) $ edges
    edgePair [x, y, z] = [(x, y), (y, z)]
    edgePair _         = undefined
 
problem_79 = do
    f<-readFile  "keylog.txt"
    print $p79 f

Problem 80

Calculating the digital sum of the decimal digits of irrational square roots.

Solution: 

import Data.Char
problem_80=
    sum [f x |
    a <- [1..100],
    x <- [intSqrt $ a * t],
    x * x /= a * t
    ]
    where
    t=10^202
    f = (sum . take 100 . map (flip (-) (ord '0') .ord) . show)
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