Personal tools

Euler problems/71 to 80

From HaskellWiki

< Euler problems(Difference between revisions)
Jump to: navigation, search
Line 1: Line 1:
== [http://projecteuler.net/index.php?section=view&id=71 Problem 71] ==
+
Do them on your own!
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
 
|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)
 
</haskell>
 
 
== [http://projecteuler.net/index.php?section=view&id=72 Problem 72] ==
 
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]]
 
</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) $
 
((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
 
</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)
 
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]
 
</haskell>
 
== [http://projecteuler.net/index.php?section=view&id=75 Problem 75] ==
 
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 |
 
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..]
 
</haskell>
 
 
== [http://projecteuler.net/index.php?section=view&id=76 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.
 
<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] ==
 
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
 
with = poor ++
 
zipWith (+) with rich
 
in with
 
) (1 : repeat 0)
 
 
problem_77 =
 
find ((>5000) . (ways !!)) $ [1..]
 
where
 
ways = counter $ take 100 primes
 
</haskell>
 
 
== [http://projecteuler.net/index.php?section=view&id=78 Problem 78] ==
 
Investigating the number of ways in which coins can be separated into piles.
 
 
Solution:
 
<haskell>
 
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..]
 
</haskell>
 
 
== [http://projecteuler.net/index.php?section=view&id=79 Problem 79] ==
 
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=
 
(+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
 
</haskell>
 
 
== [http://projecteuler.net/index.php?section=view&id=80 Problem 80] ==
 
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 21:46, 29 January 2008

Do them on your own!