Difference between revisions of "Euler problems/71 to 80"
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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 | ||
+ | |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 04:59, 30 January 2008
Contents
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)