Difference between revisions of "Euler problems/61 to 70"

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Do them on your own!
  
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== [http://projecteuler.net/index.php?section=problems&id=61 Problem 61] ==
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Find the sum of the only set of six 4-digit figurate numbers with a cyclic property.
  +
  +
Solution:
  +
<haskell>
  +
import Data.List
  +
  +
permute [] = [[]]
  +
permute xs = concatMap (\x -> map (x:) $ permute $ delete x xs) xs
  +
  +
figurates n xs = extract $ concatMap (gather (map poly xs)) $ map (:[]) $ poly n
  +
where gather [xs] (v:vs)
  +
= let v' = match xs v
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in if v' == [] then [] else map (:v:vs) v'
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gather (xs:xss) (v:vs)
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= let v' = match xs v
  +
in if v' == [] then [] else concatMap (gather xss) $ map (:v:vs) v'
  +
match xs (_,v) = let p = (v `mod` 100)*100 in sublist (p+10,p+100) xs
  +
sublist (s,e) = takeWhile (\(_,x) -> x<e) . dropWhile (\(_,x) -> x<s)
  +
link ((_,x):xs) = x `mod` 100 == (snd $ last xs) `div` 100
  +
diff (x:y:xs) = if fst x /= fst y then diff (y:xs) else False
  +
diff [x] = True
  +
extract = filter diff . filter link
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poly m = [(n, x) | (n, x) <- zip [1..] $ takeWhile (<10000)
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$ scanl (+) 1 [m-1,2*m-3..],
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1010 < x, x `mod` 100 > 9]
  +
  +
problem_61 = sum $ map snd $ head $ concatMap (figurates 3) $ permute [4..8]
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</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=62 Problem 62] ==
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Find the smallest cube for which exactly five permutations of its digits are cube.
  +
  +
Solution:
  +
<haskell>
  +
import Data.List
  +
import Data.Maybe
  +
a = map (^3) [0..10000]
  +
b = map (sort . show) a
  +
c = (filter ((==5) . length) . group . sort) b
  +
d = findIndex (==(head (head c))) b
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problem_62 = (toInteger (fromJust d))^3
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=63 Problem 63] ==
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How many n-digit positive integers exist which are also an nth power?
  +
  +
Solution:
  +
<haskell>
  +
problem_63=length[x^y|x<-[1..9],y<-[1..22],y==(length$show$x^y)]
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=64 Problem 64] ==
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How many continued fractions for N ≤ 10000 have an odd period?
  +
  +
Solution:
  +
<haskell>
  +
import Data.List
  +
  +
problem_64 =length $ filter id $ map solve $ [2..9999] \\ (map (^2) [2..100])
  +
  +
solve n = even $ length $ cont n 0 1
  +
  +
cont :: Int -> Int -> Int -> [Int]
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cont r n d = m : rest
  +
where
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m = truncate ((sqrt (fromIntegral r) + fromIntegral n ) / fromIntegral d)
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a = n - d * m
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rest = if d == 1 && n /= 0
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then []
  +
else cont r (-a) ((r - a ^ 2) `div` d)
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</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=65 Problem 65] ==
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Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e.
  +
  +
Solution:
  +
<haskell>
  +
import Data.Char
  +
import Data.Ratio
  +
  +
e = [2] ++ concat [ [1, 2*i, 1] | i <- [1..] ]
  +
  +
fraction [x] = x%1
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fraction (x:xs) = x%1 + 1/(fraction xs)
  +
  +
problem_65 = sum $ map digitToInt $ show $ numerator $ fraction $ take 100 e
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=66 Problem 66] ==
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Investigate the Diophantine equation x<sup>2</sup> − Dy<sup>2</sup> = 1.
  +
  +
Solution:
  +
<haskell>
  +
intSqrt :: Integral a => a -> a
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intSqrt n
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| n < 0 = error "intSqrt: negative n"
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| otherwise = f n
  +
where
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f x = if y < x then f y else x
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where y = (x + (n `quot` x)) `quot` 2
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problem_66 =
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snd$maximum [ (x,d) |
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d <- [1..1000],
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let b = intSqrt d,
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b*b /= d, -- d can't be a perfect square
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let (x,_) = pell d b b
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]
  +
  +
pell d wd b = piter d wd b 0 1 0 1 1 0
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piter d wd b i c l k m n
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| cn == 1 = (x, y)
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| otherwise = piter d wd bn (i+1) cn k u n v
  +
where
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yb = (wd+b) `div` c
  +
bn = yb*c-b
  +
cn = (d-(bn*bn)) `div` c
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yn | i == 0 = wd
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| otherwise = yb
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u = k*yn+l -- u/v is the i-th convergent of sqrt(d)
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v = n*yn+m
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(x,y) | odd (i+1) = (u*u+d*v*v, 2*u*v)
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| otherwise = (u,v)
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</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=67 Problem 67] ==
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Using an efficient algorithm find the maximal sum in the triangle?
  +
  +
Solution:
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<haskell>
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problem_67 = readFile "triangle.txt" >>= print . solve . parse
  +
parse = map (map read . words) . lines
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solve = head . foldr1 step
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step [] [z] = [z]
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step (x:xs) (y:z:zs) = x + max y z : step xs (z:zs)
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</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=68 Problem 68] ==
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What is the maximum 16-digit string for a "magic" 5-gon ring?
  +
  +
Solution:
  +
<haskell>
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import Data.List
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permute [] = [[]]
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permute list =
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concat $ map (\(x:xs) -> map (x:) (permute xs))
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(take (length list)
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(unfoldr (\x -> Just (x, tail x ++ [head x])) list))
  +
problem_68 =
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maximum $ map (concat . map show) poel
  +
where
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gon68 = [1..10]
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knip = (length gon68) `div` 2
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(is,es) = splitAt knip gon68
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extnodes = map (\x -> [head es]++x) $ permute $ tail es
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intnodes = map (\(p:ps) -> zipWith (\ x y -> [x]++[y])
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(p:ps) (ps++[p])) $ permute is
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poel = [ concat hs | hs <- [ zipWith (\x y -> [x]++y) uitsteeksels organen |
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uitsteeksels <- extnodes, organen <- intnodes ],
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let subsom = map (sum) hs, length (nub subsom) == 1 ]
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</haskell>
  +
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== [http://projecteuler.net/index.php?section=problems&id=69 Problem 69] ==
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Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum.
  +
  +
Solution:
  +
<haskell>
  +
{-phi(n) = n*(1-1/p1)*(1-1/p2)*...*(1-1/pn)
  +
n/phi(n) = 1/(1-1/p1)*(1-1/p2)*...*(1-1/pn)
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(1-1/p) will be minimal for a small p and 1/(1-1/p) will then be maximal
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-}
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primes=[2,3,5,7,11,13,17,19,23]
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problem_69=
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maximum [c|
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a<-[1..length primes],
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let b=take a primes,
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let c=product b,
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c<10^6
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]
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</haskell>
  +
  +
Note: credit for arithmetic functions is due to [http://www.polyomino.f2s.com/ David Amos].
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=70 Problem 70] ==
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Investigate values of n for which φ(n) is a permutation of n.
  +
  +
Solution:
  +
<haskell>
  +
import Data.List
  +
isPerm a b = (show a) \\ (show b)==[]
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flsqr n x=x<(floor.sqrt.fromInteger) n
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pairs n1 =
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maximum[m|a<-gena ,b<-genb,let m=a*b,n>m,isPerm m$ m-a-b+1]
  +
where
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n=fromInteger n1
  +
gena = dropWhile (flsqr n)$ takeWhile (flsqr (2*n)) primes
  +
genb = dropWhile (flsqr (div n 2))$ takeWhile (flsqr n) primes
  +
  +
problem_70= pairs (10^7)
  +
</haskell>

Revision as of 04:58, 30 January 2008

Problem 61

Find the sum of the only set of six 4-digit figurate numbers with a cyclic property.

Solution: 

import Data.List
 
permute [] = [[]]
permute xs = concatMap (\x -> map (x:) $ permute $ delete x xs) xs
 
figurates n xs = extract $ concatMap (gather (map poly xs)) $ map (:[]) $ poly n
  where gather [xs] (v:vs) 
          = let v' = match xs v
            in if v' == [] then [] else map (:v:vs) v'
        gather (xs:xss) (v:vs) 
          = let v' = match xs v
            in if v' == [] then [] else concatMap (gather xss) $ map (:v:vs) v'
        match xs (_,v) = let p = (v `mod` 100)*100 in sublist (p+10,p+100) xs
        sublist (s,e) = takeWhile (\(_,x) -> x<e) . dropWhile (\(_,x) -> x<s)
        link ((_,x):xs) = x `mod` 100 == (snd $ last xs) `div` 100
        diff (x:y:xs) = if fst x /= fst y then diff (y:xs) else False
        diff [x]      = True
        extract = filter diff . filter link
        poly m = [(n, x) | (n, x) <- zip [1..] $ takeWhile (<10000) 
                                               $ scanl (+) 1 [m-1,2*m-3..], 
                                     1010 < x, x `mod` 100 > 9]
 
problem_61 = sum $ map snd $ head $ concatMap (figurates 3) $ permute [4..8]

Problem 62

Find the smallest cube for which exactly five permutations of its digits are cube.

Solution: 

import Data.List
import Data.Maybe
a = map (^3) [0..10000]
b = map (sort . show) a
c = (filter ((==5) . length) . group . sort) b
d = findIndex (==(head (head c))) b
problem_62 = (toInteger (fromJust d))^3

Problem 63

How many n-digit positive integers exist which are also an nth power?

Solution: 

problem_63=length[x^y|x<-[1..9],y<-[1..22],y==(length$show$x^y)]

Problem 64

How many continued fractions for N ≤ 10000 have an odd period?

Solution: 

import Data.List
 
problem_64  =length $ filter id $ map solve $ [2..9999] \\ (map (^2) [2..100])
 
solve n = even $ length $ cont n 0 1
 
cont :: Int -> Int -> Int -> [Int]
cont r n d = m : rest
    where
    m = truncate ((sqrt (fromIntegral r) + fromIntegral n ) / fromIntegral d)
    a = n - d * m
    rest = if d == 1 && n /= 0
           then []
           else cont r (-a) ((r - a ^ 2) `div` d)

Problem 65

Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e.

Solution: 

import Data.Char
import Data.Ratio
 
e = [2] ++ concat [ [1, 2*i, 1] | i <- [1..] ]
 
fraction [x] = x%1
fraction (x:xs) = x%1 + 1/(fraction xs)

problem_65 = sum $ map digitToInt $ show $ numerator $ fraction $ take 100 e

Problem 66

Investigate the Diophantine equation x2 − Dy2 = 1.

Solution: 

intSqrt :: Integral a => a -> a
intSqrt n
    | n < 0 = error "intSqrt: negative n"
    | otherwise = f n
    where
        f x = if y < x then f y else x
            where y = (x + (n `quot` x)) `quot` 2
problem_66 = 
    snd$maximum [ (x,d) | 
    d <- [1..1000],
    let b = intSqrt d,
    b*b /= d, -- d can't be a perfect square
    let (x,_) = pell d b b 
    ]

pell d wd b = piter d wd b 0 1 0 1 1 0
piter d wd b i c l k m n 
    | cn == 1 = (x, y)
    | otherwise = piter d wd bn (i+1) cn k u n v
    where 
    yb = (wd+b) `div` c
    bn = yb*c-b
    cn = (d-(bn*bn)) `div` c
    yn  | i == 0 = wd
        | otherwise = yb
    u = k*yn+l -- u/v is the i-th convergent of sqrt(d)
    v = n*yn+m
    (x,y)   | odd (i+1) = (u*u+d*v*v, 2*u*v)
            | otherwise = (u,v)

Problem 67

Using an efficient algorithm find the maximal sum in the triangle?

Solution: 

problem_67 = readFile "triangle.txt" >>= print . solve . parse
parse = map (map read . words) . lines
solve = head . foldr1 step
step [] [z] = [z]
step (x:xs) (y:z:zs) = x + max y z : step xs (z:zs)

Problem 68

What is the maximum 16-digit string for a "magic" 5-gon ring?

Solution: 

import Data.List
permute []      = [[]]
permute list = 
    concat $ map (\(x:xs) -> map (x:) (permute xs))
    (take (length list) 
    (unfoldr (\x -> Just (x, tail x ++ [head x])) list))
problem_68 = 
    maximum $ map (concat . map show) poel 
    where
    gon68 = [1..10]
    knip = (length gon68) `div` 2
    (is,es) = splitAt knip gon68
    extnodes = map (\x -> [head es]++x) $ permute $ tail es
    intnodes = map (\(p:ps) -> zipWith (\ x y -> [x]++[y])
        (p:ps) (ps++[p])) $ permute is
    poel = [ concat hs | hs <- [ zipWith (\x y -> [x]++y) uitsteeksels organen |
        uitsteeksels <- extnodes, organen <- intnodes ],
        let subsom = map (sum) hs, length (nub subsom) == 1 ]

Problem 69

Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum.

Solution: 

{-phi(n) = n*(1-1/p1)*(1-1/p2)*...*(1-1/pn)
n/phi(n) = 1/(1-1/p1)*(1-1/p2)*...*(1-1/pn)
(1-1/p) will be minimal for a small p and 1/(1-1/p) will then be maximal
 -}
primes=[2,3,5,7,11,13,17,19,23]
problem_69=
    maximum [c|
    a<-[1..length primes],
    let b=take a primes,
    let c=product b,
    c<10^6
    ]

Note: credit for arithmetic functions is due to David Amos.

Problem 70

Investigate values of n for which φ(n) is a permutation of n.

Solution: 

import Data.List
isPerm a b = (show a) \\ (show b)==[]
flsqr n x=x<(floor.sqrt.fromInteger) n
pairs n1 = 
    maximum[m|a<-gena ,b<-genb,let m=a*b,n>m,isPerm m$ m-a-b+1]
    where
    n=fromInteger n1
    gena = dropWhile (flsqr n)$  takeWhile (flsqr (2*n))  primes
    genb = dropWhile (flsqr (div n 2))$  takeWhile (flsqr n)  primes

problem_70= pairs (10^7)
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