Euler problems/61 to 70

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
Just d = elemIndex (head (head c)) b
problem_62 = toInteger 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 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)) + n) `div` d
    a = n - d * m
    rest | d == 1 && n /= 0 = []
         | otherwise = 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 | y < x = f y 
            | otherwise = 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 :: String -> [[Int]] -- restrict output type for 'read' function
parse = map (map read . words) . lines

solve = head . foldr1 step -- reverse pairewise addition from bottom to top
step [] _ = [] -- returen empty list to avoid eception warning
step (x:xs) (y:z:zs) = x + max y z : step xs (z:zs)

Alternatively, one could add lists pairewise from top to bottom. However, this would require a check on maximum in the list, after the final step of additions.

Problem 68

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

Solution:

import Data.List
permute []      = [[]]
permute list = 
    concatMap (\(x:xs) -> map (x:) (permute xs))
    (take (length list) 
    (unfoldr (\l@(x:xs) -> Just (l, xs ++ [x])) list))
problem_68 = 
    maximum $ map (concatMap show) poel 
    where
    gon68 = [1..10]
    knip = (length gon68) `div` 2
    (is,e:es) = splitAt knip gon68
    extnodes = map (e:) $ permute es
    intnodes = map (\(p:ps) -> zipWith (\ x y -> [x, y])
        (p:ps) (ps++[p])) $ permute is
    poel = [ concat hs |
            uitsteeksels <- extnodes,
            organen <- intnodes,
            let hs = zipWith (:) uitsteeksels organen,
            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|
    b<-tail $ inits 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
import Data.Function
isPerm a b = null $ show a \\ show b
flsqr n x=x<(floor.sqrt.fromInteger) n
pairs n1 = 
    fst . minimumBy (compare `on` fn) $ [(m,pm)|a<-gena,b<-genb,let m=a*b,n>m,let pm=m-a-b+1,isPerm m pm]
    where
    n=fromInteger n1
    gena = dropWhile (flsqr n)$  takeWhile (flsqr (2*n))  primes
    genb = dropWhile (flsqr (n `div` 2))$  takeWhile (flsqr n)  primes
    fn (x,px) = fromIntegral x / (fromIntegral px) 

problem_70= pairs (10^7)