# Euler problems/61 to 70

### From HaskellWiki

## Contents |

## 1 Problem 61

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

Solution:

import Data.List triangle = [n*(n+1)`div`2 | n <- [1..]] square = [n^2 | n <- [1..]] pentagonal = [n*(3*n-1)`div`2 | n <- [1..]] hexagonal = [n*(2*n-1) | n <- [1..]] heptagonal = [n*(5*n-3)`div`2 | n <- [1..]] octagonal = [n*(3*n-2) | n <- [1..]] triangle4 = fourDigs triangle square4 = fourDigs square pentagonal4 = fourDigs pentagonal hexagonal4 = fourDigs hexagonal heptagonal4 = fourDigs heptagonal octagonal4 = fourDigs octagonal fourDigs = takeWhile (<10000) . dropWhile (<1000) solve = do (l1:l2:l3:l4:l5:l6:_) <- permute [triangle4, square4, pentagonal4, hexagonal4, heptagonal4, octagonal4] a <- l1 let m = filter (g a) l2 b <- m let n = filter (g b) l3 c <- n let o = filter (g c) l4 d <- o let p = filter (g d) l5 e <- p let q = filter (g e) l6 f <- q if g f a then return (sum [a,b,c,d,e,f]) else fail "burp" where g x y = x `mod` 100 == y `div` 100 permute :: [a] -> [[a]] permute [] = [[]] permute list = concat $ map (\(x:xs) -> map (x:) (permute xs)) (take (length list) (unfoldr (\x -> Just (x, tail x ++ [head x])) list)) problem_61 = head $ solve

## 2 Problem 62

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

Solution:

problem_62 = undefined

## 3 Problem 63

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

Solution:
Since d^{n} has at least n+1 digits for any d≥10, we need only consider 1 through 9. If d^{n} has fewer than n digits, every higher power of d will also be too small since d < 10. We will also never have n+1 digits for our nth powers. All we have to do is check d^{n} for each d in {1,...,9}, trying n=1,2,... and stopping when d^{n} has fewer than n digits.

problem_63 = length . concatMap (takeWhile (\(n,p) -> n == nDigits p)) $ [powers d | d <- [1..9]] where powers d = [(n, d^n) | n <- [1..]] nDigits n = length (show n)

## 4 Problem 64

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

Solution:

problem_64 = undefined

## 5 Problem 65

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

Solution:

import Data.Ratio problem_65 = dsum . numerator . contFrac . take 100 $ e where dsum 0 = 0 dsum n = let ( d, m ) = n `divMod` 10 in m + ( dsum d ) contFrac = foldr1 (\x y -> x + 1/y) e = 2 : 1 : insOnes [2,4..] insOnes (x:xs) = x : 1 : 1 : insOnes xs

## 6 Problem 66

Investigate the Diophantine equation x^{2} − Dy^{2} = 1.

Solution:

problem_66 = undefined

## 7 Problem 67

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

Solution:

import System.Process import IO slurpURL url = do (_,out,_,_) <- runInteractiveCommand $ "curl " ++ url hGetContents out problem_67 = do src <- slurpURL "http://projecteuler.net/project/triangle.txt" print $ head $ foldr1 g $ parse src where parse :: String -> [[Int]] parse s = map ((map read).words) $ lines s f x y z = x + max y z g xs ys = zipWith3 f xs ys $ tail ys

## 8 Problem 68

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

Solution:

problem_68 = undefined

## 9 Problem 69

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

Solution:

problem_69 = undefined

## 10 Problem 70

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

Solution:

problem_70 = undefined