Euler problems/71 to 80

Problem 71

Listing reduced proper fractions in ascending order of size.

Solution:

import Data.Ratio (Ratio, (%), numerator)

fractions :: [Ratio Integer]
fractions = [f | d <- [1..1000000], let n = (d * 3) `div` 7, let f = n%d, f /= 3%7]

problem_71 :: Integer
problem_71 = numerator $ maximum $ fractions

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.

See problem 69 for phi function

problem_72 = sum [phi x|x <- [1..1000000]]

Problem 73

How many fractions lie between 1/3 and 1/2 in a sorted set of reduced proper fractions?

Solution:

import Data.Ratio (Ratio, (%), numerator, denominator)

median :: Ratio Int -> Ratio Int -> Ratio Int
median a b = ((numerator a) + (numerator b)) % ((denominator a) + (denominator b))

count :: Ratio Int -> Ratio Int -> Int
count a b
    | d > 10000 = 1
    | otherwise   = count a m + count m b
    where
        m = median a b
        d = denominator m

problem_73 :: Int
problem_73 = (count (1%3) (1%2)) - 1

Problem 74

Determine the number of factorial chains that contain exactly sixty non-repeating terms.

Solution:

problem_74 = undefined

Problem 75

Find the number of different lengths of wire can that can form a right angle triangle in only one way.

Solution: This is only slightly harder than problem 39. The search condition is simpler but the search space is larger.

problem_75 = length . filter ((== 1) . length) $ group perims
    where  perims = sort [scale*p | p <- pTriples, scale <- [1..10^6 `div` p]]
           pTriples = [p |
                       n <- [1..1000],
                       m <- [n+1..1000],
                       even n || even m,
                       gcd n m == 1,
                       let a = m^2 - n^2,
                       let b = 2*m*n,
                       let c = m^2 + n^2,
                       let p = a + b + c,
                       p <= 10^6]

Problem 76

How many different ways can one hundred be written as a sum of at least two positive integers?

Solution:

problem_76 = undefined

Problem 77

What is the first value which can be written as the sum of primes in over five thousand different ways?

Solution:

problem_77 = undefined

Problem 78

Investigating the number of ways in which coins can be separated into piles.

Solution:

problem_78 = undefined

Problem 79

By analysing a user's login attempts, can you determine the secret numeric passcode?

Solution:

problem_79 = undefined

Problem 80

Calculating the digital sum of the decimal digits of irrational square roots.

Solution:

problem_80 = undefined