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