# Euler problems/71 to 80

## 1 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```

## 2 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]]`

## 3 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```

## 4 Problem 74

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

Solution:

```import Data.Array (Array, array, (!), elems)
import Data.Char (ord)
import Data.List (foldl1')
import Prelude hiding (cycle)

fact :: Integer -> Integer
fact 0 = 1
fact n = foldl1' (*) [1..n]

factorDigits :: Array Integer Integer
factorDigits = array (0,2177281) [(x,n)|x <- [0..2177281], let n = sum \$ map (\y -> fact (toInteger \$ ord y - 48)) \$ show x]

cycle :: Integer -> Integer
cycle 145    = 1
cycle 169    = 3
cycle 363601 = 3
cycle 1454   = 3
cycle 871    = 2
cycle 45361  = 2
cycle 872    = 2
cycle 45362  = 2
cycle _      = 0

isChainLength :: Integer -> Integer -> Bool
isChainLength len n
| len < 0   = False
| t         = isChainLength (len-1) n'
| otherwise = (len - c) == 0
where
c = cycle n
t = c == 0
n' = factorDigits ! n

-- | strict version of the maximum function
maximum' :: (Ord a) => [a] -> a
maximum' [] = undefined
maximum' [x] = x
maximum' (a:b:xs) = let m = max a b in m `seq` maximum' (m : xs)

problem_74 :: Int
problem_74 = length \$ filter (\(_,b) -> isChainLength 59 b) \$ zip ([0..] :: [Integer]) \$ take 1000000 \$ elems factorDigits```

## 5 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]```

## 6 Problem 76

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

Solution:

Calculated using Euler's pentagonal formula and a list for memoization.

```partitions = 1 : [sum [s * partitions !! p| (s,p) <- zip signs \$ parts n]| n <- [1..]]
where
signs = cycle [1,1,(-1),(-1)]
suite = map penta \$ concat [[n,(-n)]|n <- [1..]]
penta n = n*(3*n - 1) `div` 2
parts n = takeWhile (>= 0) [n-x| x <- suite]

problem_76 = partitions !! 100 - 1```

## 7 Problem 77

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

Solution:

Brute force but still finds the solution in less than one second.

```combinations acc 0 _ = [acc]
combinations acc _ [] = []
combinations acc value prim@(x:xs) = combinations (acc ++ [x]) value' prim' ++ combinations acc value xs
where
value' = value - x
prim' = dropWhile (>value') prim

problem_77 :: Integer
problem_77 = head \$ filter f [1..]
where
f n = (length \$ combinations [] n \$ takeWhile (<n) primes) > 5000```

## 8 Problem 78

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

Solution:

Same as problem 76 but using array instead of lists to speedup things.

```import Data.Array

partitions :: Array Int Integer
partitions = array (0,1000000) \$ (0,1) : [(n,sum [s * partitions ! p| (s,p) <- zip signs \$ parts n])| n <- [1..1000000]]
where
signs = cycle [1,1,(-1),(-1)]
suite = map penta \$ concat [[n,(-n)]|n <- [1..]]
penta n = n*(3*n - 1) `div` 2
parts n = takeWhile (>= 0) [n-x| x <- suite]

problem_78 :: Int
problem_78 = head \$ filter (\x -> (partitions ! x) `mod` 1000000 == 0) [1..]```

## 9 Problem 79

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

Solution:

`problem_79 = undefined`

## 10 Problem 80

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

Solution:

`problem_80 = undefined`