Euler problems/91 to 100

Problem 91

Find the number of right angle triangles in the quadrant.

Solution:

reduce x y = (quot x d, quot y d)
  where d = gcd x y

problem_91 n = 3*n*n + 2* sum others
  where
    others = do
      x1 <- [1..n]
      y1 <- [1..n]
      let (yi,xi) = reduce x1 y1
      let yc = quot (n-y1) yi
      let xc = quot x1 xi
      return (min xc yc)

Problem 92

Investigating a square digits number chain with a surprising property.

Solution:

problem_92 = undefined

Problem 93

Using four distinct digits and the rules of arithmetic, find the longest sequence of target numbers.

Solution:

problem_93 = undefined

Problem 94

Investigating almost equilateral triangles with integral sides and area.

Solution:

problem_94 = undefined

Problem 95

Find the smallest member of the longest amicable chain with no element exceeding one million.

Solution which avoid visiting a number more than one time :

import Data.Array.Unboxed
import qualified Data.IntSet as S
import Data.List

takeUntil _ [] = []
takeUntil pred (x:xs) = x : if pred x then takeUntil pred xs else []

chain n s =  lgo [n] $ properDivisorsSum ! n
    where lgo xs x | x > 1000000 || S.notMember x s = (xs,[])
                   | x `elem` xs = (xs,x : takeUntil (/= x) xs)
                   | otherwise = lgo (x:xs) $ properDivisorsSum ! x

properDivisorsSum :: UArray Int Int
properDivisorsSum = accumArray (+) 1 (0,1000000) 
                    $ (0,-1):[(k,factor)| 
                               factor<-[2..1000000 `div` 2]
                             , k<-[2*factor,2*factor+factor..1000000]
                             ]

base = S.fromList [1..1000000]

problem_95 = fst $ until (S.null . snd) f ((0,0),base)
    where 
      f (p@(n,m), s) = (p', s')
          where 
            setMin = head $ S.toAscList s
            (explored, chn) = chain setMin s
            len = length chn
            p' = if len > m then (minimum chn, len) else p
            s' = foldl' (flip S.delete) s explored

---

Here is a more straightforward solution, without optimization. Yet it solves the problem in a few seconds when compiled with GHC 6.6.1 with the -O2 flag. I like to let the compiler do the optimization, without cluttering my code.

This solution avoids using unboxed arrays, which many consider to be somewhat of an imperitive-style hack. In fact, no memoization at all is required.

import Data.List (foldl1', group)

-- The sum of all proper divisors of n.
d n = product [(p * product g - 1) `div` (p - 1) |
                 g <- group $ primeFactors n, let p = head g
              ] - n

primeFactors = pf primes
 where
   pf ps@(p:ps') n
    | p * p > n = [n]
    | r == 0    = p : pf ps q
    | otherwise = pf ps' n
    where
      (q, r) = n `divMod` p

primes = 2 : filter (null . tail . primeFactors) [3,5..]

-- The longest chain of numbers is (n, k), where
-- n is the smallest number in the chain, and k is the length
-- of the chain. We limit the search to chains whose
-- smallest number is no more than m and, optionally, whose
-- largest number is no more than m'.
longestChain m m' = (n, k)
  where
    (n, Just k) = foldl1' cmpChain [(n, findChain n) | n <- [2..m]]
    findChain n = f [] n $ d n
    f s n n'
     | n' == n               = Just $ 1 + length s
     | n' < n                = Nothing
     | maybe False (< n') m' = Nothing
     | n' `elem` s           = Nothing
     | otherwise             = f (n' : s) n $ d n'
    cmpChain p@(n, k) q@(n', k')
     | (k, negate n) < (k', negate n') = q
     | otherwise                       = p

problem_95_v2 = longestChain 1000000 (Just 1000000)

Problem 96

Devise an algorithm for solving Su Doku puzzles.

Solution:

problem_96 = undefined

Problem 97

Find the last ten digits of the non-Mersenne prime: 28433 × 2^7830457 + 1.

Solution:

problem_97 = (28433 * 2^7830457 + 1) `mod` (10^10)

Problem 98

Investigating words, and their anagrams, which can represent square numbers.

Solution:

problem_98 = undefined

Problem 99

Which base/exponent pair in the file has the greatest numerical value?

Solution:

problem_99 = undefined

Problem 100

Finding the number of blue discs for which there is 50% chance of taking two blue.

Solution:

problem_100 = undefined