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Euler problems/181 to 190

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1 Problem 181

Investigating in how many ways objects of two different colours can be grouped.

2 Problem 182

RSA encryption.

Solution:

fun a1 b1 = sum [ e | e <- [2..a*b-1],
                      gcd e (a*b) == 1,
                      gcd (e-1) a == 2,
                      gcd (e-1) b == 2 ]
  where a = a1-1
        b = b1-1
 
problem_182 = fun 1009 3643

3 Problem 183

Maximum product of parts.

Solution:

-- Does the decimal expansion of p/q terminate?
terminating p q = 1 == reduce [2,5] (q `div` gcd p q)
        where reduce   []   n = n
              reduce (x:xs) n | n `mod` x == 0 = reduce (x:xs) (n `div` x)
                              | otherwise      = reduce xs n
 
-- The expression (round $ fromIntegral n / e) computes the integer k
-- for which (n/k)^k is at a maximum. Also note that, given a rational number
-- r and a natural number k, the decimal expansion of r^k terminates if
-- and only if the decimal expansion of r does.
answer = sum [if terminating n (round $ fromIntegral n / e) then -n else n
                | n <- [5 .. 10^4]]
        where e = exp 1
 
main = print answer

4 Problem 184

Triangles containing the origin.

Solution:

problem_184 = undefined

5 Problem 185

Number Mind

Solution:

This approach does NOT solve the problem in under a minute, unless of course you are extremely lucky. The best time I've seen so far has been about 76 seconds. Before I came up with this code, I tried to search for the solution by generating a list of all possible solutions based on the information given in the guesses. This was feasible with the 5 digit problem, but completely intractable with the 16 digit problem. The approach here, simple yet effective, is to make a random guess, and then vary each digit in the guess from [0..9], generating a score of how well the guess matched the given numbermind clues. You then improve the guess by selecting those digits that had a unique low score. It turns out this approach converges rather quickly, but can often be stuck in cycles, so we test for this and try a differenct random first guess if a cycle is detected. Once you run the program, you might have time for a cup of coffee, or maybe even a dinner. HenryLaxen 2008-03-12

import Data.List
import Control.Monad
import Data.Char
import System.Random
 
type Mind = [([Char],Int)]
 
values :: [Char]
values = "0123456789"
 
 
score :: [Char] -> [Char] -> Int
score guess answer = foldr (\(a,b) y -> if a == b then y+1 else y) 0 
      (zip guess answer)
 
scores :: Mind -> [Char] -> [Int]
scores m g = map (\x -> abs ((snd x) - score (fst x) g)) m
 
scoreMind :: Mind -> [Char] -> Int
scoreMind m g = sum $ scores m g
 
 
ex1 :: Mind
ex1 = 
  [("90342",2),
   ("39458",2),
   ("51545",2),
   ("34109",1),
   ("12531",1),
   ("70794",0)]
 
ex2 :: Mind   
ex2 = 
  [
   ("5616185650518293",2),
   ("3847439647293047",1),
   ("5855462940810587",3),
   ("9742855507068353",3),
   ("4296849643607543",3),
   ("3174248439465858",1),
   ("4513559094146117",2),
   ("7890971548908067",3),
   ("8157356344118483",1),
   ("2615250744386899",2),
   ("8690095851526254",3),
   ("6375711915077050",1),
   ("6913859173121360",1),
   ("6442889055042768",2),
   ("2321386104303845",0),
   ("2326509471271448",2),
   ("5251583379644322",2),
   ("1748270476758276",3),
   ("4895722652190306",1),
   ("3041631117224635",3),
   ("1841236454324589",3),
   ("2659862637316867",2)]
 
 
guesses :: [Char] -> Int -> [[Char]]
guesses str pos = [ left ++ n:(tail right) | n<-values]
  where (left,right) = splitAt pos str
 
bestGuess :: Mind -> [[Char]] -> [Int]
bestGuess mind guesses =
  let scores = map (scoreMind mind) guesses
      bestScore = minimum scores
      bestGuesses = findIndices (==bestScore) scores
  in bestGuesses
 
iterateGuesses :: Mind -> [Char] -> [Char]
iterateGuesses mind value = 
   let allguesses = map (guesses value) [0..(length value)-1]
       mins = map (bestGuess mind) allguesses
   in nextguess value mins
 
nextguess :: [Char] -> [[Int]] -> [Char]
nextguess prev mins = 
  let choose x = if length (snd x) == 1 then intToDigit ((snd x)!!0) else fst x
      both = zip prev mins
  in  foldr (\x y -> (choose x) : y) "" both
 
 
iterateMind :: Mind -> [Char] -> [([Char], Int)]
iterateMind mind n =
  let a = drop 2 $ inits $ iterate (iterateGuesses mind) n
      b = last $ takeWhile (\x -> (last x) `notElem` (init x)) a
      c = map (scoreMind mind) b
  in zip b c      
 
 
randomStart :: (Num a, Enum a) => a -> IO [Char]
randomStart n = mapM (\_ -> getStdRandom (randomR ('0','9'))) [1..n]
 
main :: IO ()
main = do
  let ex = ex1
  x <- randomStart (length (fst (head ex)))
  let y = iterateMind ex x
  let done = (snd (last  y) == 0)
  when done (putStrLn $ (fst.last) y)
  unless done main

Here's another solution, and this one squeaks by in just under a minute on my machine. The basic idea is to just do a back-tracking search, but with some semi-smart pruning and guess ordering. The code is in pretty much the order I wrote it, so most prefixes of this code should also compile. This also means you should be able to figure out what each function does one at a time.

import Control.Monad
import Data.List
import qualified Data.Set as S
 
ensure p x = guard (p x) >> return x
selectDistinct 0 _  = [[]]
selectDistinct n [] = []
selectDistinct n (x:xs) = map (x:) (selectDistinct (n - 1) xs) ++ selectDistinct n xs
 
data Environment a = Environment { guesses      :: [(Int, [a])]
                                 , restrictions :: [S.Set a]
                                 , assignmentsLeft :: Int
                                 } deriving (Eq, Ord, Show)
 
reorder e = e { guesses = sort . guesses $ e }
domain  = S.fromList "0123456789"
initial = Environment gs (replicate a S.empty) a where
    a   = length . snd . head $ gs
    gs  = [(2, "5616185650518293"), (1, "3847439647293047"), (3, "5855462940810587"), (3, "9742855507068353"), (3, "4296849643607543"), (1, "3174248439465858"), (2, "4513559094146117"), (3, "7890971548908067"), (1, "8157356344118483"), (2, "2615250744386899"), (3, "8690095851526254"), (1, "6375711915077050"), (1, "6913859173121360"), (2, "6442889055042768"), (0, "2321386104303845"), (2, "2326509471271448"), (2, "5251583379644322"), (3, "1748270476758276"), (1, "4895722652190306"), (3, "3041631117224635"), (3, "1841236454324589"), (2, "2659862637316867")]
 
acceptableCounts e = small >= 0 && big <= assignmentsLeft e where
    ns    = (0:) . map fst . guesses $ e
    small = minimum ns
    big   = maximum ns
 
positions s = map fst . filter (not . snd) . zip [0..] . zipWith S.member s
acceptableRestriction  r (n, s) = length (positions s r) >= n
acceptableRestrictions e = all (acceptableRestriction (restrictions e)) (guesses e)
 
firstGuess = head . guesses
sanityCheck e = acceptableRestrictions e && acceptableCounts e
 
solve e@(Environment _  _ 0) = [[]]
solve e@(Environment [] r _) = sequence $ map (S.toList . (domain S.\\)) r
solve e' = do
    is   <- m
    newE <- f is
    rest <- solve newE
    return $ interleaveAscIndices is (l is) rest
        where
    f = ensure sanityCheck . update e
    m = selectDistinct n (positions g (restrictions e)) 
    e = reorder e'
    l = fst . flip splitAscIndices g
    (n, g) = firstGuess e
 
splitAscIndices = indices 0 where
    indices _ [] xs = ([], xs)
    indices n (i:is) (x:xs)
        | i == n = let (b, e) = indices (n + 1) is     xs in (x:b, e)
        | True   = let (b, e) = indices (n + 1) (i:is) xs in (b, x:e)
 
interleaveAscIndices = indices 0 where
    indices _ [] [] ys = ys
    indices n (i:is) (x:xs) ys
        | i == n = x : indices (n + 1) is xs ys
        | True   = head ys : indices (n + 1) (i:is) (x:xs) (tail ys)
 
update (Environment ((_, a):gs) r l) is = Environment newGs restriction (l - length is) where
    (assignment, newRestriction)    = splitAscIndices is a
    (_, oldRestriction)             = splitAscIndices is r
    restriction                     = zipWith S.insert newRestriction oldRestriction
    newGs                           = map updateEntry gs
    updateEntry (n', a')            = (newN, newA) where
        (dropped, newA) = splitAscIndices is a'
        newN            = n' - length (filter id $ zipWith (==) assignment dropped)
 
problem_185 = head . solve $ initial