# Euler problems/181 to 190

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## Contents |

## 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