# Euler problems/181 to 190

## Problem 181

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

## 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
```

## 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
```

## Problem 184

Triangles containing the origin.

Solution:

```
problem_184 = undefined
```

## 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
```