Difference between revisions of "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

```

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 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

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
```