Euler problems/51 to 60

Problem 51

Find the smallest prime which, by changing the same part of the number, can form eight different primes.

Solution:

millerRabinPrimality on the Prime_numbers page

isPrime x
    |x==3=True
    |otherwise=millerRabinPrimality x 2
ch='1'
numChar n= sum [1|x<-show(n),x==ch]
replace d c|c==ch=d
           |otherwise=c
nextN repl n= (+0)$read $map repl $show n  
same n= [if isPrime$nextN (replace a) n then 1 else 0|a<-['1'..'9']]
problem_51=head [n|
    n<-[100003,100005..999999],
    numChar n==3,
    (sum $same n)==8
    ]

Problem 52

Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits in some order.

Solution:

import List
 
has_same_digits a b = (show a) \\ (show b) == []
 
check n = all (has_same_digits n) (map (n*) [2..6])
 
problem_52 = head $ filter check [1..]

Problem 53

How many values of C(n,r), for 1 ≤ n ≤ 100, exceed one-million?

Solution:

facs = scanl (*) 1 [1..100]
comb (r,n) = facs!!n `div` (facs!!r * facs!!(n-r))
perms = [(n,x) | x<-[1..100], n<-[1..x]]
problem_53 = length $ filter (>1000000) $ map comb $ perms

Problem 54

How many hands did player one win in the poker games?

Solution:

probably not the most straight forward way to do it.

import Data.List
import Data.Maybe
import Control.Monad

readCard [r,s] = (parseRank r, parseSuit s)
 where parseSuit = translate "SHDC"
       parseRank = translate "23456789TJQKA"
       translate from x = fromJust $ elemIndex x from

solveHand hand = (handRank,tiebreak)
 where
 handRank
  | flush && straight   = 9
  | hasKinds 4          = 8
  | all hasKinds [2,3]  = 7
  | flush               = 6
  | straight            = 5
  | hasKinds 3          = 4
  | 1 < length (kind 2) = 3
  | hasKinds 2          = 2
  | otherwise           = 1
 tiebreak = kind =<< [4,3,2,1]
 hasKinds = not . null . kind
 kind n = map head $ filter ((n==).length) $ group ranks
 ranks  = sortBy (flip compare) $ map fst hand
 flush  = 1 == length (nub (map snd hand))
 straight = length (kind 1) == 5 && 4 == head ranks - last ranks
 
gameLineToHands = splitAt 5 . map readCard . words
p1won (a,b) = solveHand a > solveHand b

problem_54 = do
    f <- readFile "poker.txt"
    let games = map gameLineToHands $ lines f
        wins = filter p1won games
    print $ length wins

Problem 55

How many Lychrel numbers are there below ten-thousand?

Solution:

reverseNum = read . reverse . show
 
palindrome x =
    sx == reverse sx
    where
    sx = show x 

lychrel = 
    not . any palindrome . take 50 . tail . iterate next
    where
    next x = x + reverseNum x
 
problem_55 = length $ filter lychrel [1..10000]

Problem 56

Considering natural numbers of the form, a^b, finding the maximum digital sum.

Solution:

digitalSum 0 = 0
digitalSum n = 
    let (d,m) = quotRem n 10 in m + digitalSum d 
 
problem_56 = 
    maximum [digitalSum (a^b) | a <- [99], b <- [90..99]]

Alternate solution:

import Data.Char (digitToInt)

digiSum :: Integer -> Int
digiSum = sum . map digitToInt . show

problem_56 :: Int
problem_56 = maximum $ map digiSum [a^b | a <- [1..100], b <- [1..100]]

Problem 57

Investigate the expansion of the continued fraction for the square root of two.

Solution:

twoex = zip ns ds 
    where
    ns = 3 : zipWith (\x y -> x + 2 * y) ns ds
    ds = 2 : zipWith (+) ns ds
 
len = length . show
 
problem_57 = 
    length $ filter (\(n,d) -> len n > len d) $ take 1000 twoex

The following solution is based on the observation that the fractions needed appear regularly in the repeating pattern _______$____$ where underscores are ignored and dollars are interesting fractions.

calc :: Int -> Int
calc n = nd13 * 2 + ((n-nd13*13) `div` 8)
    where
      nd13 = n `div` 13

problem_57 :: Int
problem_57 = calc 1000

Problem 58

Investigate the number of primes that lie on the diagonals of the spiral grid.

Solution:

isPrime x
    |x==3=True
    |otherwise=and [millerRabinPrimality x n|n<-[2,3]]
diag = 1:3:5:7:zipWith (+) diag [8,10..]
problem_58 = 
    result $ dropWhile tooBig $ drop 2 $ scanl primeRatio (0,0) diag
    where
    primeRatio (n,d) num = (if d `mod` 4 /= 0 && isPrime num then n+1 else n,d+1)
    tooBig (n,d) = n*10 >= d
    result ((_,d):_) = (d+2) `div` 4 * 2 + 1

Problem 59

Using a brute force attack, can you decrypt the cipher using XOR encryption?

Solution:

import Data.Bits
import Data.Char
import Data.List
import Data.Ord (comparing)
 
keys = [ [a,b,c] | a <- [97..122], b <- [97..122], c <- [97..122] ]
allAlpha = all (\k -> let a = ord k in (a >= 32 && a <= 122))
howManySpaces = length . filter (==' ')
 
problem_59 = do
	s <- readFile "cipher1.txt"
	let 
		cipher = (read ("[" ++ s ++ "]") :: [Int])
		decrypts = [ map chr (zipWith xor (cycle key) cipher) | key <- keys ]
		alphaDecrypts = filter allAlpha decrypts
		message = maximumBy (comparing howManySpaces) alphaDecrypts
		asciisum = sum (map ord message)
	print asciisum

Problem 60

Find a set of five primes for which any two primes concatenate to produce another prime.

Solution:

Breadth first search that works on infinite lists. Breaks the 60 secs rule. This program finds the solution in 185 sec on my Dell D620 Laptop.

problem_60 = print$sum $head solve 
isPrime x = x==3 || millerRabinPrimality x 2 
 
solve = do
 a <- primesTo10000
 let m = f a $ dropWhile (<= a) primesTo10000
 b <- m
 let n = f b $ dropWhile (<= b) m
 c <- n
 let o = f c $ dropWhile (<= c) n
 d <- o
 let p = f d $ dropWhile (<= d) o
 e <- p
 return [a,b,c,d,e]
 where
 f x = filter (\y -> and [isPrime $read $shows x $show y,
                 isPrime $read $shows y $show x])
primesTo10000 = 2:filter isPrime [3,5..9999]