Euler problems/51 to 60
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Latest revision as of 00:17, 17 February 2010
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[edit] 1 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 [1x<show(n),x==ch] replace d cc==ch=d otherwise=c nextN repl n= (+0)$read $map repl $show n same n= [if isPrime$nextN (replace a) n then 1 else 0a<['1'..'9']] problem_51=head [n n<[100003,100005..999999], numChar n==3, (sum $same n)==8 ]
[edit] 2 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..]
[edit] 3 Problem 53
How many values of C(n,r), for 1 ≤ n ≤ 100, exceed onemillion?
Solution:
facs = scanl (*) 1 [1..100] comb (r,n) = facs!!n `div` (facs!!r * facs!!(nr)) perms = [(n,x)  x<[1..100], n<[1..x]] problem_53 = length $ filter (>1000000) $ map comb $ perms
[edit] 4 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
[edit] 5 Problem 55
How many Lychrel numbers are there below tenthousand?
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]
[edit] 6 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]]
[edit] 7 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 + ((nnd13*13) `div` 8) where nd13 = n `div` 13 problem_57 :: Int problem_57 = calc 1000
[edit] 8 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 nn<[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
[edit] 9 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
[edit] 10 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]