# Euler problems/51 to 60

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## Latest revision as of 00:17, 17 February 2010

## Contents |

## [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 [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 ]

## [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 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

## [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 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]

## [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 + ((n-nd13*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 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

## [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]