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Euler problems/51 to 60

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== [http://projecteuler.net/index.php?section=view&id=51 Problem 51] ==
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== [http://projecteuler.net/index.php?section=problems&id=51 Problem 51] ==
 
Find the smallest prime which, by changing the same part of the number, can form eight different primes.
 
Find the smallest prime which, by changing the same part of the number, can form eight different primes.
   
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</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=52 Problem 52] ==
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== [http://projecteuler.net/index.php?section=problems&id=52 Problem 52] ==
 
Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits in some order.
 
Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits in some order.
   
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</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=53 Problem 53] ==
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== [http://projecteuler.net/index.php?section=problems&id=53 Problem 53] ==
 
How many values of C(n,r), for 1 ≤ n ≤ 100, exceed one-million?
 
How many values of C(n,r), for 1 ≤ n ≤ 100, exceed one-million?
   
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</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=54 Problem 54] ==
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== [http://projecteuler.net/index.php?section=problems&id=54 Problem 54] ==
 
How many hands did player one win in the [http://www.pokerroom.com poker games]?
 
How many hands did player one win in the [http://www.pokerroom.com poker games]?
   
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</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=55 Problem 55] ==
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== [http://projecteuler.net/index.php?section=problems&id=55 Problem 55] ==
 
How many Lychrel numbers are there below ten-thousand?
 
How many Lychrel numbers are there below ten-thousand?
   
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</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=56 Problem 56] ==
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== [http://projecteuler.net/index.php?section=problems&id=56 Problem 56] ==
 
Considering natural numbers of the form, a<sup>b</sup>, finding the maximum digital sum.
 
Considering natural numbers of the form, a<sup>b</sup>, finding the maximum digital sum.
   
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</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=57 Problem 57] ==
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== [http://projecteuler.net/index.php?section=problems&id=57 Problem 57] ==
 
Investigate the expansion of the continued fraction for the square root of two.
 
Investigate the expansion of the continued fraction for the square root of two.
   
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</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=58 Problem 58] ==
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== [http://projecteuler.net/index.php?section=problems&id=58 Problem 58] ==
 
Investigate the number of primes that lie on the diagonals of the spiral grid.
 
Investigate the number of primes that lie on the diagonals of the spiral grid.
   
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</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=59 Problem 59] ==
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== [http://projecteuler.net/index.php?section=problems&id=59 Problem 59] ==
 
Using a brute force attack, can you decrypt the cipher using XOR encryption?
 
Using a brute force attack, can you decrypt the cipher using XOR encryption?
   
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</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=60 Problem 60] ==
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== [http://projecteuler.net/index.php?section=problems&id=60 Problem 60] ==
 
Find a set of five primes for which any two primes concatenate to produce another prime.
 
Find a set of five primes for which any two primes concatenate to produce another prime.
   

Revision as of 13:51, 22 January 2008

Contents

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
    ]

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..]

3 Problem 53

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

Solution:

facs = reverse $ foldl (\y x->(head y) * x : y) [1] [1..100]
comb (r,n) = facs!!n `div` (facs!!r * facs!!(n-r))
perms = concat $ map (\x -> [(n,x) | n<-[1..x]]) [1..100]
problem_53 = length $ filter (>1000000) $ map comb $ perms

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 $ findIndex (==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  = reverse $ sort $ 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

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]

6 Problem 56

Considering natural numbers of the form, ab, 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]]

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

8 Problem 58

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

Solution:

isPrime x
    |x==3=True
    |otherwise=all id [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

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
 
keys = [ [a,b,c] | a <- [97..122], b <- [97..122], c <- [97..122] ]
allAlpha a = all (\k -> let a = ord k in (a >= 32 && a <= 122)) a
howManySpaces x = length (elemIndices ' ' x)
compareBy f x y = compare (f x) (f y)
 
problem_59 = do
	s <- readFile "cipher1.txt"
	let 
		cipher = (read ("[" ++ s ++ "]") :: [Int])
		decrypts = [ (map chr (zipWith xor (cycle key) cipher), map chr key) | key <- keys ]
		alphaDecrypts = filter (\(x,y) -> allAlpha x) decrypts
		message = maximumBy (\(x,y) (x',y') -> compareBy howManySpaces x x') alphaDecrypts
		asciisum = sum (map ord (fst message))
	putStrLn (show asciisum)

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=True
    |otherwise=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 -> all id[isPrime $read $shows x $show y,
                 isPrime $read $shows y $show x])
primesTo10000 = 2:filter (isPrime) [3,5..9999]