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Euler problems/51 to 60

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Do them on your own!
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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.
 +
 
 +
Solution:
 +
 
 +
millerRabinPrimality on the [[Prime_numbers]] page
 +
 
 +
<haskell>
 +
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
 +
    ]
 +
</haskell>
 +
 
 +
== [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.
 +
 
 +
Solution:
 +
<haskell>
 +
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..]
 +
</haskell>
 +
 
 +
== [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?
 +
 
 +
Solution:
 +
<haskell>
 +
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
 +
</haskell>
 +
 
 +
== [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]?
 +
 
 +
Solution:
 +
 
 +
probably not the most straight forward way to do it.
 +
 
 +
<haskell>
 +
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
 +
</haskell>
 +
 
 +
== [http://projecteuler.net/index.php?section=problems&id=55 Problem 55] ==
 +
How many Lychrel numbers are there below ten-thousand?
 +
 
 +
Solution:
 +
<haskell>
 +
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]
 +
</haskell>
 +
 
 +
== [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.
 +
 
 +
Solution:
 +
<haskell>
 +
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]]
 +
</haskell>
 +
 
 +
== [http://projecteuler.net/index.php?section=problems&id=57 Problem 57] ==
 +
Investigate the expansion of the continued fraction for the square root of two.
 +
 
 +
Solution:
 +
<haskell>
 +
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
 +
</haskell>
 +
 
 +
== [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.
 +
 
 +
Solution:
 +
<haskell>
 +
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
 +
</haskell>
 +
 
 +
== [http://projecteuler.net/index.php?section=problems&id=59 Problem 59] ==
 +
Using a brute force attack, can you decrypt the cipher using XOR encryption?
 +
 
 +
Solution:
 +
<haskell>
 +
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)
 +
</haskell>
 +
 
 +
== [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.
 +
 
 +
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.
 +
<haskell>
 +
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]
 +
</haskell>

Revision as of 04:58, 30 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]