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Euler problems/21 to 30

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Line 96: Line 96:
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
  +
next n d = (n `mod` d):next (10*n`mod`d) d
  +
  +
idigs n = tail $ take (1+n) $ next 1 n
  +
  +
pos x = map fst . filter ((==x) . snd) . zip [1..]
  +
  +
periods n = let d = idigs n in pos (head d) (tail d)
  +
 
problem_26 =
 
problem_26 =
fst $ maximumBy (\a b -> snd a `compare` snd b)
+
snd$maximum [(m,a)|
[(n,recurringCycle n) | n <- [1..999]]
+
a<-[800..1000] ,
where
+
let k=periods a,
recurringCycle d = remainders d 10 []
+
not$null k,
remainders d 0 rs = 0
+
let m=head k
remainders d r rs = let r' = r `mod` d
+
]
in case findIndex (== r') rs of
 
Just i -> i + 1
 
Nothing -> remainders d (10*r') (r':rs)
 
 
</haskell>
 
</haskell>
   
Line 109: Line 117:
   
 
Solution:
 
Solution:
 
The following is written in [http://haskell.org/haskellwiki/Literate_programming#Haskell_and_literate_programming literate Haskell]:
 
 
<haskell>
 
<haskell>
> import Data.List
+
eulerCoefficients n
+
= [((len, a*b), (a, b))
To be sure we get the maximum type checking of the compiler,
+
| b <- takeWhile (<n) primes, a <- [-b+1..n-1],
we switch off the default type
+
let len = length $ takeWhile (isPrime . (\x -> x^2 + a*x + b)) [0..],
+
if b == 2 then even a else odd a, len > 39]
> default ()
+
+
problem_27 = snd . fst . maximum . eulerCoefficients $ 1000
Generate a list of primes.
 
It works by filtering out numbers that are
 
divisable by a previously found prime
 
 
> primes :: [Int]
 
> primes = sieve (2 : [3, 5..])
 
> where
 
> sieve (p:xs) = p : sieve (filter (\x -> x `mod` p > 0) xs)
 
 
> isPrime :: Int -> Bool
 
> isPrime x = x `elem` (takeWhile (<= x) primes)
 
 
 
The lists of values we are going to try for a and b;
 
b must be a prime, as n² + an + b is equal to b when n = 0
 
 
> testRangeA :: [Int]
 
> testRangeA = [-1000 .. 1000]
 
 
> testRangeB :: [Int]
 
> testRangeB = takeWhile (< 1000) primes
 
 
 
The search
 
 
> bestCoefficients :: (Int, Int, Int)
 
> bestCoefficients =
 
> maximumBy (\(x, _, _) (y, _, _) -> compare x y) $
 
> [f a b | a <- testRangeA, b <- testRangeB]
 
> where
 
 
Generate a list of results of the quadratic formula
 
(only the contiguous primes)
 
wrap the result in a triple, together with a and b
 
 
> f :: Int -> Int -> (Int, Int, Int)
 
> f a b = ( length $ contiguousPrimes a b
 
> , a
 
> , b
 
> )
 
 
> contiguousPrimes :: Int -> Int -> [Int]
 
> contiguousPrimes a b = takeWhile isPrime (map (quadratic a b) [0..])
 
 
 
The quadratic formula
 
 
> quadratic :: Int -> Int -> Int -> Int
 
> quadratic a b n = n * n + a * n + b
 
 
 
> problem_27 =
 
> do
 
> let (l, a, b) = bestCoefficients
 
>
 
> putStrLn $ ""
 
> putStrLn $ "Problem Euler 27"
 
> putStrLn $ ""
 
> putStrLn $ "The best quadratic formula found is:"
 
> putStrLn $ " n * n + " ++ show a ++ " * n + " ++ show b
 
> putStrLn $ ""
 
> putStrLn $ "The number of primes is: " ++ (show l)
 
> putStrLn $ ""
 
> putStrLn $ "The primes are:"
 
> print $ take l $ contiguousPrimes a b
 
> putStrLn $ ""
 
 
 
 
</haskell>
 
</haskell>
   
Line 126: Line 132:
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
corners :: Int -> (Int, Int, Int, Int)
+
problem_28 = sum (map (\n -> 4*(n-2)^2+10*(n-1)) [3,5..1001]) + 1
corners i = (n*n, 1+(n*(2*m)), 2+(n*(2*m-1)), 3+(n*(2*m-2)))
 
where m = (i-1) `div` 2
 
n = 2*m+1
 
 
sumcorners :: Int -> Int
 
sumcorners i = a+b+c+d where (a, b, c, d) = corners i
 
 
sumdiags :: Int -> Int
 
sumdiags i | even i = error "not a spiral"
 
| i == 3 = s + 1
 
| otherwise = s + sumdiags (i-2)
 
where s = sumcorners i
 
 
problem_28 = sumdiags 1001
 
 
</haskell>
 
</haskell>
 
You can note that from 1 to 3 there's (+2), and such too for 5, 7 and 9, it then goes up to (+4) 4 times, and so on, adding 2 to the number to add for each level of the spiral. You can so avoid all need for multiplications and just do additions with the following code :
 
<haskell>problem_28 = sum . scanl (+) 1 . concatMap (replicate 4) $ [2,4..1000]</haskell>
 
   
 
== [http://projecteuler.net/index.php?section=view&id=29 Problem 29] ==
 
== [http://projecteuler.net/index.php?section=view&id=29 Problem 29] ==
Line 137: Line 140:
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
problem_29 = length . group . sort $ [a^b | a <- [2..100], b <- [2..100]]
+
import Control.Monad
  +
problem_29 = length . group . sort $ liftM2 (^) [2..100] [2..100]
 
</haskell>
 
</haskell>
   
Line 145: Line 148:
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
  +
import Data.Array
 
import Data.Char
 
import Data.Char
limit = snd $ head $ dropWhile (\(a,b) -> a > b)
 
$ zip (map (9^5*) [1..]) (map (10^) [1..])
 
 
 
fifth n = foldr (\a b -> (toInteger(ord a) - 48)^5 + b) 0 $ show n
+
p = listArray (0,9) $ map (^5) [0..9]
  +
  +
upperLimit = 295277
 
 
problem_30 = sum $ filter (\n -> n == fifth n) [2..limit]
+
candidates =
  +
[ n |
  +
n <- [10..upperLimit],
  +
(sum $ digits n) `mod` 10 == last(digits n),
  +
powersum n == n
  +
]
  +
where
  +
digits n = map digitToInt $ show n
  +
powersum n = sum $ map (p!) $ digits n
  +
  +
problem_30 = sum candidates
 
</haskell>
 
</haskell>

Revision as of 03:46, 22 January 2008

Contents

1 Problem 21

Evaluate the sum of all amicable pairs under 10000.

Solution:

problem_21 = 
    sum [n |
    n <- [2..9999],
    let m = eulerTotient  n,
    m > 1,
    m < 10000,
    n ==  eulerTotient  m
    ]

2 Problem 22

What is the total of all the name scores in the file of first names?

Solution:

import Data.List
import Data.Char
problem_22 = do
    input <- readFile "names.txt"
    let names = sort $ read$"["++ input++"]"
    let scores = zipWith score names [1..]
    print $ show $ sum $ scores
    where
    score w i = (i *) $ sum $ map (\c -> ord c - ord 'A' + 1) w

3 Problem 23

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

Solution:

import Data.Array 
n = 28124
abundant n = eulerTotient n - n > n
abunds_array = listArray (1,n) $ map abundant [1..n]
abunds = filter (abunds_array !) [1..n]
 
rests x = map (x-) $ takeWhile (<= x `div` 2) abunds
isSum = any (abunds_array !) . rests
 
problem_23 = putStrLn $ show $ foldl1 (+) $ filter (not . isSum) [1..n]

4 Problem 24

What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?

Solution:

import Data.List 
 
fac 0 = 1
fac n = n * fac (n - 1)
perms [] _= []
perms xs n=
    x:( perms ( delete x $ xs ) (mod n m))
    where
    m=fac$(length(xs) -1)
    y=div n m
    x = xs!!y
 
problem_24 =  perms "0123456789"  999999

5 Problem 25

What is the first term in the Fibonacci sequence to contain 1000 digits?

Solution:

import Data.List
fib x
    |x==0=0
    |x==1=1
    |x==2=1
    |odd x=(fib (d+1))^2+(fib d)^2
    |otherwise=(fib (d+1))^2-(fib (d-1))^2
    where
    d=div x 2
 
phi=(1+sqrt 5)/2
dig x=floor( (fromInteger x-1) * log 10 /log phi)
problem_25 = 
    head[a|a<-[dig num..],(>=limit)$fib a]
    where
    num=1000
    limit=10^(num-1)

6 Problem 26

Find the value of d < 1000 for which 1/d contains the longest recurring cycle.

Solution:

next n d = (n `mod` d):next (10*n`mod`d) d
 
idigs n = tail $ take (1+n) $ next 1 n
 
pos x = map fst . filter ((==x) . snd) . zip [1..]
 
periods n = let d = idigs n in pos (head d) (tail d)
 
problem_26 = 
    snd$maximum [(m,a)|
    a<-[800..1000] ,
    let k=periods a,
    not$null k,
    let m=head k
    ]

7 Problem 27

Find a quadratic formula that produces the maximum number of primes for consecutive values of n.

Solution:

eulerCoefficients n 
  = [((len, a*b), (a, b)) 
      | b <- takeWhile (<n) primes, a <- [-b+1..n-1],
        let len = length $ takeWhile (isPrime . (\x -> x^2 + a*x + b)) [0..],
        if b == 2 then even a else odd a, len > 39]
 
problem_27 = snd . fst . maximum . eulerCoefficients $ 1000

8 Problem 28

What is the sum of both diagonals in a 1001 by 1001 spiral?

Solution:

problem_28 = sum (map (\n -> 4*(n-2)^2+10*(n-1)) [3,5..1001]) + 1

9 Problem 29

How many distinct terms are in the sequence generated by ab for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?

Solution:

import Control.Monad
problem_29 = length . group . sort $ liftM2 (^) [2..100] [2..100]

10 Problem 30

Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.

Solution:

import Data.Array
import Data.Char
 
p = listArray (0,9) $ map (^5) [0..9]
 
upperLimit = 295277
 
candidates = 
    [ n |
    n <- [10..upperLimit],
    (sum $ digits n) `mod` 10 == last(digits n),
    powersum n == n
    ]
    where 
    digits n = map digitToInt $ show n
    powersum n = sum $ map (p!) $ digits n
 
problem_30 = sum candidates