Difference between revisions of "Euler problems/21 to 30"
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| + | Do them on your own! |
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| − | == [http://projecteuler.net/index.php?section=problems&id=21 Problem 21] == |
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| − | Evaluate the sum of all amicable pairs under 10000. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | problem_21 = |
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| − | sum [n | |
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| − | n <- [2..9999], |
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| − | let m = eulerTotient n, |
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| − | m > 1, |
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| − | m < 10000, |
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| − | n == eulerTotient m |
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| − | ] |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=22 Problem 22] == |
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| − | What is the total of all the name scores in the file of first names? |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Data.List |
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| − | import Data.Char |
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| − | problem_22 = do |
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| − | input <- readFile "names.txt" |
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| − | let names = sort $ read$"["++ input++"]" |
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| − | let scores = zipWith score names [1..] |
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| − | print $ show $ sum $ scores |
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| − | where |
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| − | score w i = (i *) $ sum $ map (\c -> ord c - ord 'A' + 1) w |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=23 Problem 23] == |
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| − | Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Data.Array |
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| − | n = 28124 |
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| − | abundant n = eulerTotient n - n > n |
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| − | abunds_array = listArray (1,n) $ map abundant [1..n] |
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| − | abunds = filter (abunds_array !) [1..n] |
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| − | |||
| − | rests x = map (x-) $ takeWhile (<= x `div` 2) abunds |
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| − | isSum = any (abunds_array !) . rests |
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| − | |||
| − | problem_23 = putStrLn $ show $ foldl1 (+) $ filter (not . isSum) [1..n] |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=24 Problem 24] == |
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| − | What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9? |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Data.List |
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| − | |||
| − | fac 0 = 1 |
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| − | fac n = n * fac (n - 1) |
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| − | perms [] _= [] |
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| − | perms xs n= |
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| − | x:( perms ( delete x $ xs ) (mod n m)) |
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| − | where |
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| − | m=fac$(length(xs) -1) |
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| − | y=div n m |
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| − | x = xs!!y |
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| − | |||
| − | problem_24 = perms "0123456789" 999999 |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=25 Problem 25] == |
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| − | What is the first term in the Fibonacci sequence to contain 1000 digits? |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Data.List |
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| − | fib x |
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| − | |x==0=0 |
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| − | |x==1=1 |
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| − | |x==2=1 |
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| − | |odd x=(fib (d+1))^2+(fib d)^2 |
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| − | |otherwise=(fib (d+1))^2-(fib (d-1))^2 |
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| − | where |
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| − | d=div x 2 |
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| − | |||
| − | phi=(1+sqrt 5)/2 |
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| − | dig x=floor( (fromInteger x-1) * log 10 /log phi) |
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| − | problem_25 = |
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| − | head[a|a<-[dig num..],(>=limit)$fib a] |
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| − | where |
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| − | num=1000 |
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| − | limit=10^(num-1) |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=26 Problem 26] == |
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| − | Find the value of d < 1000 for which 1/d contains the longest recurring cycle. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | next n d = (n `mod` d):next (10*n`mod`d) d |
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| − | |||
| − | idigs n = tail $ take (1+n) $ next 1 n |
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| − | |||
| − | pos x = map fst . filter ((==x) . snd) . zip [1..] |
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| − | |||
| − | periods n = let d = idigs n in pos (head d) (tail d) |
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| − | |||
| − | problem_26 = |
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| − | snd$maximum [(m,a)| |
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| − | a<-[800..1000] , |
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| − | let k=periods a, |
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| − | not$null k, |
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| − | let m=head k |
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| − | ] |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=27 Problem 27] == |
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| − | Find a quadratic formula that produces the maximum number of primes for consecutive values of n. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | eulerCoefficients n |
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| − | = [((len, a*b), (a, b)) |
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| − | | b <- takeWhile (<n) primes, a <- [-b+1..n-1], |
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| − | let len = length $ takeWhile (isPrime . (\x -> x^2 + a*x + b)) [0..], |
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| − | if b == 2 then even a else odd a, len > 39] |
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| − | |||
| − | problem_27 = snd . fst . maximum . eulerCoefficients $ 1000 |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=28 Problem 28] == |
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| − | What is the sum of both diagonals in a 1001 by 1001 spiral? |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | problem_28 = sum (map (\n -> 4*(n-2)^2+10*(n-1)) [3,5..1001]) + 1 |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=29 Problem 29] == |
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| − | How many distinct terms are in the sequence generated by a<sup>b</sup> for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100? |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Control.Monad |
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| − | problem_29 = length . group . sort $ liftM2 (^) [2..100] [2..100] |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=problems&id=30 Problem 30] == |
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| − | Find the sum of all the numbers that can be written as the sum of fifth powers of their digits. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Data.Array |
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| − | import Data.Char |
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| − | |||
| − | p = listArray (0,9) $ map (^5) [0..9] |
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| − | |||
| − | upperLimit = 295277 |
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| − | |||
| − | candidates = |
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| − | [ n | |
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| − | n <- [10..upperLimit], |
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| − | (sum $ digits n) `mod` 10 == last(digits n), |
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| − | powersum n == n |
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| − | ] |
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| − | where |
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| − | digits n = map digitToInt $ show n |
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| − | powersum n = sum $ map (p!) $ digits n |
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| − | |||
| − | problem_30 = sum candidates |
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| − | </haskell> |
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Revision as of 21:42, 29 January 2008
Do them on your own!