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| − | == [http://projecteuler.net/index.php?section=problems&id=101 Problem 101] ==
| + | Do them on your own! |
| − | Investigate the optimum polynomial function to model the first k terms of a given sequence.
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| − | | |
| − | Solution:
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| − | <haskell>
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| − | problem_101 = undefined
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| − | </haskell>
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| − | | |
| − | == [http://projecteuler.net/index.php?section=problems&id=102 Problem 102] ==
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| − | For how many triangles in the text file does the interior contain the origin?
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| − | | |
| − | Solution:
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| − | <haskell>
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| − | import Text.Regex
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| − | --ghc -M p102.hs
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| − | isOrig (x1:y1:x2:y2:x3:y3:[])=
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| − | t1*t2>=0 && t3*t4>=0 && t5*t6>=0
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| − | where
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| − | x4=0
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| − | y4=0
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| − | t1=(y2-y1)*(x4-x1)+(x1-x2)*(y4-y1)
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| − | t2=(y2-y1)*(x3-x1)+(x1-x2)*(y3-y1)
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| − | t3=(y3-y1)*(x4-x1)+(x1-x3)*(y4-y1)
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| − | t4=(y3-y1)*(x2-x1)+(x1-x3)*(y2-y1)
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| − | t5=(y3-y2)*(x4-x2)+(x2-x3)*(y4-y2)
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| − | t6=(y3-y2)*(x1-x2)+(x2-x3)*(y1-y2)
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| − | buildTriangle s = map read (splitRegex (mkRegex ",") s) :: [Integer]
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| − | problem_102=do
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| − | x<-readFile "triangles.txt"
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| − | let y=map buildTriangle$lines x
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| − | print $length$ filter isOrig y
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| − | </haskell>
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| − | | |
| − | == [http://projecteuler.net/index.php?section=problems&id=103 Problem 103] ==
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| − | Investigating sets with a special subset sum property.
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| − | | |
| − | Solution:
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| − | <haskell>
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| − | six=[11,18,19,20,22,25]
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| − | seven=[mid+a|let mid=six!!3,a<-0:six]
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| − | problem_103=foldl (++) "" $map show seven
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| − | </haskell>
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| − | | |
| − | == [http://projecteuler.net/index.php?section=problems&id=104 Problem 104] ==
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| − | Finding Fibonacci numbers for which the first and last nine digits are pandigital.
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| − | | |
| − | Solution:
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| − | | |
| − | Very nice problem. I didnt realize you could deal with the precision problem.
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| − | Therefore I used this identity to speed up the fibonacci calculation:
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| − | f_(2*n+k)
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| − | = f_k*(f_(n+1))^2
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| − | + 2*f_(k-1)*f_(n+1)*f_n
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| − | + f_(k-2)*(f_n)^2
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| − | | |
| − | <haskell>
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| − | import Data.List
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| − | import Data.Char
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| − |
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| − | fibos = rec 0 1
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| − | where
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| − | rec a b = a:rec b (a+b)
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| − |
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| − | fibo_2nk n k =
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| − | let
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| − | fkm1 = fibo (k-1)
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| − | fkm2 = fibo (k-2)
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| − | fk = fkm1 + fkm2
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| − | fnp1 = fibo (n+1)
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| − | fnp1sq = fnp1^2
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| − | fn = fibo n
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| − | fnsq = fn^2
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| − | in
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| − | fk*fnp1sq + 2*fkm1*fnp1*fn + fkm2*fnsq
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| − |
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| − | fibo x =
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| − | let
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| − | threshold = 30000
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| − | n = div x 3
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| − | k = n+mod x 3
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| − | in
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| − | if x < threshold
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| − | then fibos !! x
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| − | else fibo_2nk n k
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| − |
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| − | findCandidates = rec 0 1 0
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| − | where
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| − | m = 10^9
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| − | rec a b n =
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| − | let
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| − | continue = rec b (mod (a+b) m) (n+1)
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| − | isBackPan a = (sort $ show a) == "123456789"
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| − | in
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| − | if isBackPan a
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| − | then n:continue
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| − | else continue
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| − | search =
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| − | let
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| − | isFrontPan x = (sort $ take 9 $ show x) == "123456789"
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| − | in
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| − | map fst
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| − | $ take 1
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| − | $ dropWhile (not.snd)
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| − | $ zip findCandidates
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| − | $ map (isFrontPan.fibo) findCandidates
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| − |
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| − | problem_104 = search
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| − | </haskell>
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| − | It took 8 sec on a 2.2Ghz machine.
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| − | | |
| − | The lesson I learned fom this challenge, is: know mathematical identities and exploit them. They allow you take short cuts.
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| − | Normally you compute all previous fibonacci numbers to compute a random fibonacci number. Which has linear costs. The aforementioned identity builds the number not from its two predecessors but from 4 much smaller ones. This makes the algorithm logarithmic in its complexity. It really shines if you want to compute a random very large fibonacci number. f.i. the 10mio.th fibonacci number which is over 2mio characters long, took 20sec to compute on my 2.2ghz laptop.
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| − | == [http://projecteuler.net/index.php?section=problems&id=105 Problem 105] ==
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| − | Find the sum of the special sum sets in the file.
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| − | | |
| − | Solution:
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| − | <haskell>
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| − | import Data.List
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| − | import Control.Monad
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| − | import Text.Regex
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| − |
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| − | solNum=map solve [7..12]
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| − | solve n = twoSetsOf [0..n-1] =<< [2..div n 2]
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| − | twoSetsOf xs n = do
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| − | firstSet <- setsOf n xs
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| − | let rest = dropWhile (/= head firstSet) xs \\ firstSet
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| − | secondSet <- setsOf n rest
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| − | let f = firstSet >>= enumFromTo 1
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| − | s = secondSet >>= enumFromTo 1
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| − | guard $ not $ null (f \\ s) || null (s \\ f)
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| − | [(firstSet,secondSet)]
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| − | | |
| − | setsOf 0 _ = [[]]
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| − | setsOf (n+1) xs = concat [map (y:) (setsOf n ys) | (y:ys) <- tails xs]
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| − | comp lst a b=
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| − | a1/=b1
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| − | where
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| − | a1=sum$map (lst!!) a
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| − | b1=sum$map (lst!!) b
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| − | notEqu lst =
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| − | all id[comp slst a b|(a,b)<-solNum!!s]
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| − | where
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| − | s=length lst-7
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| − | slst=sort lst
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| − | moreElem lst =
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| − | all id maE
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| − | where
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| − | le=length lst
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| − | sortLst=sort lst
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| − | maxElem =
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| − | (-1):[sum $drop (le-a) sortLst|
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| − | a<-[0..le]
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| − | ]
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| − | minElem =
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| − | [sum $take a sortLst|
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| − | a<-[0..le]
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| − | ]
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| − | maE=[a<b|(a,b)<-zip maxElem minElem]
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| − | stoInt s=map read (splitRegex (mkRegex ",") s) :: [Integer]
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| − | check x=moreElem x && notEqu x
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| − | main = do
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| − | f <- readFile "sets.txt"
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| − | let sets = map stoInt$ lines f
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| − | let ssets = filter check sets
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| − | print $ sum $ concat ssets
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| − | </haskell>
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| − | | |
| − | == [http://projecteuler.net/index.php?section=problems&id=106 Problem 106] ==
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| − | Find the minimum number of comparisons needed to identify special sum sets.
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| − | | |
| − | Solution:
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| − | <haskell>
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| − | binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y))
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| − | prodxy x y=product[x..y]
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| − | -- http://mathworld.wolfram.com/DyckPath.html
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| − | catalan n=flip div (n+1) $binomial (2*n) n
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| − | calc n=
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| − | sum[e*(c-d)|
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| − | a<-[1..di2],
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| − | let mu2=a*2,
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| − | let c=flip div 2 $ binomial mu2 a,
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| − | let d=catalan a,
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| − | let e=binomial n mu2]
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| − | where
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| − | di2=div n 2
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| − | problem_106 = calc 12
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| − | </haskell>
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| − | | |
| − | == [http://projecteuler.net/index.php?section=problems&id=107 Problem 107] ==
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| − | Determining the most efficient way to connect the network.
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| − | | |
| − | Solution:
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| − | <haskell>
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| − | import Control.Monad.ST
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| − | import Control.Monad
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| − | import Data.Array.MArray
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| − | import Data.Array.ST
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| − | import Data.List
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| − | import Data.Map (fromList,(!))
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| − | import Text.Regex
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| − | makeArr x=map zero (splitRegex (mkRegex ",") x)
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| − | makeNet x lst y=[((a,b),m)|a<-[0..x-1],b<-[0..a-1],let m=lst!!a!!b,m/=y]
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| − | zero x
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| − | |'-' `elem` x=0
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| − | |otherwise=read x::Int
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| − | problem_107 =do
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| − | a<-readFile "network.txt"
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| − | let b=map makeArr $lines a
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| − | network = makeNet 40 b 0
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| − | edges = sortBy (\x y->compare (snd x) (snd y)) network
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| − | eedges =map fst edges
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| − | mape=fromList edges
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| − | d=sum $ map snd edges
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| − | e=sum$map (mape!)$kruskal eedges
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| − | print (d-e)
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| − | kruskal es = runST ( do
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| − | let hi = maximum $ map (uncurry max) es
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| − | lo = minimum $ map (uncurry min) es
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| − | djs <- makeDjs (lo,hi)
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| − | filterM (kruskalST djs) es)
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| − |
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| − | kruskalST djs (u,v) = do
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| − | disjoint <- djsDisjoint u v djs
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| − | when disjoint $ djsUnion u v djs
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| − | return disjoint
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| − |
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| − | type DisjointSet s = STArray s Int (Maybe Int)
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| − |
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| − | makeDjs :: (Int,Int) -> ST s (DisjointSet s)
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| − | makeDjs b = newArray b Nothing
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| − | | |
| − | djsUnion a b djs = do
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| − | root <- djsFind a djs
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| − | writeArray djs root $ Just b
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| − | | |
| − | djsFind a djs = maybe (return a) f =<< readArray djs a
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| − | where f p = do p' <- djsFind p djs
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| − | writeArray djs a (Just p')
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| − | return p'
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| − |
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| − | djsDisjoint a b uf = liftM2 (/=) (djsFind a uf) (djsFind b uf)
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| − | </haskell>
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| − | | |
| − | == [http://projecteuler.net/index.php?section=problems&id=108 Problem 108] ==
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| − | Solving the Diophantine equation 1/x + 1/y = 1/n.
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| − | Solution:
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| − | <haskell>
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| − | import List
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| − | primes=[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73]
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| − | series _ 1 =[[0]]
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| − | series xs n =[x:ps|x<-xs,ps<-series [0..x] (n-1) ]
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| − | distinct=product. map (+1)
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| − | sumpri x=product $map (\(x,y)->x^y)$zip primes x
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| − | prob x y =head$sort[(sumpri m ,m)|m<-series [1..3] x,(>y)$distinct$map (*2) m]
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| − | problem_108=prob 7 2000
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| − | </haskell>
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| − | | |
| − | == [http://projecteuler.net/index.php?section=problems&id=109 Problem 109] ==
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| − | How many distinct ways can a player checkout in the game of darts with a score of less than 100?
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| − | Solution:
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| − | <haskell>
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| − | problem_109 = undefined
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| − | </haskell>
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| − | | |
| − | == [http://projecteuler.net/index.php?section=problems&id=110 Problem 110] ==
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| − | Find an efficient algorithm to analyse the number of solutions of the equation 1/x + 1/y = 1/n.
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| − | Solution:
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| − | <haskell>
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| − | -- prob in problem_108
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| − | problem_110 = prob 13 (8*10^6)
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| − | </haskell>
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