Euler problems/101 to 110

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Problem 101

Investigate the optimum polynomial function to model the first k terms of a given sequence.


import Data.List
f s n = sum $ zipWith (*) (iterate (*n) 1) s
fits t = sum $ map (p101 . map (f t)) $ inits [1..toInteger $ length t - 1]
problem_101 = fits (1 : (concat $ replicate 5 [-1,1]))
diff s = zipWith (-) (drop 1 s) s
p101 = sum . map last . takeWhile (not . null) . iterate diff

Problem 102

For how many triangles in the text file does the interior contain the origin?


import Text.Regex 
--ghc -M p102.hs
isOrig (x1:y1:x2:y2:x3:y3:[])=
    t1*t2>=0 && t3*t4>=0 && t5*t6>=0
buildTriangle s = map read (splitRegex (mkRegex ",") s) :: [Integer] 
    x<-readFile "triangles.txt"
    let y=map buildTriangle$lines x
    print $length$ filter isOrig y

Problem 103

Investigating sets with a special subset sum property.


seven=[mid+a|let mid=six!!3,a<-0:six]
problem_103=concatMap show seven

Problem 104

Finding Fibonacci numbers for which the first and last nine digits are pandigital.


Very nice problem. I didnt realize you could deal with the precision problem. Therefore I used this identity to speed up the fibonacci calculation: f_(2*n+k) = f_k*(f_(n+1))^2 + 2*f_(k-1)*f_(n+1)*f_n + f_(k-2)*(f_n)^2

import Data.List
import Data.Char
fibos = rec 0 1
        rec a b = a:rec b (a+b)
fibo_2nk n k = 
        fkm1 = fibo (k-1)
        fkm2 = fibo (k-2)
        fk = fkm1 + fkm2
        fnp1 = fibo (n+1)
        fnp1sq = fnp1^2
        fn = fibo n
        fnsq = fn^2
        fk*fnp1sq + 2*fkm1*fnp1*fn + fkm2*fnsq
fibo x = 
        threshold = 30000
        n = div x 3
        k = n+mod x 3
        if x < threshold 
        then fibos !! x
        else fibo_2nk n k
findCandidates = rec 0 1 0
        m = 10^9
        rec a b n  =
                continue = rec b (mod (a+b) m) (n+1)
                isBackPan a = (sort $ show a) == "123456789"
                if isBackPan a 
                then n:continue
                else continue
search = 
        isFrontPan x = (sort $ take 9 $ show x) == "123456789"
        map fst
            $ take 1
            $ dropWhile (not.snd)            
            $ zip findCandidates
            $ map (isFrontPan.fibo) findCandidates
problem_104 = search

It took 8 sec on a 2.2Ghz machine.

The lesson I learned fom this challenge, is: know mathematical identities and exploit them. They allow you take short cuts. 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 fibonacci number which is over 2mio characters long, took 20sec to compute on my 2.2ghz laptop.

I have a slightly simpler solution, which I think is worth posting. It runs in about 6 seconds. HenryLaxen June 2, 2008

fibs = 1 : 1 : zipWith (+) fibs (tail fibs)

isFibPan n =
  let a = n `mod` 1000000000
      b = sort (show a)
      c = sort $ take 9 $ show n
  in  b == "123456789" && c == "123456789"

ex_104 = snd $ head $ dropWhile (\(x,y) -> (not . isFibPan) x) (zip fibs [1..])

Problem 105

Find the sum of the special sum sets in the file.


import Data.List
import Control.Monad
solNum=map solve [7..12] 
solve n =  twoSetsOf [0..n-1] =<< [2..div n 2]          
twoSetsOf xs n = do
        firstSet <- setsOf n xs
        let rest = dropWhile (/= head firstSet) xs \\ firstSet
        secondSet <- setsOf n rest
        let f = firstSet  >>= enumFromTo 1
            s = secondSet >>= enumFromTo 1
        guard $ not $ null (f \\ s) || null (s \\ f)
        return (firstSet,secondSet)

setsOf 0 _ = [[]]
setsOf (n+1) xs = concat [map (y:) (setsOf n ys) | (y:ys) <- tails xs]
comp lst a b=
    a1=sum$map (lst!!) a
    b1=sum$map (lst!!) b
notEqu lst =
    and [comp slst a b|(a,b)<-solNum!!s]
    s=length lst-7
    slst=sort lst
moreElem lst =
    and maE
    le=length lst
    sortLst=sort lst
    maxElem = 
        (-1):[sum $drop (le-a) sortLst|
    minElem = 
        [sum $take a sortLst|
    maE=zipWith (<) maxElem minElem
stoInt s=read "["++s++"]" :: [Integer]  
check x=moreElem x && notEqu x
main = do
    f <- readFile "sets.txt"
    let sets = map stoInt$ lines f
    let ssets = filter check sets
    print $ sum $ concat ssets

Problem 106

Find the minimum number of comparisons needed to identify special sum sets.


binomial x y =(prodxy (y+1) x) `div` (prodxy 1 (x-y))
prodxy x y=product[x..y]
catalan n=(`div` (n+1)) $binomial (2*n) n
calc n=
    let mu2=a*2,
    let c=(`div` 2) $ binomial mu2 a,
    let d=catalan a,
    let e=binomial n mu2]
    di2=n `div` 2
problem_106 = calc 12

Problem 107

Determining the most efficient way to connect the network.


import Control.Monad.ST
import Control.Monad
import Data.Array.MArray
import Data.Array.ST
import Data.List
import Data.Map (fromList,(!))
import Text.Regex 
import Data.Ord (comparing)
makeArr x=map zero (splitRegex (mkRegex ",") x)  
makeNet x lst y=[((a,b),m)|a<-[0..x-1],b<-[0..a-1],let m=lst!!a!!b,m/=y]
zero x
    |'-' `elem` x=0
    |otherwise=read x::Int
problem_107 =do
    a<-readFile "network.txt"
    let b=map makeArr $lines a
        network = makeNet 40 b 0
        edges = sortBy (comparing snd) network 
        eedges =map fst edges
        mape=fromList edges
        d=sum $ map snd edges 
        e=sum$map (mape!)$kruskal eedges
    print (d-e)
kruskal es = runST ( do
    let hi = maximum $ map (uncurry max) es
        lo = minimum $ map (uncurry min) es
    djs <- makeDjs (lo,hi)
    filterM (kruskalST djs) es)
kruskalST djs (u,v) = do
    disjoint <- djsDisjoint u v djs
    when disjoint $ djsUnion u v djs
    return disjoint
type DisjointSet s = STArray s Int (Maybe Int)
makeDjs :: (Int,Int) -> ST s (DisjointSet s)
makeDjs b = newArray b Nothing

djsUnion a b djs = do
    root <- djsFind a djs
    writeArray djs root $ Just b

djsFind a djs = maybe (return a) f =<< readArray djs a
 where f p = do p' <- djsFind p djs
                writeArray djs a (Just p')
                return p'
djsDisjoint  a b uf = liftM2 (/=) (djsFind a uf) (djsFind b uf)

Problem 108

Solving the Diophantine equation 1/x + 1/y = 1/n.


import List
series _ 1 =[[0]]
series xs n =[x:ps|x<-xs,ps<-series [0..x] (n-1) ]
distinct=product. map (+1)
sumpri x=product $zipWith (^) primes x
prob x y =minimum[(sumpri m ,m)|m<-series [1..3] x,(>y)$distinct$map (*2) m]
problem_108=prob 7 2000

Problem 109

How many distinct ways can a player checkout in the game of darts with a score of less than 100?


import Data.Array
wedges = [1..20]
zones = listArray (0,62) $ 0:25:50:wedges++map (2*) wedges++map (3*) wedges
checkouts = 
    [[a,b,c] |
    a <- 2:[23..42],
    b <- [0..62],
    c <- [b..62]
score = (zones!)    
problem_109 = length $ filter ((<100).score) checkouts

Problem 110

Find an efficient algorithm to analyse the number of solutions of the equation 1/x + 1/y = 1/n.


-- prob in problem_108
problem_110 = prob 13 (8*10^6)