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== [http://projecteuler.net/index.php?section=view&id=101 Problem 101] ==
+
== [http://projecteuler.net/index.php?section=problems&id=101 Problem 101] ==
 
Investigate the optimum polynomial function to model the first k terms of a given sequence.
 
Investigate the optimum polynomial function to model the first k terms of a given sequence.
   
Line 7: Line 7:
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=102 Problem 102] ==
+
== [http://projecteuler.net/index.php?section=problems&id=102 Problem 102] ==
 
For how many triangles in the text file does the interior contain the origin?
 
For how many triangles in the text file does the interior contain the origin?
   
Line 32: Line 32:
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=103 Problem 103] ==
+
== [http://projecteuler.net/index.php?section=problems&id=103 Problem 103] ==
 
Investigating sets with a special subset sum property.
 
Investigating sets with a special subset sum property.
   
Line 42: Line 42:
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=104 Problem 104] ==
+
== [http://projecteuler.net/index.php?section=problems&id=104 Problem 104] ==
 
Finding Fibonacci numbers for which the first and last nine digits are pandigital.
 
Finding Fibonacci numbers for which the first and last nine digits are pandigital.
   
Line 111: Line 111:
 
The lesson I learned fom this challenge, is: know mathematical identities and exploit them. They allow you take short cuts.
 
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 10mio.th fibonacci number which is over 2mio characters long, took 20sec to compute on my 2.2ghz laptop.
 
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.
== [http://projecteuler.net/index.php?section=view&id=105 Problem 105] ==
+
== [http://projecteuler.net/index.php?section=problems&id=105 Problem 105] ==
 
Find the sum of the special sum sets in the file.
 
Find the sum of the special sum sets in the file.
   
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
import List
+
import Data.List
  +
import Control.Monad
 
import Text.Regex
 
import Text.Regex
--ghc -M p105.hs
+
digitstwo n k
+
solNum=map solve [7..12]
|k==0=[]
+
solve n = twoSetsOf [0..n-1] =<< [2..div n 2]
|otherwise= y:digitstwo x (k-1)
+
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)
  +
[(firstSet,secondSet)]
  +
  +
setsOf 0 _ = [[]]
  +
setsOf (n+1) xs = concat [map (y:) (setsOf n ys) | (y:ys) <- tails xs]
  +
comp lst a b=
  +
a1/=b1
 
where
 
where
(x,y)=divMod n 2
+
a1=sum$map (lst!!) a
subset lst=
+
b1=sum$map (lst!!) b
[sl|
+
notEqu lst =
n<-[0..2^len-1],
+
all id[comp slst a b|(a,b)<-solNum!!s]
let a=digitstwo n len,
 
sum a<=mlen,
 
let t=zip lst a,
 
let sl=delzero t
 
]
 
 
where
 
where
len=length lst
+
s=length lst-7
mlen= div len 2
+
slst=sort lst
delzero x=[a|(a,b)<-x,b/=0]
 
notEqu lst su=
 
length sm ==(length.nub) sm
 
where
 
sm=[sum a|a<-su]
 
 
moreElem lst =
 
moreElem lst =
foldl (&&) True maE
+
all id maE
 
where
 
where
 
le=length lst
 
le=length lst
Line 144: Line 144:
 
]
 
]
 
maE=[a<b|(a,b)<-zip maxElem minElem]
 
maE=[a<b|(a,b)<-zip maxElem minElem]
buildTriangle s = map read (splitRegex (mkRegex ",") s) :: [Integer]
+
stoInt s=map read (splitRegex (mkRegex ",") s) :: [Integer]
problem_105=do
+
check x=moreElem x && notEqu x
x<-readFile "sets.txt"
+
main = do
let sm=
+
f <- readFile "sets.txt"
[a|
+
let sets = map stoInt$ lines f
a<-map buildTriangle $ lines x,
+
let ssets = filter check sets
moreElem a ,
+
print $ sum $ concat ssets
notEqu a $subset a
 
]
 
print $sum$concat sm
 
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=106 Problem 106] ==
+
== [http://projecteuler.net/index.php?section=problems&id=106 Problem 106] ==
 
Find the minimum number of comparisons needed to identify special sum sets.
 
Find the minimum number of comparisons needed to identify special sum sets.
   
Line 174: Line 174:
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=107 Problem 107] ==
+
== [http://projecteuler.net/index.php?section=problems&id=107 Problem 107] ==
 
Determining the most efficient way to connect the network.
 
Determining the most efficient way to connect the network.
   
Line 182: Line 182:
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=108 Problem 108] ==
+
== [http://projecteuler.net/index.php?section=problems&id=108 Problem 108] ==
 
Solving the Diophantine equation 1/x + 1/y = 1/n.
 
Solving the Diophantine equation 1/x + 1/y = 1/n.
   
Line 197: Line 197:
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=109 Problem 109] ==
+
== [http://projecteuler.net/index.php?section=problems&id=109 Problem 109] ==
 
How many distinct ways can a player checkout in the game of darts with a score of less than 100?
 
How many distinct ways can a player checkout in the game of darts with a score of less than 100?
   
Line 205: Line 205:
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=110 Problem 110] ==
+
== [http://projecteuler.net/index.php?section=problems&id=110 Problem 110] ==
 
Find an efficient algorithm to analyse the number of solutions of the equation 1/x + 1/y = 1/n.
 
Find an efficient algorithm to analyse the number of solutions of the equation 1/x + 1/y = 1/n.
   

Revision as of 13:37, 26 January 2008

Contents

1 Problem 101

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

Solution:

problem_101 = undefined

2 Problem 102

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

Solution:

import Text.Regex 
--ghc -M p102.hs
isOrig (x1:y1:x2:y2:x3:y3:[])=
    t1*t2>=0 && t3*t4>=0 && t5*t6>=0
    where
    x4=0
    y4=0
    t1=(y2-y1)*(x4-x1)+(x1-x2)*(y4-y1)
    t2=(y2-y1)*(x3-x1)+(x1-x2)*(y3-y1)
    t3=(y3-y1)*(x4-x1)+(x1-x3)*(y4-y1)
    t4=(y3-y1)*(x2-x1)+(x1-x3)*(y2-y1)
    t5=(y3-y2)*(x4-x2)+(x2-x3)*(y4-y2)
    t6=(y3-y2)*(x1-x2)+(x2-x3)*(y1-y2)
buildTriangle s = map read (splitRegex (mkRegex ",") s) :: [Integer] 
problem_102=do
    x<-readFile "triangles.txt"
    let y=map buildTriangle$lines x
    print $length$ filter isOrig y

3 Problem 103

Investigating sets with a special subset sum property.

Solution:

six=[11,18,19,20,22,25]
seven=[mid+a|let mid=six!!3,a<-0:six]
problem_103=foldl (++) "" $map show seven

4 Problem 104

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

Solution:

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
    where
        rec a b = a:rec b (a+b)
 
fibo_2nk n k = 
    let        
        fkm1 = fibo (k-1)
        fkm2 = fibo (k-2)
        fk = fkm1 + fkm2
        fnp1 = fibo (n+1)
        fnp1sq = fnp1^2
        fn = fibo n
        fnsq = fn^2
    in
        fk*fnp1sq + 2*fkm1*fnp1*fn + fkm2*fnsq
 
fibo x = 
    let
        threshold = 30000
        n = div x 3
        k = n+mod x 3
    in
        if x < threshold 
        then fibos !! x
        else fibo_2nk n k
 
findCandidates = rec 0 1 0
    where
        m = 10^9
        rec a b n  =
            let
                continue = rec b (mod (a+b) m) (n+1)
                isBackPan a = (sort $ show a) == "123456789"
            in
                if isBackPan a 
                then n:continue
                else continue
search = 
    let
        isFrontPan x = (sort $ take 9 $ show x) == "123456789"
    in
        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 10mio.th fibonacci number which is over 2mio characters long, took 20sec to compute on my 2.2ghz laptop.

5 Problem 105

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

Solution:

import Data.List
import Control.Monad
import Text.Regex 
 
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)
        [(firstSet,secondSet)] 
 
setsOf 0 _ = [[]]
setsOf (n+1) xs = concat [map (y:) (setsOf n ys) | (y:ys) <- tails xs]
comp lst a b=
    a1/=b1
    where
    a1=sum$map (lst!!) a
    b1=sum$map (lst!!) b
notEqu lst =
    all id[comp slst a b|(a,b)<-solNum!!s]
    where
    s=length lst-7
    slst=sort lst
moreElem lst =
    all id maE
    where
    le=length lst
    sortLst=sort lst
    maxElem = 
        (-1):[sum $drop (le-a) sortLst|
        a<-[0..le]
        ]
    minElem = 
        [sum $take a sortLst|
        a<-[0..le]
        ]
    maE=[a<b|(a,b)<-zip maxElem minElem]
stoInt s=map read (splitRegex (mkRegex ",") s) :: [Integer]  
check x=moreElem x && notEqu x
main = do
    f <- readFile "sets.txt"
    let sets = map stoInt$ lines f
    let ssets = filter check sets
    print $ sum $ concat ssets

6 Problem 106

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

Solution:

binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y))
prodxy x y=product[x..y]
-- http://mathworld.wolfram.com/DyckPath.html
catalan n=flip div (n+1) $binomial (2*n) n
calc n=
    sum[e*(c-d)|
    a<-[1..di2],
    let mu2=a*2,
    let c=flip div 2 $ binomial mu2 a,
    let d=catalan a,
    let e=binomial n mu2]
    where
    di2=div n 2
problem_106 = calc 12

7 Problem 107

Determining the most efficient way to connect the network.

Solution:

problem_107 = undefined

8 Problem 108

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

Solution:

import List
primes=[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73]
series _ 1 =[[0]]
series xs n =[x:ps|x<-xs,ps<-series [0..x] (n-1) ]
distinct=product. map (+1)
sumpri x=product $map (\(x,y)->x^y)$zip  primes x
prob x y =head$sort[(sumpri m ,m)|m<-series [1..3] x,(>y)$distinct$map (*2) m]
problem_108=prob 7 2000

9 Problem 109

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

Solution:

problem_109 = undefined

10 Problem 110

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

Solution:

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