Difference between revisions of "Euler problems/101 to 110"

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  +
== [http://projecteuler.net/index.php?section=problems&id=101 Problem 101] ==
−
== How to Tie and Wear a Womans Scarf Many Fashionable Ways ==
  
  +
Investigate the optimum polynomial function to model the first k terms of a given sequence.
−
When it comes to accessorizing that special outfit, the scarf is the one item you want to have in your wardrobe. The scarf is the perfect accessory as it can add pizzaz to any outfit from the perfect little black dress to a business suit or your favorite T-shirt and jeans outfit. The scarf can add a sensational look making even the plainest outfit into a fabulous fashion hit.
  
   
  +
Solution:
−
Scarves Can be Worn Many Places on the Body Besides Your Neck
  
  +
<haskell>
  +
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
   
  +
</haskell>
−
There are numerous ways to wear a [http://www.hand-painted-silk-scarves.com '''silk scarf'''], making it a very versatile accessory. Wear it around your neck, tied, knotted or draped. Wear a scarf around head to keep your hair in place or around your waist as a belt. Wear a large scarf around your shoulders as a shawl.If you really feel like “tying one on” try tying a silk scarf around your hips. Just by tying a scarf around your hips, like a scarf belt, you can make your waist look smaller. You can achieve this look very easily by folding a long oblong scarf into the desired width, draping it around your hips and positioning it as a sash. Adjust it as you like. Let it hang down asymmetrically drawing the eye downward to produce a smaller looking waist
  
   
  +
== [http://projecteuler.net/index.php?section=problems&id=102 Problem 102] ==
−
How to Tie a Silk Scarf Around Your Neck
  
  +
For how many triangles in the text file does the interior contain the origin?
   
  +
Solution:
−
Of course, the traditional way to wear a silk scarf is around the neck. A fashion statement can be made by tying your scarf in creative ways. Try tying a long silk scarf can be knotted in front and hang straight down. Use a crystal brooch, ring or scarf pin that is all the rage now to hold your scarf in the front. The scarf clips are great to use. If you don't have a scarf pin or brooch, try wearing that long silk scarf in the the style by holding it across the front of our neck and then bringing one end around to fall done your right front side and then bring other end around your neck to fall down your left front side. This is a great way to wear those special hand painted silk scarves that are so light and airy now days. They are great for summer and spring. Another way to wear the long silk scarf is just drape it around your neck and shoulders and then let it hang down dramatically in front.
 
  +
<haskell>
  +
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
  +
</haskell>
   
  +
== [http://projecteuler.net/index.php?section=problems&id=103 Problem 103] ==
  +
Investigating sets with a special subset sum property.
   
  +
Solution:
−
These beautiful silk scarves are my hand painted art work, with franch ink and elegant Chinese 100% pure silk, for more pictures and information about them, pls check with my website:
  
  +
<haskell>
  +
six=[11,18,19,20,22,25]
  +
seven=[mid+a|let mid=six!!3,a<-0:six]
  +
problem_103=foldl (++) "" $map show seven
  +
</haskell>
   
  +
== [http://projecteuler.net/index.php?section=problems&id=104 Problem 104] ==
−
[http://www.hand-painted-silk-scarves.com www.hand-painted-silk-scarves.com]
  
  +
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
  +
  +
<haskell>
  +
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
  +
</haskell>
  +
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.
  +
  +
I have a slightly simpler solution, which I think is worth posting. It runs in about 6 seconds. HenryLaxen June 2, 2008
  +
  +
<haskell>
  +
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..])
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=105 Problem 105] ==
  +
Find the sum of the special sum sets in the file.
  +
  +
Solution:
  +
<haskell>
  +
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
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=106 Problem 106] ==
  +
Find the minimum number of comparisons needed to identify special sum sets.
  +
  +
Solution:
  +
<haskell>
  +
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
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=107 Problem 107] ==
  +
Determining the most efficient way to connect the network.
  +
  +
Solution:
  +
<haskell>
  +
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
  +
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 (\x y->compare (snd x) (snd y)) 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)
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=108 Problem 108] ==
  +
Solving the Diophantine equation 1/x + 1/y = 1/n.
  +
  +
Solution:
  +
<haskell>
  +
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
  +
</haskell>
  +
  +
== [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?
  +
  +
Solution:
  +
<haskell>
  +
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 = sum.map (zones!)
  +
problem_109 = length $ filter ((<100).score) checkouts
  +
</haskell>
  +
  +
== [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.
  +
  +
Solution:
  +
<haskell>
  +
-- prob in problem_108
  +
problem_110 = prob 13 (8*10^6)
  +
</haskell>

Revision as of 07:54, 17 August 2008

Problem 101

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

Solution: 

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?

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

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

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.

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.

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

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

Problem 107

Determining the most efficient way to connect the network.

Solution: 

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 
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 (\x y->compare (snd x) (snd y)) 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.

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

Problem 109

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

Solution: 

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 = sum.map (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.

Solution: 

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