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Euler problems/141 to 150

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Line 209: Line 209:
Solution:
Solution:
<haskell>
<haskell>
-
problem_144 = undefined
+
type Point = (Double, Double)
 +
type Vector = (Double, Double)
 +
type Normal = (Double, Double)
 +
 +
sub :: Vector -> Vector -> Vector
 +
sub (x,y) (a,b) = (x-a, y-b)
 +
 +
mull :: Double -> Vector -> Vector
 +
mull s (x,y) = (s*x, s*y)
 +
 +
mulr :: Vector -> Double -> Vector
 +
mulr v s = mull s v
 +
 +
dot :: Vector -> Vector -> Double
 +
dot (x,y) (a,b) = x*a + y*b
 +
 +
normSq :: Vector -> Double
 +
normSq v = dot v v
 +
 +
normalize :: Vector -> Vector
 +
normalize v
 +
|len /= 0 =mulr v (1.0/len)
 +
|otherwise=error "Vettore nullo.\n"
 +
where
 +
len = (sqrt . normSq) v
 +
 +
proj :: Vector -> Vector -> Vector
 +
proj a b = mull ((dot a b)/normSq b) b
 +
 
 +
reflect :: Vector -> Normal -> Vector
 +
reflect i n = sub i $ mulr (proj i n) 2.0
 +
 +
type Ray = (Point, Vector)
 +
 +
makeRay :: Point -> Vector -> Ray
 +
makeRay p v = (p, v)
 +
 +
getPoint :: Ray -> Double -> Point
 +
getPoint ((px,py),(vx,vy)) t = (px + t*vx, py + t*vy)
 +
 +
type Ellipse = (Double, Double)
 +
 +
getNormal :: Ellipse -> Point -> Normal
 +
getNormal (a,b) (x,y) = ((-b/a)*x, (-a/b)*y)
 +
 
 +
rayFromPoint :: Ellipse -> Vector -> Point -> Ray
 +
rayFromPoint e v p = makeRay p (reflect v (getNormal e p))
 +
 +
test :: Point -> Bool
 +
test (x,y) = y > 0 && x >= -0.01 && x <= 0.01
 +
 +
intersect :: Ellipse -> Ray -> Point
 +
intersect (e@(a,b)) (r@((px,py),(vx,vy))) =
 +
getPoint r t1
 +
where
 +
c0 = normSq (vx/a, vy/b)
 +
c1 = 2.0 * dot (vx/a, vy/b) (px/a, py/b)
 +
c2 = (normSq (px/a, py/b)) - 1.0
 +
(t0, t1) = quadratic c0 c1 c2
 +
 +
quadratic :: Double -> Double -> Double -> (Double, Double)
 +
quadratic a b c
 +
|d < 0= error "Discriminante minore di zero"
 +
|otherwise= if (t0 < t1) then (t0, t1) else (t1, t0)
 +
where
 +
d = b * b - 4.0 * a * c
 +
sqrtD = sqrt d
 +
q = if b < 0 then -0.5*(b - sqrtD) else 0.5*(b + sqrtD)
 +
t0 = q / a
 +
t1 = c / q
 +
 +
calculate :: Ellipse -> Ray -> Int -> IO ()
 +
calculate e (r@(o,d)) n
 +
|test p=print n
 +
|otherwise=do
 +
putStrLn $ "\rHit " ++ show n
 +
calculate e (rayFromPoint e d p) (n+1)
 +
where
 +
p = intersect e r
 +
 +
origin = (0.0,10.1)
 +
direction = sub (1.4,-9.6) origin
 +
ellipse = (5.0,10.0)
 +
 +
problem_144 = do
 +
calculate ellipse (makeRay origin direction) 0
</haskell>
</haskell>
Line 313: Line 398:
Solution:
Solution:
<haskell>
<haskell>
-
problem_147 = undefined
+
numPar w h
 +
= h*(h+1)*w*(w+1) `div` 4
 +
numDiag w h
 +
| w < h = numDiag h w
 +
| otherwise =
 +
w*diff - h*(diff+1) `div` 2
 +
where
 +
diff = (2*h-1)*h*(2*h+1) `div` 3
 +
problem_147 =
 +
sum [2*(numPar w h + numDiag w h) |
 +
w <- [2 .. 43], h <- [1 .. w-1]]
 +
+ sum [numPar w w + numDiag w w |
 +
w <- [1 .. 43]]
 +
+ sum [numPar w h + numDiag w h |
 +
w <- [44 .. 47],
 +
h <- [1 .. 43]]
</haskell>
</haskell>

Revision as of 05:34, 10 February 2008

Contents

1 Problem 141

Investigating progressive numbers, n, which are also square.

Solution: 

import Data.List
intSqrt :: Integral a => a -> a
intSqrt n
    | n < 0 = error "intSqrt: negative n"
    | otherwise = f n
    where
        f x = if y < x then f y else x
            where y = (x + (n `quot` x)) `quot` 2
isSqrt n = n==((^2).intSqrt) n
takec a b =
    two++takeWhile (<=e12) 
    [sq| c1<-[1..], let c=c1*c1,let sq=(c^2*a^3*b+b^2*c) ]
    where
    e12=10^12
    two=[sq|c<-[b,2*b],let sq=(c^2*a^3*b+b^2*c) ]
problem_141=
    sum$nub[c|
    (a,b)<-takeWhile (\(a,b)->a^3*b+b^2<e12) 
        [(a,b)|
        a<-[2..e4],
        b<-[1..(a-1)]
        ],
    gcd a b==1,
    c<-takec a b,
    isSqrt c
    ]
    where
    e4=120
    e12=10^12

2 Problem 142

Perfect Square Collection

Solution: 

import List
isSquare n = (round . sqrt $ fromIntegral n) ^ 2 == n
aToX (a,b,c)=[x,y,z]
    where
    x=div (a+b) 2
    y=div (a-b) 2
    z=c-x
{-
 -                                2    2    2
 -                               a  = c  + d
 -                                2    2    2
 -                               a  = e  + f
 -                                2    2    2
 -                               c  = e  + b
 -   let b=x*y  then 
 -                                             (y + xb)
 -                                          c= ---------
 -                                                 2
 -                                             (-y + xb)
 -                                          e= ---------
 -                                                 2
 -                                             (-x + yb)
 -                                          d= ---------
 -                                                 2
 -                                             (x + yb)
 -                                          f= ---------
 -                                                 2
 -
 - and 
 -                                2    2    2
 -                               a  = c  + d
 - then 
 -                                   2    2    2  2
 -                              2  (y  + x ) (x  y  + 1)
 -                             a = ---------------------
 -                                           4
 -
 -}
problem_142 = sum$head[aToX(t,t2 ,t3)|
    a<-[3,5..50],
    b<-[(a+2),(a+4)..50],
    let a2=a^2,
    let b2=b^2,
    let n=(a2+b2)*(a2*b2+1),
    isSquare n,
    let t=div n 4,
    let t2=a2*b2,
    let t3=div (a2*(b2+1)^2) 4
    ]

3 Problem 143

Investigating the Torricelli point of a triangle

Solution: 

import Data.List
import Data.Array.ST
import Data.Array
import qualified Data.Array.Unboxed as U
import Control.Monad
 
mkCan :: [Int] -> [(Int,Int)]
mkCan lst = map func $ group $ insert 3 lst
            where
              func ps@(p:_)
                | p == 3    = (3,2*l-1)
                | otherwise = (p, 2*l)
                  where
                    l = length ps
 
spfArray :: U.UArray Int Int
spfArray
    = runSTUArray
    (do ar <- newArray (2,13397) 0
        let loop k
                | k > 13397 = return ()
                | otherwise = do writeArray ar k 2
                                 loop (k+2)
        loop 2
        let go i
              | i > 13397 = return ar
              | otherwise
                = do p <- readArray ar i
                     if (p == 0)
                        then do writeArray ar i i
                                let run k
                                      | k > 13397 = go (i+2)
                                      | otherwise
                                        = do q <- readArray ar k
                                             when (q == 0)
                                                  (writeArray ar k i)
                                             run (k+2*i)
                                run (i*i)
                        else go (i+2)
        go 3)
 
factArray :: Array Int [Int]
factArray
    = runSTArray
    (do ar <- newArray (1,13397) []
        let go i
              | i > 13397 = return ar
              | otherwise = do let p = spfArray U.! i
                                   q = i `div` p
                               fs <- readArray ar q
                               writeArray ar i (p:fs)
                               go (i+1)
        go 2)
 
sdivs :: Int -> [(Int,Int)]
sdivs s
    = filter ((<= 100000) . uncurry (+)) $ zip sds' lds'
      where
        bd = 3*s*s
        pks = mkCan $ factArray ! s
        fun (p,k) = take (k+1) $ iterate (*p) 1
        ds = map fun pks
        (sds,lds) = span ((< bd) . (^2)) . sort $ foldr (liftM2 (*)) [1] ds
        sds' = map (+ 2*s) sds
        lds' = reverse $ map (+ 2*s) lds
 
pairArray :: Array Int [Int]
pairArray
    = runSTArray
    (do ar <- newArray (3,50000) []
        let go s
              | s > 13397 = return ar
              | otherwise
                = do let run [] = go (s+1)
                         run ((r,q):ds)
                            = do lst <- readArray ar r
                                 let nlst = insert q lst
                                 writeArray ar r nlst
                                 run ds
                     run $ sdivs s
        go 1)
 
select2 :: [Int] -> [(Int,Int)]
select2 []     = []
select2 (a:bs) = [(a,b) | b <- bs] ++ select2 bs
 
sumArray :: U.UArray Int Bool
sumArray
    = runSTUArray
    (do ar <- newArray (12,100000) False
        let go r
              | r > 33332 = return ar
              | otherwise
                = do let run [] = go (r+1)
                         run ((q,p):xs)
                            = do when (p `elem` (pairArray!q))
                                      (writeArray ar (p+q+r) True)
                                 run xs
                     run $ filter ((<= 100000) . (+r) . uncurry (+)) $
                             select2 $ pairArray!r
        go 3)
 
main :: IO ()
main = writeFile "p143.log"$show$ sum [s | (s,True) <- U.assocs sumArray]
problem_143 = main

4 Problem 144

Investigating multiple reflections of a laser beam.

Solution: 

type Point = (Double, Double)
type Vector = (Double, Double)
type Normal = (Double, Double)
 
sub :: Vector -> Vector -> Vector
sub (x,y) (a,b) = (x-a, y-b)
 
mull :: Double -> Vector -> Vector
mull s (x,y) = (s*x, s*y)
 
mulr :: Vector -> Double -> Vector
mulr v s = mull s v
 
dot :: Vector -> Vector -> Double
dot (x,y) (a,b) = x*a + y*b
 
normSq :: Vector -> Double
normSq v = dot v v
 
normalize :: Vector -> Vector
normalize v 
    |len /= 0 =mulr v (1.0/len)
    |otherwise=error "Vettore nullo.\n"  
    where
    len = (sqrt . normSq) v 
 
proj :: Vector -> Vector -> Vector
proj a b = mull ((dot a b)/normSq b) b
 
reflect :: Vector -> Normal -> Vector
reflect i n = sub i $ mulr (proj i n) 2.0
 
type Ray = (Point, Vector)
 
makeRay :: Point -> Vector -> Ray
makeRay p v = (p, v)
 
getPoint :: Ray -> Double -> Point
getPoint ((px,py),(vx,vy)) t = (px + t*vx, py + t*vy)
 
type Ellipse = (Double, Double)
 
getNormal :: Ellipse -> Point -> Normal
getNormal (a,b) (x,y) = ((-b/a)*x, (-a/b)*y)
 
rayFromPoint :: Ellipse -> Vector -> Point -> Ray
rayFromPoint e v p = makeRay p (reflect v (getNormal e p))
 
test :: Point -> Bool
test (x,y) = y > 0 && x >= -0.01 && x <= 0.01
 
intersect :: Ellipse -> Ray -> Point
intersect (e@(a,b)) (r@((px,py),(vx,vy))) =
    getPoint r t1
    where
    c0 = normSq (vx/a, vy/b)
    c1 = 2.0 * dot (vx/a, vy/b) (px/a, py/b)
    c2 = (normSq (px/a, py/b)) - 1.0
    (t0, t1) = quadratic c0 c1 c2 
 
quadratic :: Double -> Double -> Double -> (Double, Double)
quadratic a b c  
    |d < 0= error "Discriminante minore di zero"
    |otherwise= if (t0 < t1) then (t0, t1) else (t1, t0)
    where
    d = b * b - 4.0 * a * c
    sqrtD = sqrt d
    q = if b < 0 then -0.5*(b - sqrtD) else 0.5*(b + sqrtD)
    t0 = q / a
    t1 = c / q 
 
calculate :: Ellipse -> Ray -> Int -> IO ()
calculate e (r@(o,d)) n 
    |test p=print n 
    |otherwise=do
         putStrLn $ "\rHit " ++ show n
         calculate e (rayFromPoint e d p) (n+1)
    where
    p = intersect e r 
 
origin = (0.0,10.1)
direction = sub (1.4,-9.6) origin
ellipse = (5.0,10.0)
 
problem_144 = do
    calculate ellipse (makeRay origin direction) 0

5 Problem 145

How many reversible numbers are there below one-billion?

Solution: 

import List
 
digits n 
{-  123->[3,2,1]
 -}
    |n<10=[n]
    |otherwise= y:digits x 
    where
    (x,y)=divMod n 10
-- 123 ->321
dmm=(\x y->x*10+y)
palind n=foldl dmm 0 (digits n) 
 
isOdd x=(length$takeWhile odd x)==(length x)
isOdig x=isOdd m && s<=h
    where
    k=x+palind x
    m=digits k
    y=floor$logBase 10 $fromInteger x
    ten=10^y
    s=mod x 10
    h=div x ten
 
a2=[i|i<-[10..99],isOdig i]
aa2=[i|i<-[10..99],isOdig i,mod i 10/=0]
a3=[i|i<-[100..999],isOdig i]
m5=[i|i1<-[0..99],i2<-[0..99],
      let i3=i1*1000+3*100+i2,
      let i=10^6*   8+i3*10+5,
      isOdig i
   ]
 
fun i
    |i==2  =2*le aa2
    |even i=(fun 2)*d^(m-1)
    |i==3  =2*le a3
    |i==7  =fun 3*le m5
    |otherwise=0
    where
    le=length
    m=div i 2
    d=2*le a2
 
problem_145 = sum[fun a|a<-[1..9]]

6 Problem 146

Investigating a Prime Pattern

Solution: 

import List
isPrime x=millerRabinPrimality x 2
--isPrime x=foldl   (&& )True [millerRabinPrimality x y|y<-[2,3,7,61,24251]]
six=[1,3,7,9,13,27]
allPrime x=foldl (&&) True [isPrime k|a<-six,let k=x^2+a]
linkPrime [x]=filterPrime x
linkPrime (x:xs)=[y|
    a<-linkPrime xs,
    b<-[0..(x-1)],
    let y=b*prxs+a,
    let c=mod y x,
    elem c d]
    where
    prxs=product xs
    d=filterPrime x
 
filterPrime p=
    [a|
    a<-[0..(p-1)],
    length[b|b<-six,mod (a^2+b) p/=0]==6
    ]
testPrimes=[2,3,5,7,11,13,17,23]
primes=[2,3,5,7,11,13,17,23,29]
test =
    sum[y|
    y<-linkPrime testPrimes,
    y<1000000,
    allPrime (y)
    ]==1242490
p146 =[y|y<-linkPrime primes,y<150000000,allPrime (y)]
problem_146=[a|a<-p146, allNext a]
allNext x=
    sum [1|(x,y)<-zip a b,x==y]==6
    where
    a=[x^2+b|b<-six]
    b=head a:(map nextPrime a)
nextPrime x=head [a|a<-[(x+1)..],isPrime a]
main=writeFile "p146.log" $show $sum problem_146

7 Problem 147

Rectangles in cross-hatched grids

Solution: 

numPar w h
    = h*(h+1)*w*(w+1) `div` 4
numDiag w h
    | w < h     = numDiag h w
    | otherwise = 
        w*diff - h*(diff+1) `div` 2
        where
        diff = (2*h-1)*h*(2*h+1) `div` 3
problem_147 = 
    sum [2*(numPar w h + numDiag w h) |
    w <- [2 .. 43], h <- [1 .. w-1]]
    + sum [numPar w w + numDiag w w |
    w <- [1 .. 43]]
    + sum [numPar w h + numDiag w h |
    w <- [44 .. 47],
    h <- [1 .. 43]]

8 Problem 148

Exploring Pascal's triangle.

Solution: 

triangel 0 = 0
triangel n 
    |n <7 =n+triangel (n-1)  
    |n==k7 =28^k 
    |otherwise=(triangel i) + j*(triangel (n-i))
    where
    i=k7*((n-1)`div`k7)
    j= -(n`div`(-k7))
    k7=7^k
    k=floor(log (fromIntegral n)/log 7)
problem_148=triangel (10^9)

9 Problem 149

Searching for a maximum-sum subsequence.

Solution: 

#include<stdio.h>
#define N 2000
#define   max(a,b)   ((a)   >   (b)   ?   (a)   :   (b))
int s[4000001];
int MaxSubsequenceSum(int s[] , int n) {
    int j;
    int ThisSum, MaxSum ;
    ThisSum = MaxSum = 0;
    for ( j=0; j<n ; j++)
    {
        ThisSum += s[j];
        if (ThisSum> MaxSum)
            MaxSum = ThisSum;
        else if (ThisSum < 0)
            ThisSum = 0;
    }
    return MaxSum;
}
long long Generate(int ind){
    long long k = ind;
    if (ind <= 55) 
        return  ((100003 - 200003*k + 300007*k*k*k) % 1000000) - 500000;
    return (s[k-24]+s[k-55]+1000000)%1000000-500000;
 
}
int main()
{
    int sums=0;
    int maxx=0;
    for (int i=1;i<4000001;i++){
        s[i]=(int)(Generate(i));
    }
    printf("%d  %d \n",s[10],s[100]);
    int ks[N],kss[N];
    for (int k=0;k<N;k++){
        for(int b=0;b<N;b++)
        {  
            ks[b]=s[k*N+b+1]; 
            kss[b]=s[b*N+k+1]; 
        }
        sums=MaxSubsequenceSum(ks,N);
        sums=max(sums,MaxSubsequenceSum(kss,N));
        maxx=max (maxx,sums);
    }
    int ksi,ksj, x,y,y1;
    for (int k=-N+1;k<N;k++){
        ksi=ksj=0;
        for(int b=0;b<N;b++)
        {  
            x=k+b;
            y=b;
            y1=N-1-b;
            if (x>-1 && x<N && y>-1 && y<N)
                ks[ksi++]=s[x*N+y+1];
            if (x>-1 && x<N && y1>-1 && y1<N)
                kss[ksj++]=s[x*N+y1+1];
        }
        sums=MaxSubsequenceSum(ks,ksi);
        sums=max(sums,MaxSubsequenceSum(kss,ksj));
        maxx=max (maxx,sums);
    }
    printf("%d\n",maxx);
}
problem_149 = main

10 Problem 150

Searching a triangular array for a sub-triangle having minimum-sum.

Solution: 

problem_150 = undefined
Retrieved from "http://www.haskell.org/haskellwiki/Euler_problems/141_to_150"