Euler problems/141 to 150

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

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=(a+b)`div`2
    y=(a-b)`div`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=n`div`4,
    let t2=a2*b2,
    let t3=(a2*(b2+1)^2)`div`4
    ]

Problem 143

Investigating the Torricelli point of a triangle

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

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=x`mod`10
    h=x`div`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]]

Problem 146

Investigating a Prime Pattern

Solution:

import List
isPrime x=millerRabinPrimality x 2
--isPrime x=all (millerRabinPrimality x) [2,3,7,61,24251]
six=[1,3,7,9,13,27]
allPrime x=all (\a -> isPrime (x^2+a)) six
linkPrime [x]=filterPrime x
linkPrime (x:xs)=[y|
    a<-linkPrime xs,
    b<-[0..(x-1)],
    let y=b*prxs+a,
    let c=y`mod`x,
    elem c d]
    where
    prxs=product xs
    d=filterPrime x

filterPrime p=
    [a|
    a<-[0..(p-1)],
    length[b|b<-six,(a^2+b)`mod`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

Problem 147

Rectangles in cross-hatched grids

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 . logBase 7 . fromIntegral $ n
problem_148=triangel (10^9)

Problem 149

Searching for a maximum-sum subsequence.

Solution:

import Data.Array
import Data.List (foldl')
 
n = 2000
 
res = maximum' $ concat [rows, cols, diags, diags']
    where
        rows   = map (maxSumInRow . getRow   laggedFibArray) [0 .. n-1]
        cols   = map (maxSumInRow . getCol   laggedFibArray) [0 .. n-1]
        diags  = map (maxSumInRow . getDiag  laggedFibArray) [-(n-2) .. (n-2)]
        diags' = map (maxSumInRow . getDiag' laggedFibArray) [-(n-2) .. (n-2)]
 
 
laggedFibArray :: Array Integer Integer
laggedFibArray = listArray (0, n^2-1) $ map f [1..n^2]
    where
        f k = norm $ if k < 56
              then 100003 - (200003*k) + (300007*(k^3))
              else (laggedFibArray ! (k-25)) + (laggedFibArray ! (k-56)) + (10^6)
 
        norm x = mod x (10^6) - 500000
 
 
getRow   a i = map (a!) [i*n .. (i+1)*n-1]
getCol   a i = map (a!) [i,n+i .. n*(n-1)+i]
getDiag  a i = map (a!) $
    if i >= 0
    then [(i*n) + (k*(n+1)) | k <- [0..n-i-1]]
    else [k + n*(k+i) | k <- [-i .. n-1]]
getDiag' a i = map (a!) $
    if i >= 0
    then [(n*k) + n-k-i-1 | k <- [0..n-i-1]]
    else [n*(k-i) + n-k-1 | k <- [0..n+i-1]]
 
 
maxSumInRow = snd . foldl' f (0,0)
    where
        f (line_sum, line_max) x = (line_sum', max line_max line_sum')
            where line_sum' = max (line_sum+x) 0

-- strict version of maximum
maximum' (x:xs) = foldl' max x xs

main = print res

Problem 150

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