Euler problems/141 to 150: Difference between revisions
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== [http://projecteuler.net/index.php?section= | == [http://projecteuler.net/index.php?section=problems&id=141 Problem 141] == | ||
Investigating progressive numbers, n, which are also square. | Investigating progressive numbers, n, which are also square. | ||
Solution: | Solution: | ||
<haskell> | <haskell> | ||
problem_141 = | 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 | |||
</haskell> | </haskell> | ||
== [http://projecteuler.net/index.php?section= | == [http://projecteuler.net/index.php?section=problems&id=142 Problem 142] == | ||
Perfect Square Collection | Perfect Square Collection | ||
Line 16: | Line 44: | ||
aToX (a,b,c)=[x,y,z] | aToX (a,b,c)=[x,y,z] | ||
where | where | ||
x= | x=(a+b)`div`2 | ||
y= | y=(a-b)`div`2 | ||
z=c-x | z=c-x | ||
{- | {- | ||
Line 57: | Line 85: | ||
let n=(a2+b2)*(a2*b2+1), | let n=(a2+b2)*(a2*b2+1), | ||
isSquare n, | isSquare n, | ||
let t=div | let t=n`div`4, | ||
let t2=a2*b2, | let t2=a2*b2, | ||
let t3= | let t3=(a2*(b2+1)^2)`div`4 | ||
] | ] | ||
</haskell> | </haskell> | ||
== [http://projecteuler.net/index.php?section= | == [http://projecteuler.net/index.php?section=problems&id=143 Problem 143] == | ||
Investigating the Torricelli point of a triangle | Investigating the Torricelli point of a triangle | ||
== [http://projecteuler.net/index.php?section=problems&id=144 Problem 144] == | |||
Investigating multiple reflections of a laser beam. | |||
Solution: | Solution: | ||
<haskell> | <haskell> | ||
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)) | ||
problem_144 = | |||
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> | ||
== [http://projecteuler.net/index.php?section= | == [http://projecteuler.net/index.php?section=problems&id=145 Problem 145] == | ||
How many reversible numbers are there below one-billion? | How many reversible numbers are there below one-billion? | ||
Line 105: | Line 213: | ||
y=floor$logBase 10 $fromInteger x | y=floor$logBase 10 $fromInteger x | ||
ten=10^y | ten=10^y | ||
s=mod | s=x`mod`10 | ||
h=div | h=x`div`ten | ||
a2=[i|i<-[10..99],isOdig i] | a2=[i|i<-[10..99],isOdig i] | ||
Line 131: | Line 239: | ||
</haskell> | </haskell> | ||
== [http://projecteuler.net/index.php?section= | == [http://projecteuler.net/index.php?section=problems&id=146 Problem 146] == | ||
Investigating a Prime Pattern | Investigating a Prime Pattern | ||
Solution: | Solution: | ||
<haskell> | <haskell> | ||
problem_146 = | 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 | |||
</haskell> | </haskell> | ||
== [http://projecteuler.net/index.php?section= | == [http://projecteuler.net/index.php?section=problems&id=147 Problem 147] == | ||
Rectangles in cross-hatched grids | Rectangles in cross-hatched grids | ||
== [http://projecteuler.net/index.php?section=problems&id=148 Problem 148] == | |||
Exploring Pascal's triangle. | |||
Solution: | Solution: | ||
<haskell> | <haskell> | ||
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) | |||
</haskell> | </haskell> | ||
== [http://projecteuler.net/index.php?section= | == [http://projecteuler.net/index.php?section=problems&id=149 Problem 149] == | ||
Searching for a maximum-sum subsequence. | |||
Solution: | Solution: | ||
<haskell> | <haskell> | ||
import List | 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 | 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 | |||
</haskell> | </haskell> | ||
== [http://projecteuler.net/index.php?section= | == [http://projecteuler.net/index.php?section=problems&id=150 Problem 150] == | ||
Searching a triangular array for a sub-triangle having minimum-sum. | Searching a triangular array for a sub-triangle having minimum-sum. | ||
{{sect-stub}} | |||
Latest revision as of 10:51, 12 February 2010
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.