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Euler problems/141 to 150

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<haskell>
 
<haskell>
 
import Data.List
 
import Data.List
import Data.Map((!),fromList,member)
+
import Data.Array.ST
merge xs@(x:xt) ys@(y:yt) = case compare x y of
+
import Data.Array
LT -> x : (merge xt ys)
+
import qualified Data.Array.Unboxed as U
EQ -> x : (merge xt yt)
+
import Control.Monad
GT -> y : (merge xs yt)
 
 
diff xs@(x:xt) ys@(y:yt) = case compare x y of
 
LT -> x : (diff xt ys)
 
EQ -> diff xt yt
 
GT -> diff xs yt
 
 
 
primes, nonprimes :: [Integer]
+
mkCan :: [Int] -> [(Int,Int)]
primes = [2,3,5] ++ (diff [7,9..] nonprimes)
+
mkCan lst = map func $ group $ insert 3 lst
nonprimes = foldr1 f . map g $ tail primes
+
where
where f (x:xt) ys = x : (merge xt ys)
+
func ps@(p:_)
g p = [ n*p | n <- [p,p+2..]]
+
| p == 3 = (3,2*l-1)
primeFactors :: Integer -> [Integer]
+
| otherwise = (p, 2*l)
primeFactors n = factor n primes
+
where
where
+
l = length ps
factor _ [] = []
+
factor m (p:ps) | p*p > m = [m]
+
spfArray :: U.UArray Int Int
| m `mod` p == 0 = p : factor (m `div` p) (p:ps)
+
spfArray
| otherwise = factor m ps
+
= 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)
 
 
fstfac x = [(head a ,length a)|a<-group$primeFactors x]
+
factArray :: Array Int [Int]
fac [(x,y)]=[x^a|a<-[0..y]]
+
factArray
fac (x:xs)=[a*b|a<-fac [x],b<-fac xs]
+
= runSTArray
factors x=fac$fstfac x
+
(do ar <- newArray (1,13397) []
intSqrt :: Integral a => a -> a
+
let go i
intSqrt n
+
| i > 13397 = return ar
| n < 0 = error "intSqrt: negative n"
+
| otherwise = do let p = spfArray U.! i
| otherwise = f n
+
q = i `div` p
where
+
fs <- readArray ar q
f x = if y < x then f y else x
+
writeArray ar i (p:fs)
where y = (x + (n `quot` x)) `quot` 2
+
go (i+1)
prim40=tail$take 40 primes
+
go 2)
primeSqr=
+
fromList[(a,fromList$zip b [1..])|
+
sdivs :: Int -> [(Int,Int)]
a<-prim40,
+
sdivs s
let b=nub[t|c<-[0..a-1],
+
= filter ((<= 100000) . uncurry (+)) $ zip sds' lds'
let t=mod (c*c) a]
+
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
   
isSqrt n
+
pairArray :: Array Int [Int]
|k= n==((^2).intSqrt) n
+
pairArray
|otherwise=False
+
= runSTArray
where
+
(do ar <- newArray (3,50000) []
k=foldl (&&) True [member k ma |
+
let go s
a<-prim40,
+
| s > 13397 = return ar
let ma=(primeSqr !a),
+
| otherwise
let k=mod n a
+
= do let run [] = go (s+1)
]
+
run ((r,q):ds)
getOne a = [c|
+
= do lst <- readArray ar r
x<-factors t,
+
let nlst = insert q lst
a>x,
+
writeArray ar r nlst
let y=(a-x)*(3*a+x),
+
run ds
let k=4*x,
+
run $ sdivs s
let (c,m)=divMod y k,
+
go 1)
m==0
 
]
 
where
 
t=(3*a^2)
 
   
getThree a = [[a,m,n]|
+
select2 :: [Int] -> [(Int,Int)]
m<-t,
+
select2 [] = []
n<-[k|k<-t,mod k 5/=0],
+
select2 (a:bs) = [(a,b) | b <- bs] ++ select2 bs
let z=(2*m+n)^2+3*n*n,
 
isSqrt z
 
]
 
where
 
t=getOne a
 
gcdlst [x,y]=gcd x y
 
gcdlst (x:xs)=gcd x$gcdlst xs
 
 
p143 k=[c|
 
a<-[1+k*groups..groups*(k+1)],
 
c<-getThree (a*5),
 
gcdlst c==1
 
]
 
-- run test find test==[],so one of a b c is 5*x
 
test=[(a,b,c)|a<-t,b<-t,c<-t,
 
f a b,
 
f b c,
 
f c a]
 
where
 
t=[1..4]
 
f a b=elem (mod (a^2+b^2+a*b) 5) [0,1,4]
 
 
groups=200
 
 
 
google num
+
sumArray :: U.UArray Int Bool
-- write file to change bignum to small num
+
sumArray
=if (num>33)
+
= runSTUArray
then return()
+
(do ar <- newArray (12,100000) False
else do let k=p143 num
+
let go r
appendFile "file.log" $(show$k) ++" "++(show num) ++"\n"
+
| r > 33332 = return ar
appendFile "files.log" $(show$map sum k) ++" "++(show num) ++"\n"
+
| otherwise
google (num+1)
+
= do let run [] = go (r+1)
-- first use main to make file.log
+
run ((q,p):xs)
-- then run problem_143
+
= do when (p `elem` (pairArray!q))
main=google 0
+
(writeArray ar (p+q+r) True)
split :: Char -> String -> [String]
+
run xs
split = unfoldr . split'
+
run $ filter ((<= 100000) . (+r) . uncurry (+)) $
  +
select2 $ pairArray!r
  +
go 3)
 
 
split' :: Char -> String -> Maybe (String, String)
+
main :: IO ()
split' c l
+
main = writeFile "p143.log"$show$ sum [s | (s,True) <- U.assocs sumArray]
| null l = Nothing
+
problem_143 = main
| otherwise = Just (h, drop 1 t)
 
where (h, t) = span (/=c) l
 
 
sToInt x=((++[-1]).read) $head$split ' ' x
 
 
filer x
 
|x<0=False
 
|x>100000=False
 
|otherwise=True
 
problem_143=do
 
x<-readFile "files.log"
 
let y=concat$map sToInt $lines x
 
let z= filter filer y
 
let t=[b|a<-z,b<-takeWhile (<=100000) [a*b|b<-[1..]]]
 
print$ sum$nub t
 
 
 
</haskell>
 
</haskell>
   

Revision as of 01:49, 14 January 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:

problem_144 = undefined

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
find2km :: Integral a => a -> (a,a)
find2km n = f 0 n
    where 
        f k m
            | r == 1 = (k,m)
            | otherwise = f (k+1) q
            where (q,r) = quotRem m 2        
 
millerRabinPrimality :: Integer -> Integer -> Bool
millerRabinPrimality n a
    | a <= 1 || a >= n-1 = 
        error $ "millerRabinPrimality: a out of range (" 
              ++ show a ++ " for "++ show n ++ ")" 
    | n < 2 = False
    | even n = False
    | b0 == 1 || b0 == n' = True
    | otherwise = iter (tail b)
    where
        n' = n-1
        (k,m) = find2km n'
        b0 = powMod n a m
        b = take (fromIntegral k) $ iterate (squareMod n) b0
        iter [] = False
        iter (x:xs)
            | x == 1 = False
            | x == n' = True
            | otherwise = iter xs
 
pow' :: (Num a, Integral b) => (a -> a -> a) -> (a -> a) -> a -> b -> a
pow' _ _ _ 0 = 1
pow' mul sq x' n' = f x' n' 1
    where 
        f x n y
            | n == 1 = x `mul` y
            | r == 0 = f x2 q y
            | otherwise = f x2 q (x `mul` y)
            where
                (q,r) = quotRem n 2
                x2 = sq x
 
mulMod :: Integral a => a -> a -> a -> a
mulMod a b c = (b * c) `mod` a
squareMod :: Integral a => a -> a -> a
squareMod a b = (b * b) `rem` a
powMod :: Integral a => a -> a -> a -> a
powMod m = pow' (mulMod m) (squareMod m)
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:

problem_147 = undefined

8 Problem 148

Exploring Pascal's triangle.

Solution:

import List
digits n 
{-  123->[3,2,1]
 -   -}
     |n<7=[n]
     |otherwise= y:digits x 
     where
     (x,y)=divMod n 7
 
notDivX x=product$map (+1) $digits x
 
array::[Integer]
array=
    [a*b*c*d*e*f|
    let t=[1..7],
    a<-t,
    b<-t,
    c<-t,
    d<-t,
    e<-t,
    f<-t
    ]
 
fastNotDivX::Integer->Integer
fastNotDivX x=sum[k*a|a<-array]
    where
    k=product$map (+1) $digits x
 
sumNotDivX x=sum[notDivX a|a<-[0..x]]
 
-- sum[fastNotDivX x|x<-[0..b]]=sumNotDivX ((b+1)*7^6-1)
 
moreNotDivX =sum[notDivX a|a<-[1000000000.. 1000016499 ]]
 
google num
-- write file to change bignum to small num
  =if (num>8499)
      then return()
      else do appendFile "file.log" $(show$fastNotDivX num)  ++"  "++(show num) ++"\n"
              google (num+1)
-- first use main to make file.log
-- then run problem_148
main=google 0
 
split :: Char -> String -> [String]
split = unfoldr . split'
 
split' :: Char -> String -> Maybe (String, String)
split' c l
  | null l = Nothing
  | otherwise = Just (h, drop 1 t)
  where (h, t) = span (/=c) l
 
sToInt x=((+0).read) $head$split ' ' x
 
problem_148=do
    x<-readFile "file.log"
    let y=sum$map sToInt $lines x
    print (  y-(fromInteger moreNotDivX))

9 Problem 149

Searching for a maximum-sum subsequence.

Solution:

problem_149 = undefined

10 Problem 150

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

Solution:

problem_150 = undefined