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Euler problems/121 to 130

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(add problem 128)
Line 51: Line 51:
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
primes = 2 : filter ((==1) . length . primeFactors) [3,5..]
 
 
primeFactors n = factor n primes
 
where
 
factor _ [] = []
 
factor m (p:ps) | p*p > m = [m]
 
| m `mod` p == 0 = p : [m `div` p]
 
| otherwise = factor m ps
 
 
isPrime :: Integer -> Bool
 
isPrime 1 = False
 
isPrime n = case (primeFactors n) of
 
(_:_:_) -> False
 
_ -> True
 
 
 
problem_123 =
 
problem_123 =
 
head[a+1|a<-[20000,20002..22000],
 
head[a+1|a<-[20000,20002..22000],
Line 79: Line 64:
 
<haskell>
 
<haskell>
 
import List
 
import List
primes :: [Integer]
+
problem_124=
primes = 2 : filter ((==1) . length . primeFactors) [3,5..]
+
snd$(!!9999)$sort[(product$nub$primeFactors x,x)|x<-[1..100000]]
 
primeFactors :: Integer -> [Integer]
 
primeFactors n = factor n primes
 
where
 
factor _ [] = []
 
factor m (p:ps) | p*p > m = [m]
 
| m `mod` p == 0 = p : factor (m `div` p) (p:ps)
 
| otherwise = factor m ps
 
problem_124=snd$(!!9999)$sort[(product$nub$primeFactors x,x)|x<-[1..100000]]
 
   
 
</haskell>
 
</haskell>
Line 174: Line 159:
 
rads = listArray (1,n) $ map rad [1..n]
 
rads = listArray (1,n) $ map rad [1..n]
 
invrads = sort $ map (\(a,b) -> (b, a)) $ assocs rads
 
invrads = sort $ map (\(a,b) -> (b, a)) $ assocs rads
primeFactors :: Integer -> [Integer]
 
primeFactors n = factor n primes
 
where
 
factor _ [] = []
 
factor m (p:ps) | p*p > m = [m]
 
| m `mod` p == 0 = p : factor (m `div` p) (p:ps)
 
| otherwise = factor m ps
 
merge xs@(x:xt) ys@(y:yt) = case compare x y of
 
LT -> x : (merge xt ys)
 
EQ -> x : (merge xt yt)
 
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]
 
primes = [2,3,5] ++ (diff [7,9..] nonprimes)
 
nonprimes = foldr1 f . map g $ tail primes
 
where f (x:xt) ys = x : (merge xt ys)
 
g p = [ n*p | n <- [p,p+2..]]
 
 
problem_127 = main
 
problem_127 = main
 
</haskell>
 
</haskell>
Line 204: Line 167:
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
problem_128 = undefined
+
p128=
  +
concat[m|a<-[0..70000],let m=middle a++right a,not$null m]
  +
where
  +
middle n
  +
|all isPrime [11+6*n,13+6*n,29+12*n]=[2+3*(n+1)*(n+2)]
  +
|otherwise=[]
  +
right n
  +
|all isPrime [11+6*n,17+6*n,17+12*n]=[1+3*(n+2)*(n+3)]
  +
|otherwise=[]
  +
problem_128=do
  +
print(p128!!1997)
  +
isPrime x
  +
|x<100=isPrime' x
  +
|otherwise=all (millerRabinPrimality x )[2,7,61]
 
</haskell>
 
</haskell>
   
Line 213: Line 176:
 
<haskell>
 
<haskell>
 
import Data.List
 
import Data.List
mulMod :: Integral a => a -> a -> a -> a
 
mulMod a b c= (b * c) `rem` a
 
squareMod :: Integral a => a -> a -> a
 
squareMod a b = (b * b) `rem` a
 
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
 
powMod :: Integral a => a -> a -> a -> a
 
powMod m = pow' (mulMod m) (squareMod m)
 
 
merge xs@(x:xt) ys@(y:yt) = case compare x y of
 
LT -> x : (merge xt ys)
 
EQ -> x : (merge xt yt)
 
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]
 
primes = [2,3,5] ++ (diff [7,9..] nonprimes)
 
nonprimes = foldr1 f . map g $ tail primes
 
where f (x:xt) ys = x : (merge xt ys)
 
g p = [ n*p | n <- [p,p+2..]]
 
primeFactors :: Integer -> [Integer]
 
primeFactors n = factor n primes
 
where
 
factor _ [] = []
 
factor m (p:ps) | p*p > m = [m]
 
| m `mod` p == 0 = p : factor (m `div` p) (p:ps)
 
| otherwise = factor m ps
 
 
fstfac x = [(head a ,length a)|a<-group$primeFactors x]
 
fac [(x,y)]=[x^a|a<-[0..y]]
 
fac (x:xs)=[a*b|a<-fac [x],b<-fac xs]
 
 
factors x=fac$fstfac x
 
factors x=fac$fstfac x
 
funp (p,1)=head[a|a<-sort$factors (p-1),powMod p 10 a==1]
 
funp (p,1)=head[a|a<-sort$factors (p-1),powMod p 10 a==1]

Revision as of 06:37, 25 January 2008

Contents

1 Problem 121

Investigate the game of chance involving coloured discs.

Solution:

import Data.List
problem_121 = possibleGames `div` winningGames
   where
   possibleGames = product [1..16]
   winningGames = 
       (1+) $ sum $ map product $ chooseUpTo 7 [1..15]
   chooseUpTo 0     _ = []
   chooseUpTo (n+1) x = 
       [y:z | 
       (y:ys) <- tails x,
       z <- []: chooseUpTo n ys
       ]

2 Problem 122

Finding the most efficient exponentiation method.

Solution using a depth first search, pretty fast :

import Data.List
import Data.Array.Diff
import Control.Monad
 
depthAddChain 12 branch mins = mins
depthAddChain d branch mins = foldl' step mins $ nub $ filter (> head branch)
                               $ liftM2 (+) branch branch
    where
      step da e | e > 200 = da
                | otherwise =
                    case compare (da ! e) d of
                      GT -> depthAddChain (d+1) (e:branch) $ da // [(e,d)]
                      EQ -> depthAddChain (d+1) (e:branch) da
                      LT -> da
 
baseBranch = [2,1]
 
baseMins :: DiffUArray Int Int
baseMins = listArray (1,200) $ 0:1: repeat maxBound
 
problem_122 = sum . elems $ depthAddChain 2 baseBranch baseMins

3 Problem 123

Determining the remainder when (pn − 1)n + (pn + 1)n is divided by pn2.

Solution:

problem_123 = 
    head[a+1|a<-[20000,20002..22000],
    let n=2*(a+1)*primes!!(fromInteger a),
    n>10^10
    ]

4 Problem 124

Determining the kth element of the sorted radical function.

Solution:

import List
problem_124=
    snd$(!!9999)$sort[(product$nub$primeFactors x,x)|x<-[1..100000]]

5 Problem 125

Finding square sums that are palindromic.

Solution:

import Data.List 
import Data.Map(fromList,(!))
 
toFloat = (flip encodeFloat 0) 
digits n 
{-  123->[3,2,1]
 -}
    |n<10=[n]
    |otherwise= y:digits x 
    where
    (x,y)=divMod n 10
 
palind n=foldl dmm 0 (digits n) 
-- 123 ->321
dmm=(\x y->x*10+y)
 
makepalind n=(n*d+p):[c+b*d|b<-[0..9]]
    where
    a=(+1)$floor$logBase 10$fromInteger n
    d=10^a
    p=palind n
    c=n*10*d+p
 
twomakep n=(n*d+p)
    where
    a=(+1)$floor$logBase 10$fromInteger n
    d=10^a
    p=palind n
 
p125=sum[b|a<-[1..999], b<-makepalind a,not$null$ funa b]
p125a=sum[b|a<-[1000..9999], let b=twomakep a,not$null$ funa b]
p125b=sum[a|a<-[1..9], not$null$ funa a]
 
findmap=fromList[(a,2*fill_map a)|a<-[0..737]]
fill_map x
    |odd x=fastsum $div (x-1) 2
    |otherwise=fastsumodd (x-1)
    where
    fastsum  y=div (y*(y+1)*(2*y+1)) 6
    fastsumodd  y=let n=div (y+1) 2 in div (n*(4*n*n-1)) 3
 
funa x=[(a,x)|a<-takeWhile (\a->a*a*a<4*x) [2..],funb a x]
funb x n
    |odd x=d2==0 && 4*d1>=(x+1)^2 && isSq d1
    |otherwise=d4==0 && odd d3 && d3>=(x+1)^2 && isSq d3
    where
    x1=fromInteger x
    (d1,d2)=divMod ((n-findmap! x1)) (x)
    (d3,d4)=divMod ((4*n-findmap!x1)) (x)
isSq x=(floor$sqrt$toFloat x)^2==x
problem_125 = (p125+p125a+p125b)

6 Problem 126

Exploring the number of cubes required to cover every visible face on a cuboid.

Solution:

problem_126 = undefined

7 Problem 127

Investigating the number of abc-hits below a given limit.

Solution:

import Data.List
import Data.Array.IArray
import Data.Array.Unboxed
 
main = appendFile "p127.log" $show$ solve 99999
 
rad x = fromIntegral $ product $ map fst $ primePowerFactors $ fromIntegral x
primePowerFactors x = [(head a ,length a)|a<-group$primeFactors x] 
solve :: Int -> Int
solve n = sum [ c | (rc,c) <- invrads
                  , 2 * rc < c
                  , (ra, a) <- takeWhile (\(a,_)->(c > 2*rc*a)) invrads
                  , a < c `div` 2
                  , gcd ra rc == 1
                  , ra * rads ! (c - a) < c `div` rc]
    where
     rads :: UArray Int Int
     rads = listArray (1,n) $ map rad [1..n]
     invrads = sort $ map (\(a,b) -> (b, a)) $ assocs rads
problem_127 = main

8 Problem 128

Which tiles in the hexagonal arrangement have prime differences with neighbours?

Solution:

p128=
    concat[m|a<-[0..70000],let m=middle a++right a,not$null m]
    where
    middle n
        |all isPrime [11+6*n,13+6*n,29+12*n]=[2+3*(n+1)*(n+2)]
        |otherwise=[]
    right n
        |all isPrime [11+6*n,17+6*n,17+12*n]=[1+3*(n+2)*(n+3)]
        |otherwise=[]
problem_128=do
    print(p128!!1997)
isPrime x
    |x<100=isPrime' x
    |otherwise=all (millerRabinPrimality x )[2,7,61]

9 Problem 129

Investigating minimal repunits that divide by n.

Solution:

import Data.List
factors x=fac$fstfac x
funp (p,1)=head[a|a<-sort$factors (p-1),powMod p 10 a==1]
funp (p,s)=p^(s-1)*funp (p,1)
funn []=1
funn (x:xs) =lcm (funp x) (funn xs)
p129 q=head[a|a<-[q..],gcd a 10==1,let s=funn$fstfac$(*9) a,s>q,s>a]
problem_129 = p129 (10^6)

10 Problem 130

Finding composite values, n, for which n−1 is divisible by the length of the smallest repunits that divide it.

Solution:

--factors powMod in p129
fun x |(not$null a)=head a 
    |otherwise=0
    where 
    a=take 1 [n|n<-sort$factors (x-1),(powMod x 10 n)==1]
problem_130 =sum$take 25[a|a<-[1..],
    not$isPrime a,
    let b=fun a, 
    b/=0,
    mod (a-1) b==0,
    mod a 3 /=0]