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Euler problems/61 to 70

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== [http://projecteuler.net/index.php?section=problems&id=61 Problem 61] ==
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Do them on your own!
Find the sum of the only set of six 4-digit figurate numbers with a cyclic property.
 
 
Solution:
 
<haskell>
 
import Data.List
 
 
permute [] = [[]]
 
permute xs = concatMap (\x -> map (x:) $ permute $ delete x xs) xs
 
 
figurates n xs = extract $ concatMap (gather (map poly xs)) $ map (:[]) $ poly n
 
where gather [xs] (v:vs)
 
= let v' = match xs v
 
in if v' == [] then [] else map (:v:vs) v'
 
gather (xs:xss) (v:vs)
 
= let v' = match xs v
 
in if v' == [] then [] else concatMap (gather xss) $ map (:v:vs) v'
 
match xs (_,v) = let p = (v `mod` 100)*100 in sublist (p+10,p+100) xs
 
sublist (s,e) = takeWhile (\(_,x) -> x<e) . dropWhile (\(_,x) -> x<s)
 
link ((_,x):xs) = x `mod` 100 == (snd $ last xs) `div` 100
 
diff (x:y:xs) = if fst x /= fst y then diff (y:xs) else False
 
diff [x] = True
 
extract = filter diff . filter link
 
poly m = [(n, x) | (n, x) <- zip [1..] $ takeWhile (<10000)
 
$ scanl (+) 1 [m-1,2*m-3..],
 
1010 < x, x `mod` 100 > 9]
 
 
problem_61 = sum $ map snd $ head $ concatMap (figurates 3) $ permute [4..8]
 
</haskell>
 
 
== [http://projecteuler.net/index.php?section=problems&id=62 Problem 62] ==
 
Find the smallest cube for which exactly five permutations of its digits are cube.
 
 
Solution:
 
<haskell>
 
import Data.List
 
import Data.Maybe
 
a = map (^3) [0..10000]
 
b = map (sort . show) a
 
c = (filter ((==5) . length) . group . sort) b
 
d = findIndex (==(head (head c))) b
 
problem_62 = (toInteger (fromJust d))^3
 
</haskell>
 
 
== [http://projecteuler.net/index.php?section=problems&id=63 Problem 63] ==
 
How many n-digit positive integers exist which are also an nth power?
 
 
Solution:
 
<haskell>
 
problem_63=length[x^y|x<-[1..9],y<-[1..22],y==(length$show$x^y)]
 
</haskell>
 
 
== [http://projecteuler.net/index.php?section=problems&id=64 Problem 64] ==
 
How many continued fractions for N ≤ 10000 have an odd period?
 
 
Solution:
 
<haskell>
 
import Data.List
 
 
problem_64 =length $ filter id $ map solve $ [2..9999] \\ (map (^2) [2..100])
 
 
solve n = even $ length $ cont n 0 1
 
 
cont :: Int -> Int -> Int -> [Int]
 
cont r n d = m : rest
 
where
 
m = truncate ((sqrt (fromIntegral r) + fromIntegral n ) / fromIntegral d)
 
a = n - d * m
 
rest = if d == 1 && n /= 0
 
then []
 
else cont r (-a) ((r - a ^ 2) `div` d)
 
</haskell>
 
 
== [http://projecteuler.net/index.php?section=problems&id=65 Problem 65] ==
 
Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e.
 
 
Solution:
 
<haskell>
 
import Data.Char
 
import Data.Ratio
 
 
e = [2] ++ concat [ [1, 2*i, 1] | i <- [1..] ]
 
 
fraction [x] = x%1
 
fraction (x:xs) = x%1 + 1/(fraction xs)
 
 
problem_65 = sum $ map digitToInt $ show $ numerator $ fraction $ take 100 e
 
</haskell>
 
 
== [http://projecteuler.net/index.php?section=problems&id=66 Problem 66] ==
 
Investigate the Diophantine equation x<sup>2</sup> − Dy<sup>2</sup> = 1.
 
 
Solution:
 
<haskell>
 
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
 
problem_66 =
 
snd$maximum [ (x,d) |
 
d <- [1..1000],
 
let b = intSqrt d,
 
b*b /= d, -- d can't be a perfect square
 
let (x,_) = pell d b b
 
]
 
 
pell d wd b = piter d wd b 0 1 0 1 1 0
 
piter d wd b i c l k m n
 
| cn == 1 = (x, y)
 
| otherwise = piter d wd bn (i+1) cn k u n v
 
where
 
yb = (wd+b) `div` c
 
bn = yb*c-b
 
cn = (d-(bn*bn)) `div` c
 
yn | i == 0 = wd
 
| otherwise = yb
 
u = k*yn+l -- u/v is the i-th convergent of sqrt(d)
 
v = n*yn+m
 
(x,y) | odd (i+1) = (u*u+d*v*v, 2*u*v)
 
| otherwise = (u,v)
 
</haskell>
 
 
== [http://projecteuler.net/index.php?section=problems&id=67 Problem 67] ==
 
Using an efficient algorithm find the maximal sum in the triangle?
 
 
Solution:
 
<haskell>
 
problem_67 = readFile "triangle.txt" >>= print . solve . parse
 
parse = map (map read . words) . lines
 
solve = head . foldr1 step
 
step [] [z] = [z]
 
step (x:xs) (y:z:zs) = x + max y z : step xs (z:zs)
 
</haskell>
 
 
== [http://projecteuler.net/index.php?section=problems&id=68 Problem 68] ==
 
What is the maximum 16-digit string for a "magic" 5-gon ring?
 
 
Solution:
 
<haskell>
 
import Data.List
 
permute [] = [[]]
 
permute list =
 
concat $ map (\(x:xs) -> map (x:) (permute xs))
 
(take (length list)
 
(unfoldr (\x -> Just (x, tail x ++ [head x])) list))
 
problem_68 =
 
maximum $ map (concat . map show) poel
 
where
 
gon68 = [1..10]
 
knip = (length gon68) `div` 2
 
(is,es) = splitAt knip gon68
 
extnodes = map (\x -> [head es]++x) $ permute $ tail es
 
intnodes = map (\(p:ps) -> zipWith (\ x y -> [x]++[y])
 
(p:ps) (ps++[p])) $ permute is
 
poel = [ concat hs | hs <- [ zipWith (\x y -> [x]++y) uitsteeksels organen |
 
uitsteeksels <- extnodes, organen <- intnodes ],
 
let subsom = map (sum) hs, length (nub subsom) == 1 ]
 
</haskell>
 
 
== [http://projecteuler.net/index.php?section=problems&id=69 Problem 69] ==
 
Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum.
 
 
Solution:
 
<haskell>
 
{-phi(n) = n*(1-1/p1)*(1-1/p2)*...*(1-1/pn)
 
n/phi(n) = 1/(1-1/p1)*(1-1/p2)*...*(1-1/pn)
 
(1-1/p) will be minimal for a small p and 1/(1-1/p) will then be maximal
 
-}
 
primes=[2,3,5,7,11,13,17,19,23]
 
problem_69=
 
maximum [c|
 
a<-[1..length primes],
 
let b=take a primes,
 
let c=product b,
 
c<10^6
 
]
 
</haskell>
 
 
Note: credit for arithmetic functions is due to [http://www.polyomino.f2s.com/ David Amos].
 
 
== [http://projecteuler.net/index.php?section=problems&id=70 Problem 70] ==
 
Investigate values of n for which φ(n) is a permutation of n.
 
 
Solution:
 
<haskell>
 
import Data.List
 
isPerm a b = (show a) \\ (show b)==[]
 
flsqr n x=x<(floor.sqrt.fromInteger) n
 
pairs n1 =
 
maximum[m|a<-gena ,b<-genb,let m=a*b,n>m,isPerm m$ m-a-b+1]
 
where
 
n=fromInteger n1
 
gena = dropWhile (flsqr n)$ takeWhile (flsqr (2*n)) primes
 
genb = dropWhile (flsqr (div n 2))$ takeWhile (flsqr n) primes
 
 
problem_70= pairs (10^7)
 
</haskell>
 

Revision as of 21:53, 29 January 2008

Do them on your own!