Difference between revisions of "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! |
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+ | Find the sum of the only set of six 4-digit figurate numbers with a cyclic property. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | import Data.List |
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+ | |||
+ | permute [] = [[]] |
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+ | permute xs = concatMap (\x -> map (x:) $ permute $ delete x xs) xs |
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+ | |||
+ | figurates n xs = extract $ concatMap (gather (map poly xs)) $ map (:[]) $ poly n |
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+ | where gather [xs] (v:vs) |
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+ | = let v' = match xs v |
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+ | in if v' == [] then [] else map (:v:vs) v' |
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+ | gather (xs:xss) (v:vs) |
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+ | = let v' = match xs v |
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+ | in if v' == [] then [] else concatMap (gather xss) $ map (:v:vs) v' |
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+ | match xs (_,v) = let p = (v `mod` 100)*100 in sublist (p+10,p+100) xs |
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+ | sublist (s,e) = takeWhile (\(_,x) -> x<e) . dropWhile (\(_,x) -> x<s) |
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+ | link ((_,x):xs) = x `mod` 100 == (snd $ last xs) `div` 100 |
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+ | diff (x:y:xs) = if fst x /= fst y then diff (y:xs) else False |
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+ | diff [x] = True |
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+ | extract = filter diff . filter link |
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+ | poly m = [(n, x) | (n, x) <- zip [1..] $ takeWhile (<10000) |
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+ | $ scanl (+) 1 [m-1,2*m-3..], |
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+ | 1010 < x, x `mod` 100 > 9] |
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+ | |||
+ | problem_61 = sum $ map snd $ head $ concatMap (figurates 3) $ permute [4..8] |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=62 Problem 62] == |
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+ | Find the smallest cube for which exactly five permutations of its digits are cube. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | import Data.List |
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+ | import Data.Maybe |
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+ | a = map (^3) [0..10000] |
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+ | b = map (sort . show) a |
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+ | c = (filter ((==5) . length) . group . sort) b |
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+ | d = findIndex (==(head (head c))) b |
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+ | problem_62 = (toInteger (fromJust d))^3 |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=63 Problem 63] == |
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+ | How many n-digit positive integers exist which are also an nth power? |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | problem_63=length[x^y|x<-[1..9],y<-[1..22],y==(length$show$x^y)] |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=64 Problem 64] == |
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+ | How many continued fractions for N ≤ 10000 have an odd period? |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | import Data.List |
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+ | |||
+ | problem_64 =length $ filter id $ map solve $ [2..9999] \\ (map (^2) [2..100]) |
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+ | |||
+ | solve n = even $ length $ cont n 0 1 |
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+ | |||
+ | cont :: Int -> Int -> Int -> [Int] |
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+ | cont r n d = m : rest |
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+ | where |
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+ | m = truncate ((sqrt (fromIntegral r) + fromIntegral n ) / fromIntegral d) |
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+ | a = n - d * m |
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+ | rest = if d == 1 && n /= 0 |
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+ | then [] |
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+ | else cont r (-a) ((r - a ^ 2) `div` d) |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=65 Problem 65] == |
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+ | Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | import Data.Char |
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+ | import Data.Ratio |
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+ | |||
+ | e = [2] ++ concat [ [1, 2*i, 1] | i <- [1..] ] |
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+ | |||
+ | fraction [x] = x%1 |
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+ | fraction (x:xs) = x%1 + 1/(fraction xs) |
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+ | |||
+ | problem_65 = sum $ map digitToInt $ show $ numerator $ fraction $ take 100 e |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=66 Problem 66] == |
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+ | Investigate the Diophantine equation x<sup>2</sup> − Dy<sup>2</sup> = 1. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | intSqrt :: Integral a => a -> a |
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+ | intSqrt n |
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+ | | n < 0 = error "intSqrt: negative n" |
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+ | | otherwise = f n |
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+ | where |
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+ | f x = if y < x then f y else x |
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+ | where y = (x + (n `quot` x)) `quot` 2 |
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+ | problem_66 = |
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+ | snd$maximum [ (x,d) | |
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+ | d <- [1..1000], |
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+ | let b = intSqrt d, |
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+ | b*b /= d, -- d can't be a perfect square |
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+ | let (x,_) = pell d b b |
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+ | ] |
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+ | |||
+ | pell d wd b = piter d wd b 0 1 0 1 1 0 |
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+ | piter d wd b i c l k m n |
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+ | | cn == 1 = (x, y) |
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+ | | otherwise = piter d wd bn (i+1) cn k u n v |
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+ | where |
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+ | yb = (wd+b) `div` c |
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+ | bn = yb*c-b |
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+ | cn = (d-(bn*bn)) `div` c |
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+ | yn | i == 0 = wd |
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+ | | otherwise = yb |
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+ | u = k*yn+l -- u/v is the i-th convergent of sqrt(d) |
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+ | v = n*yn+m |
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+ | (x,y) | odd (i+1) = (u*u+d*v*v, 2*u*v) |
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+ | | otherwise = (u,v) |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=67 Problem 67] == |
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+ | Using an efficient algorithm find the maximal sum in the triangle? |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | problem_67 = readFile "triangle.txt" >>= print . solve . parse |
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+ | parse = map (map read . words) . lines |
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+ | solve = head . foldr1 step |
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+ | step [] [z] = [z] |
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+ | step (x:xs) (y:z:zs) = x + max y z : step xs (z:zs) |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=68 Problem 68] == |
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+ | What is the maximum 16-digit string for a "magic" 5-gon ring? |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | import Data.List |
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+ | permute [] = [[]] |
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+ | permute list = |
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+ | concat $ map (\(x:xs) -> map (x:) (permute xs)) |
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+ | (take (length list) |
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+ | (unfoldr (\x -> Just (x, tail x ++ [head x])) list)) |
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+ | problem_68 = |
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+ | maximum $ map (concat . map show) poel |
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+ | where |
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+ | gon68 = [1..10] |
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+ | knip = (length gon68) `div` 2 |
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+ | (is,es) = splitAt knip gon68 |
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+ | extnodes = map (\x -> [head es]++x) $ permute $ tail es |
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+ | intnodes = map (\(p:ps) -> zipWith (\ x y -> [x]++[y]) |
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+ | (p:ps) (ps++[p])) $ permute is |
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+ | poel = [ concat hs | hs <- [ zipWith (\x y -> [x]++y) uitsteeksels organen | |
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+ | uitsteeksels <- extnodes, organen <- intnodes ], |
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+ | let subsom = map (sum) hs, length (nub subsom) == 1 ] |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=69 Problem 69] == |
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+ | Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | {-phi(n) = n*(1-1/p1)*(1-1/p2)*...*(1-1/pn) |
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+ | n/phi(n) = 1/(1-1/p1)*(1-1/p2)*...*(1-1/pn) |
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+ | (1-1/p) will be minimal for a small p and 1/(1-1/p) will then be maximal |
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+ | -} |
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+ | primes=[2,3,5,7,11,13,17,19,23] |
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+ | problem_69= |
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+ | maximum [c| |
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+ | a<-[1..length primes], |
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+ | let b=take a primes, |
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+ | let c=product b, |
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+ | c<10^6 |
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+ | ] |
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+ | </haskell> |
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+ | |||
+ | Note: credit for arithmetic functions is due to [http://www.polyomino.f2s.com/ David Amos]. |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=70 Problem 70] == |
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+ | Investigate values of n for which φ(n) is a permutation of n. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | import Data.List |
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+ | isPerm a b = (show a) \\ (show b)==[] |
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+ | flsqr n x=x<(floor.sqrt.fromInteger) n |
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+ | pairs n1 = |
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+ | maximum[m|a<-gena ,b<-genb,let m=a*b,n>m,isPerm m$ m-a-b+1] |
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+ | where |
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+ | n=fromInteger n1 |
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+ | gena = dropWhile (flsqr n)$ takeWhile (flsqr (2*n)) primes |
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+ | genb = dropWhile (flsqr (div n 2))$ takeWhile (flsqr n) primes |
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+ | |||
+ | problem_70= pairs (10^7) |
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+ | </haskell> |
Revision as of 21:52, 29 January 2008
Problem 61
Find the sum of the only set of six 4-digit figurate numbers with a cyclic property.
Solution:
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]
Problem 62
Find the smallest cube for which exactly five permutations of its digits are cube.
Solution:
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
Problem 63
How many n-digit positive integers exist which are also an nth power?
Solution:
problem_63=length[x^y|x<-[1..9],y<-[1..22],y==(length$show$x^y)]
Problem 64
How many continued fractions for N ≤ 10000 have an odd period?
Solution:
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)
Problem 65
Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e.
Solution:
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
Problem 66
Investigate the Diophantine equation x2 − Dy2 = 1.
Solution:
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)
Problem 67
Using an efficient algorithm find the maximal sum in the triangle?
Solution:
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)
Problem 68
What is the maximum 16-digit string for a "magic" 5-gon ring?
Solution:
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 ]
Problem 69
Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum.
Solution:
{-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
]
Note: credit for arithmetic functions is due to David Amos.
Problem 70
Investigate values of n for which φ(n) is a permutation of n.
Solution:
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)