Difference between revisions of "Euler problems/111 to 120"
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+ | == [http://projecteuler.net/index.php?section=problems&id=111 Problem 111] == |
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− | Do them on your own! |
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+ | Search for 10-digit primes containing the maximum number of repeated digits. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | import Control.Monad (replicateM) |
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+ | |||
+ | -- All ways of interspersing n copies of x into a list |
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+ | intr :: Int -> a -> [a] -> [[a]] |
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+ | intr 0 _ y = [y] |
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+ | intr n x (y:ys) = concat |
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+ | [map ((replicate i x ++) . (y :)) $ intr (n-i) x ys |
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+ | | i <- [0..n]] |
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+ | intr n x _ = [replicate n x] |
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+ | |||
+ | -- All 10-digit primes containing the maximal number of the digit d |
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+ | maxDigits :: Char -> [Integer] |
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+ | maxDigits d = head $ dropWhile null |
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+ | [filter isPrime $ map read $ filter ((/='0') . head) $ |
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+ | concatMap (intr (10-n) d) $ |
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+ | replicateM n $ delete d "0123456789" |
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+ | | n <- [1..9]] |
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+ | |||
+ | problem_111 = sum $ concatMap maxDigits "0123456789" |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=112 Problem 112] == |
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+ | Investigating the density of "bouncy" numbers. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | isIncreasing' n p |
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+ | | n == 0 = True |
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+ | | p >= p1 = isIncreasing' (n `div` 10) p1 |
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+ | | otherwise = False |
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+ | where |
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+ | p1 = n `mod` 10 |
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+ | |||
+ | isIncreasing :: Int -> Bool |
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+ | isIncreasing n = isIncreasing' (n `div` 10) (n `mod` 10) |
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+ | |||
+ | isDecreasing' n p |
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+ | | n == 0 = True |
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+ | | p <= p1 = isDecreasing' (n `div` 10) p1 |
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+ | | otherwise = False |
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+ | where |
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+ | p1 = n `mod` 10 |
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+ | |||
+ | isDecreasing :: Int -> Bool |
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+ | isDecreasing n = isDecreasing' (n `div` 10) (n `mod` 10) |
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+ | |||
+ | isBouncy n = not (isIncreasing n) && not (isDecreasing n) |
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+ | nnn=1500000 |
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+ | num150 =length [x|x<-[1..nnn],isBouncy x] |
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+ | p112 n nb |
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+ | | fromIntegral nnb / fromIntegral n >= 0.99 = n |
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+ | | otherwise = prob112' (n+1) nnb |
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+ | where |
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+ | nnb = if isBouncy n then nb + 1 else nb |
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+ | |||
+ | problem_112=p112 (nnn+1) num150 |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=113 Problem 113] == |
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+ | How many numbers below a googol (10100) are not "bouncy"? |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | import Array |
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+ | |||
+ | mkArray b f = listArray b $ map f (range b) |
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+ | |||
+ | digits = 100 |
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+ | |||
+ | inc = mkArray ((1, 0), (digits, 9)) ninc |
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+ | dec = mkArray ((1, 0), (digits, 9)) ndec |
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+ | |||
+ | ninc (1, _) = 1 |
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+ | ninc (l, d) = sum [inc ! (l-1, i) | i <- [d..9]] |
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+ | |||
+ | ndec (1, _) = 1 |
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+ | ndec (l, d) = sum [dec ! (l-1, i) | i <- [0..d]] |
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+ | |||
+ | problem_113 = sum [inc ! i | i <- range ((digits, 0), (digits, 9))] |
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+ | + sum [dec ! i | i <- range ((1, 1), (digits, 9))] |
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+ | - digits*9 -- numbers like 11111 are counted in both inc and dec |
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+ | - 1 -- 0 is included in the increasing numbers |
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+ | </haskell> |
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+ | Note: inc and dec contain the same data, but it seems clearer to duplicate them. |
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+ | |||
+ | it is another way to solution this problem: |
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+ | <haskell> |
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+ | binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y)) |
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+ | prodxy x y=product[x..y] |
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+ | problem_113=sum[binomial (8+a) a+binomial (9+a) a-10|a<-[1..100]] |
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+ | </haskell> |
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+ | == [http://projecteuler.net/index.php?section=problems&id=114 Problem 114] == |
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+ | Investigating the number of ways to fill a row with separated blocks that are at least three units long. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | -- fun in p115 |
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+ | problem_114=fun 3 50 |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=115 Problem 115] == |
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+ | Finding a generalisation for the number of ways to fill a row with separated blocks. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y)) |
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+ | prodxy x y=product[x..y] |
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+ | fun m n=sum[binomial (k+a) (k-a)|a<-[0..div (n+1) (m+1)],let k=1-a*m+n] |
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+ | problem_115 = (+1)$length$takeWhile (<10^6) [fun 50 i|i<-[1..]] |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=116 Problem 116] == |
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+ | Investigating the number of ways of replacing square tiles with one of three coloured tiles. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y)) |
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+ | prodxy x y=product[x..y] |
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+ | f116 n x=sum[binomial (a+b) a|a<-[1..div n x],let b=n-a*x] |
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+ | p116 x=sum[f116 x a|a<-[2..4]] |
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+ | problem_116 = p116 50 |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=117 Problem 117] == |
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+ | Investigating the number of ways of tiling a row using different-sized tiles. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | fibs5 = 0 : 0 :1: 1:zipWith4 (\a b c d->a+b+c+d) fibs5 a1 a2 a3 |
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+ | where |
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+ | a1=tail fibs5 |
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+ | a2=tail a1 |
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+ | a3=tail a2 |
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+ | p117 x=fibs5!!(x+2) |
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+ | problem_117 = p117 50 |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=118 Problem 118] == |
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+ | Exploring the number of ways in which sets containing prime elements can be made. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | digits = ['1'..'9'] |
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+ | |||
+ | -- possible partitions voor prime number sets |
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+ | -- leave out patitions with more than 4 1's |
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+ | -- because only {2,3,5,7,..} is possible |
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+ | -- and the [9]-partition because every permutation of all |
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+ | -- nine digits is divisable by 3 |
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+ | test xs |
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+ | |len>4=False |
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+ | |xs==[9]=False |
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+ | |otherwise=True |
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+ | where |
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+ | len=length $filter (==1) xs |
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+ | parts = filter test $partitions 9 |
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+ | permutationsOf [] = [[]] |
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+ | permutationsOf xs = [x:xs' | x <- xs, xs' <- permutationsOf (delete x xs)] |
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+ | combinationsOf 0 _ = [[]] |
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+ | combinationsOf _ [] = [] |
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+ | combinationsOf k (x:xs) = |
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+ | map (x:) (combinationsOf (k-1) xs) ++ combinationsOf k xs |
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+ | |||
+ | priemPerms [] = 0 |
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+ | priemPerms ds = |
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+ | fromIntegral . length . filter (isPrime . read) . permutationsOf $ ds |
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+ | setsums [] 0 = [[]] |
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+ | setsums [] _ = [] |
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+ | setsums (x:xs) n |
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+ | | x > n = setsums xs n |
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+ | | otherwise = map (x:) (setsums (x:xs) (n-x)) ++ setsums xs n |
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+ | |||
+ | partitions n = setsums (reverse [1..n]) n |
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+ | |||
+ | fc :: [Integer] -> [Char] -> Integer |
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+ | fc (p:[]) ds = priemPerms ds |
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+ | fc (p:ps) ds = |
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+ | foldl fcmul 0 . combinationsOf p $ ds |
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+ | where |
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+ | fcmul x y |
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+ | | np y == 0 = x |
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+ | | otherwise = x + np y * fc ps (ds \\ y) |
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+ | where |
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+ | np = priemPerms |
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+ | -- here is the 'imperfection' correction method: |
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+ | -- make use of duplicate reducing factors for partitions |
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+ | -- with repeating factors, f.i. [1,1,1,1,2,3]: |
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+ | -- in this case 4 1's -> factor = 4! |
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+ | -- or for [1,1,1,3,3] : factor = 3! * 2! |
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+ | dupF :: [Integer] -> Integer |
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+ | dupF = foldl (\ x y -> x * product [1..y]) 1 . map (fromIntegral . length) . group |
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+ | |||
+ | main = do |
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+ | print . sum . map (\x -> fc x digits `div` dupF x) $ parts |
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+ | problem_118 = main |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=119 Problem 119] == |
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+ | Investigating the numbers which are equal to sum of their digits raised to some power. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | import Data.List |
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+ | digits n |
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+ | {- 123->[3,2,1] |
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+ | -} |
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+ | |n<10=[n] |
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+ | |otherwise= y:digits x |
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+ | where |
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+ | (x,y)=divMod n 10 |
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+ | problem_119 =sort [(a^b)| |
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+ | a<-[2..200], |
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+ | b<-[2..9], |
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+ | let m=a^b, |
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+ | let n=sum$digits m, |
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+ | n==a]!!29 |
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+ | </haskell> |
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+ | |||
+ | == [http://projecteuler.net/index.php?section=problems&id=120 Problem 120] == |
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+ | Finding the maximum remainder when (a − 1)n + (a + 1)n is divided by a2. |
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+ | |||
+ | Solution: |
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+ | <haskell> |
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+ | fun m=div (m*(8*m^2-3*m-5)) 3 |
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+ | problem_120 = fun 500 |
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+ | </haskell> |
Revision as of 04:59, 30 January 2008
Problem 111
Search for 10-digit primes containing the maximum number of repeated digits.
Solution:
import Control.Monad (replicateM)
-- All ways of interspersing n copies of x into a list
intr :: Int -> a -> [a] -> [[a]]
intr 0 _ y = [y]
intr n x (y:ys) = concat
[map ((replicate i x ++) . (y :)) $ intr (n-i) x ys
| i <- [0..n]]
intr n x _ = [replicate n x]
-- All 10-digit primes containing the maximal number of the digit d
maxDigits :: Char -> [Integer]
maxDigits d = head $ dropWhile null
[filter isPrime $ map read $ filter ((/='0') . head) $
concatMap (intr (10-n) d) $
replicateM n $ delete d "0123456789"
| n <- [1..9]]
problem_111 = sum $ concatMap maxDigits "0123456789"
Problem 112
Investigating the density of "bouncy" numbers.
Solution:
isIncreasing' n p
| n == 0 = True
| p >= p1 = isIncreasing' (n `div` 10) p1
| otherwise = False
where
p1 = n `mod` 10
isIncreasing :: Int -> Bool
isIncreasing n = isIncreasing' (n `div` 10) (n `mod` 10)
isDecreasing' n p
| n == 0 = True
| p <= p1 = isDecreasing' (n `div` 10) p1
| otherwise = False
where
p1 = n `mod` 10
isDecreasing :: Int -> Bool
isDecreasing n = isDecreasing' (n `div` 10) (n `mod` 10)
isBouncy n = not (isIncreasing n) && not (isDecreasing n)
nnn=1500000
num150 =length [x|x<-[1..nnn],isBouncy x]
p112 n nb
| fromIntegral nnb / fromIntegral n >= 0.99 = n
| otherwise = prob112' (n+1) nnb
where
nnb = if isBouncy n then nb + 1 else nb
problem_112=p112 (nnn+1) num150
Problem 113
How many numbers below a googol (10100) are not "bouncy"?
Solution:
import Array
mkArray b f = listArray b $ map f (range b)
digits = 100
inc = mkArray ((1, 0), (digits, 9)) ninc
dec = mkArray ((1, 0), (digits, 9)) ndec
ninc (1, _) = 1
ninc (l, d) = sum [inc ! (l-1, i) | i <- [d..9]]
ndec (1, _) = 1
ndec (l, d) = sum [dec ! (l-1, i) | i <- [0..d]]
problem_113 = sum [inc ! i | i <- range ((digits, 0), (digits, 9))]
+ sum [dec ! i | i <- range ((1, 1), (digits, 9))]
- digits*9 -- numbers like 11111 are counted in both inc and dec
- 1 -- 0 is included in the increasing numbers
Note: inc and dec contain the same data, but it seems clearer to duplicate them.
it is another way to solution this problem:
binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y))
prodxy x y=product[x..y]
problem_113=sum[binomial (8+a) a+binomial (9+a) a-10|a<-[1..100]]
Problem 114
Investigating the number of ways to fill a row with separated blocks that are at least three units long.
Solution:
-- fun in p115
problem_114=fun 3 50
Problem 115
Finding a generalisation for the number of ways to fill a row with separated blocks.
Solution:
binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y))
prodxy x y=product[x..y]
fun m n=sum[binomial (k+a) (k-a)|a<-[0..div (n+1) (m+1)],let k=1-a*m+n]
problem_115 = (+1)$length$takeWhile (<10^6) [fun 50 i|i<-[1..]]
Problem 116
Investigating the number of ways of replacing square tiles with one of three coloured tiles.
Solution:
binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y))
prodxy x y=product[x..y]
f116 n x=sum[binomial (a+b) a|a<-[1..div n x],let b=n-a*x]
p116 x=sum[f116 x a|a<-[2..4]]
problem_116 = p116 50
Problem 117
Investigating the number of ways of tiling a row using different-sized tiles.
Solution:
fibs5 = 0 : 0 :1: 1:zipWith4 (\a b c d->a+b+c+d) fibs5 a1 a2 a3
where
a1=tail fibs5
a2=tail a1
a3=tail a2
p117 x=fibs5!!(x+2)
problem_117 = p117 50
Problem 118
Exploring the number of ways in which sets containing prime elements can be made.
Solution:
digits = ['1'..'9']
-- possible partitions voor prime number sets
-- leave out patitions with more than 4 1's
-- because only {2,3,5,7,..} is possible
-- and the [9]-partition because every permutation of all
-- nine digits is divisable by 3
test xs
|len>4=False
|xs==[9]=False
|otherwise=True
where
len=length $filter (==1) xs
parts = filter test $partitions 9
permutationsOf [] = [[]]
permutationsOf xs = [x:xs' | x <- xs, xs' <- permutationsOf (delete x xs)]
combinationsOf 0 _ = [[]]
combinationsOf _ [] = []
combinationsOf k (x:xs) =
map (x:) (combinationsOf (k-1) xs) ++ combinationsOf k xs
priemPerms [] = 0
priemPerms ds =
fromIntegral . length . filter (isPrime . read) . permutationsOf $ ds
setsums [] 0 = [[]]
setsums [] _ = []
setsums (x:xs) n
| x > n = setsums xs n
| otherwise = map (x:) (setsums (x:xs) (n-x)) ++ setsums xs n
partitions n = setsums (reverse [1..n]) n
fc :: [Integer] -> [Char] -> Integer
fc (p:[]) ds = priemPerms ds
fc (p:ps) ds =
foldl fcmul 0 . combinationsOf p $ ds
where
fcmul x y
| np y == 0 = x
| otherwise = x + np y * fc ps (ds \\ y)
where
np = priemPerms
-- here is the 'imperfection' correction method:
-- make use of duplicate reducing factors for partitions
-- with repeating factors, f.i. [1,1,1,1,2,3]:
-- in this case 4 1's -> factor = 4!
-- or for [1,1,1,3,3] : factor = 3! * 2!
dupF :: [Integer] -> Integer
dupF = foldl (\ x y -> x * product [1..y]) 1 . map (fromIntegral . length) . group
main = do
print . sum . map (\x -> fc x digits `div` dupF x) $ parts
problem_118 = main
Problem 119
Investigating the numbers which are equal to sum of their digits raised to some power.
Solution:
import Data.List
digits n
{- 123->[3,2,1]
-}
|n<10=[n]
|otherwise= y:digits x
where
(x,y)=divMod n 10
problem_119 =sort [(a^b)|
a<-[2..200],
b<-[2..9],
let m=a^b,
let n=sum$digits m,
n==a]!!29
Problem 120
Finding the maximum remainder when (a − 1)n + (a + 1)n is divided by a2.
Solution:
fun m=div (m*(8*m^2-3*m-5)) 3
problem_120 = fun 500