Difference between revisions of "Euler problems/71 to 80"
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| + | Do them on your own! |
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| − | == [http://projecteuler.net/index.php?section=view&id=71 Problem 71] == |
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| − | Listing reduced proper fractions in ascending order of size. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | -- http://mathworld.wolfram.com/FareySequence.html |
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| − | import Data.Ratio ((%), numerator,denominator) |
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| − | fareySeq a b |
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| − | |da2<=10^6=fareySeq a1 b |
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| − | |otherwise=na |
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| − | where |
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| − | na=numerator a |
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| − | nb=numerator b |
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| − | da=denominator a |
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| − | db=denominator b |
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| − | a1=(na+nb)%(da+db) |
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| − | da2=denominator a1 |
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| − | problem_71=fareySeq (0%1) (3%7) |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=view&id=72 Problem 72] == |
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| − | How many elements would be contained in the set of reduced proper fractions for d ≤ 1,000,000? |
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| − | |||
| − | Solution: |
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| − | |||
| − | Using the [http://mathworld.wolfram.com/FareySequence.html Farey Sequence] method, the solution is the sum of phi (n) from 1 to 1000000. |
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| − | <haskell> |
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| − | groups=1000 |
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| − | eulerTotient n = product (map (\(p,i) -> p^(i-1) * (p-1)) factors) |
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| − | where factors = fstfac n |
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| − | fstfac x = [(head a ,length a)|a<-group$primeFactors x] |
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| − | p72 n= sum [eulerTotient x|x <- [groups*n+1..groups*(n+1)]] |
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| − | problem_72 = sum [p72 x|x <- [0..999]] |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=view&id=73 Problem 73] == |
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| − | How many fractions lie between 1/3 and 1/2 in a sorted set of reduced proper fractions? |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Data.Array |
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| − | twix k = crude k - fd2 - sum [ar!(k `div` m) | m <- [3 .. k `div` 5], odd m] |
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| − | where |
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| − | fd2 = crude (k `div` 2) |
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| − | ar = array (5,k `div` 3) $ |
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| − | ((5,1):[(j, crude j - sum [ar!(j `div` m) | m <- [2 .. j `div` 5]]) |
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| − | | j <- [6 .. k `div` 3]]) |
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| − | crude j = |
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| − | m*(3*m+r-2) + s |
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| − | where |
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| − | (m,r) = j `divMod` 6 |
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| − | s = case r of |
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| − | 5 -> 1 |
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| − | _ -> 0 |
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| − | |||
| − | problem_73 = twix 10000 |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=view&id=74 Problem 74] == |
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| − | Determine the number of factorial chains that contain exactly sixty non-repeating terms. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Data.List |
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| − | explode 0 = [] |
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| − | explode n = n `mod` 10 : explode (n `quot` 10) |
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| − | |||
| − | chain 2 = 1 |
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| − | chain 1 = 1 |
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| − | chain 145 = 1 |
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| − | chain 40585 = 1 |
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| − | chain 169 = 3 |
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| − | chain 363601 = 3 |
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| − | chain 1454 = 3 |
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| − | chain 871 = 2 |
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| − | chain 45361 = 2 |
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| − | chain 872 = 2 |
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| − | chain 45362 = 2 |
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| − | chain x = 1 + chain (sumFactDigits x) |
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| − | makeIncreas 1 minnum = [[a]|a<-[minnum..9]] |
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| − | makeIncreas digits minnum = [a:b|a<-[minnum ..9],b<-makeIncreas (digits-1) a] |
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| − | p74= |
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| − | sum[div p6 $countNum a| |
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| − | a<-tail$makeIncreas 6 1, |
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| − | let k=digitToN a, |
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| − | chain k==60 |
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| − | ] |
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| − | where |
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| − | p6=facts!! 6 |
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| − | sumFactDigits = foldl' (\a b -> a + facts !! b) 0 . explode |
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| − | factorial n = if n == 0 then 1 else n * factorial (n - 1) |
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| − | digitToN = foldl' (\a b -> 10*a + b) 0 .dropWhile (==0) |
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| − | facts = scanl (*) 1 [1..9] |
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| − | countNum xs=ys |
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| − | where |
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| − | ys=product$map (factorial.length)$group xs |
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| − | problem_74= length[k|k<-[1..9999],chain k==60]+p74 |
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| − | test = print $ [a|a<-tail$makeIncreas 6 0,let k=digitToN a,chain k==60] |
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| − | </haskell> |
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| − | == [http://projecteuler.net/index.php?section=view&id=75 Problem 75] == |
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| − | Find the number of different lengths of wire can that can form a right angle triangle in only one way. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Data.Array |
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| − | |||
| − | triplets = |
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| − | [p | |
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| − | n <- [2..706], |
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| − | m <- [1..n-1], |
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| − | gcd m n == 1, |
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| − | let p = 2 * (n^2 + m*n), |
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| − | odd (m + n), |
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| − | p <= 10^6 |
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| − | ] |
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| − | |||
| − | hist bnds ns = |
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| − | accumArray (+) 0 bnds [(n, 1) | |
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| − | n <- ns, |
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| − | inRange bnds n |
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| − | ] |
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| − | |||
| − | problem_75 = |
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| − | length $ filter (\(_,b) -> b == 1) $ assocs arr |
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| − | where |
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| − | arr = hist (12,10^6) $ concatMap multiples triplets |
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| − | multiples n = takeWhile (<=10^6) [n, 2*n..] |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=view&id=76 Problem 76] == |
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| − | How many different ways can one hundred be written as a sum of at least two positive integers? |
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| − | |||
| − | Solution: |
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| − | |||
| − | Here is a simpler solution: For each n, we create the list of the number of partitions of n |
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| − | whose lowest number is i, for i=1..n. We build up the list of these lists for n=0..100. |
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| − | <haskell> |
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| − | build x = (map sum (zipWith drop [0..] x) ++ [1]) : x |
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| − | problem_76 = (sum $ head $ iterate build [] !! 100) - 1 |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=view&id=77 Problem 77] == |
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| − | What is the first value which can be written as the sum of primes in over five thousand different ways? |
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| − | |||
| − | Solution: |
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| − | |||
| − | Brute force but still finds the solution in less than one second. |
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| − | <haskell> |
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| − | counter = foldl (\without p -> |
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| − | let (poor,rich) = splitAt p without |
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| − | with = poor ++ |
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| − | zipWith (+) with rich |
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| − | in with |
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| − | ) (1 : repeat 0) |
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| − | |||
| − | problem_77 = |
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| − | find ((>5000) . (ways !!)) $ [1..] |
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| − | where |
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| − | ways = counter $ take 100 primes |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=view&id=78 Problem 78] == |
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| − | Investigating the number of ways in which coins can be separated into piles. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Data.Array |
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| − | |||
| − | partitions :: Array Int Integer |
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| − | partitions = |
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| − | array (0,1000000) $ |
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| − | (0,1) : |
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| − | [(n,sum [s * partitions ! p| |
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| − | (s,p) <- zip signs $ parts n])| |
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| − | n <- [1..1000000]] |
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| − | where |
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| − | signs = cycle [1,1,(-1),(-1)] |
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| − | suite = map penta $ concat [[n,(-n)]|n <- [1..]] |
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| − | penta n = n*(3*n - 1) `div` 2 |
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| − | parts n = takeWhile (>= 0) [n-x| x <- suite] |
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| − | |||
| − | problem_78 :: Int |
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| − | problem_78 = |
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| − | head $ filter (\x -> (partitions ! x) `mod` 1000000 == 0) [1..] |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=view&id=79 Problem 79] == |
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| − | By analysing a user's login attempts, can you determine the secret numeric passcode? |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Data.Char (digitToInt, intToDigit) |
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| − | import Data.Graph (buildG, topSort) |
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| − | import Data.List (intersect) |
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| − | |||
| − | p79 file= |
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| − | (+0)$read . intersect graphWalk $ usedDigits |
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| − | where |
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| − | usedDigits = intersect "0123456789" $ file |
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| − | edges = concat . map (edgePair . map digitToInt) . words $ file |
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| − | graphWalk = map intToDigit . topSort . buildG (0, 9) $ edges |
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| − | edgePair [x, y, z] = [(x, y), (y, z)] |
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| − | edgePair _ = undefined |
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| − | |||
| − | problem_79 = do |
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| − | f<-readFile "keylog.txt" |
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| − | print $p79 f |
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| − | </haskell> |
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| − | |||
| − | == [http://projecteuler.net/index.php?section=view&id=80 Problem 80] == |
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| − | Calculating the digital sum of the decimal digits of irrational square roots. |
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| − | |||
| − | Solution: |
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| − | <haskell> |
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| − | import Data.Char |
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| − | problem_80= |
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| − | sum [f x | |
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| − | a <- [1..100], |
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| − | x <- [intSqrt $ a * t], |
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| − | x * x /= a * t |
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| − | ] |
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| − | where |
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| − | t=10^202 |
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| − | f = (sum . take 100 . map (flip (-) (ord '0') .ord) . show) |
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| − | </haskell> |
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Revision as of 21:46, 29 January 2008
Do them on your own!