Difference between revisions of "Euler problems/91 to 100"
(Euler problem 91) |
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+ | == [http://projecteuler.net/index.php?section=problems&id=91 Problem 91] == |
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− | [[Category:Programming exercise spoilers]] |
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− | == [http://projecteuler.net/index.php?section=view&id=91 Problem 91] == |
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Find the number of right angle triangles in the quadrant. |
Find the number of right angle triangles in the quadrant. |
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Line 8: | Line 7: | ||
where d = gcd x y |
where d = gcd x y |
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− | problem_91 n = |
+ | problem_91 n = |
+ | 3*n*n + 2* sum others |
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− | where |
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− | + | where |
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− | + | others =[min xc yc| |
|
− | + | x1 <- [1..n], |
|
− | + | y1 <- [1..n], |
|
− | let |
+ | let (yi,xi) = reduce x1 y1, |
− | let |
+ | let yc = quot (n-y1) yi, |
− | + | let xc = quot x1 xi |
|
+ | ] |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=92 Problem 92] == |
Investigating a square digits number chain with a surprising property. |
Investigating a square digits number chain with a surprising property. |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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+ | import Data.Array |
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− | problem_92 = undefined |
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+ | import Data.Char |
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+ | import Data.List |
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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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+ | squares :: Array Char Int |
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+ | squares = array ('0','9') [ (intToDigit x,x^2) | x <- [0..9] ] |
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+ | |||
+ | next :: Int -> Int |
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+ | next = sum . map (squares !) . show |
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+ | factorial n = if n == 0 then 1 else n * factorial (n - 1) |
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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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+ | yield :: Int -> Int |
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+ | yield = until (\x -> x == 89 || x == 1) next |
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+ | problem_92= |
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+ | sum[div p7 $countNum a| |
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+ | a<-tail$makeIncreas 7 0, |
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+ | let k=sum $map (^2) a, |
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+ | yield k==89 |
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+ | ] |
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+ | where |
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+ | p7=factorial 7 |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=93 Problem 93] == |
Using four distinct digits and the rules of arithmetic, find the longest sequence of target numbers. |
Using four distinct digits and the rules of arithmetic, find the longest sequence of target numbers. |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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+ | import Data.List |
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− | problem_93 = undefined |
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+ | import Control.Monad |
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+ | import Data.Ord (comparing) |
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+ | |||
+ | solve [] [x] = [x] |
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+ | solve ns stack = |
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+ | pushes ++ ops |
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+ | where |
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+ | pushes = do |
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+ | x <- ns |
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+ | solve (x `delete` ns) (x:stack) |
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+ | ops = do |
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+ | guard (length stack > 1) |
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+ | x <- opResults (stack!!0) (stack!!1) |
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+ | solve ns (x : drop 2 stack) |
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+ | |||
+ | opResults a b = |
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+ | [a*b,a+b,a-b] ++ (if b /= 0 then [a / b] else []) |
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+ | |||
+ | results xs = fun 1 ys |
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+ | where |
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+ | ys = nub $ sort $ map truncate $ |
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+ | filter (\x -> x > 0 && floor x == ceiling x) $ solve xs [] |
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+ | fun n (x:xs) |
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+ | |n == x =fun (n+1) xs |
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+ | |otherwise=n-1 |
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+ | |||
+ | cmp = comparing results |
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+ | |||
+ | main = |
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+ | appendFile "p93.log" $ show $ |
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+ | maximumBy cmp $ [[a,b,c,d] | |
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+ | a <- [1..10], |
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+ | b <- [a+1..10], |
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+ | c <- [b+1..10], |
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+ | d <- [c+1..10] |
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+ | ] |
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+ | problem_93 = main |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=94 Problem 94] == |
Investigating almost equilateral triangles with integral sides and area. |
Investigating almost equilateral triangles with integral sides and area. |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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+ | import List |
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− | problem_94 = undefined |
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+ | findmin d = d:head [[n,m]|m<-[1..10],n<-[1..10],n*n==d*m*m+1] |
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+ | pow 1 x=x |
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+ | pow n x =mult x $pow (n-1) x |
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+ | where |
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+ | mult [d,a, b] [_,a1, b1]=d:[a*a1+d*b*b1,a*b1+b*a1] |
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+ | --find it looks like (5-5-6) |
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+ | f556 =takeWhile (<10^9) |
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+ | [n2|i<-[1..], |
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+ | let [_,m,_]=pow i$findmin 12, |
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+ | let n=div (m-1) 6, |
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+ | let n1=4*n+1, -- sides |
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+ | let n2=3*n1+1 -- perimeter |
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+ | ] |
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+ | --find it looks like (5-6-6) |
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+ | f665 =takeWhile (<10^9) |
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+ | [n2|i<-[1..], |
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+ | let [_,m,_]=pow i$findmin 3, |
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+ | mod (m-2) 3==0, |
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+ | let n=div (m-2) 3, |
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+ | let n1=2*n, |
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+ | let n2=3*n1+2 |
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+ | ] |
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+ | problem_94=sum f556+sum f665-2 |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=95 Problem 95] == |
Find the smallest member of the longest amicable chain with no element exceeding one million. |
Find the smallest member of the longest amicable chain with no element exceeding one million. |
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+ | Here is a more straightforward solution, without optimization. |
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+ | Yet it solves the problem in a few seconds when |
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+ | compiled with GHC 6.6.1 with the -O2 flag. I like to let |
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+ | the compiler do the optimization, without cluttering my code. |
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+ | This solution avoids using unboxed arrays, which many consider to be |
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− | Solution which avoid visiting a number more than one time : |
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+ | somewhat of an imperitive-style hack. In fact, no memoization |
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− | <haskell> |
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+ | at all is required. |
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− | import Data.Array.Unboxed |
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− | import qualified Data.IntSet as S |
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− | import Data.List |
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+ | <haskell> |
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− | takeUntil _ [] = [] |
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+ | import Data.List (foldl1', group) |
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− | takeUntil pred (x:xs) = x : if pred x then takeUntil pred xs else [] |
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+ | |||
− | |||
+ | |||
− | chain n s = lgo [n] $ properDivisorsSum ! n |
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+ | -- The longest chain of numbers is (n, k), where |
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− | where lgo xs x | x > 1000000 || S.notMember x s = (xs,[]) |
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+ | -- n is the smallest number in the chain, and k is the length |
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− | | x `elem` xs = (xs,x : takeUntil (/= x) xs) |
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+ | -- of the chain. We limit the search to chains whose |
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− | | otherwise = lgo (x:xs) $ properDivisorsSum ! x |
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+ | -- smallest number is no more than m and, optionally, whose |
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− | |||
+ | -- largest number is no more than m'. |
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− | properDivisorsSum :: UArray Int Int |
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+ | chain s n n' |
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− | properDivisorsSum = accumArray (+) 1 (0,1000000) |
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− | + | | n' == n = s |
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− | + | | n' < n = [] |
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+ | | (< n') 1000000 = [] |
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− | , k<-[2*factor,2*factor+factor..1000000] |
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− | + | | n' `elem` s = [] |
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+ | | otherwise = chain(n' : s) n $ eulerTotient n' |
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− | |||
+ | findChain n = length$chain [] n $ eulerTotient n |
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− | base = S.fromList [1..1000000] |
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+ | longestChain = |
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− | |||
+ | foldl1' cmpChain [(n, findChain n) | n <- [12496..15000]] |
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− | problem_95 = fst $ until (S.null . snd) f ((0,0),base) |
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− | where |
+ | where |
− | + | cmpChain p@(n, k) q@(n', k') |
|
− | + | | (k, negate n) < (k', negate n') = q |
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− | + | | otherwise = p |
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+ | problem_95 = fst $ longestChain |
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− | (explored, chn) = chain setMin s |
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− | len = length chn |
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− | p' = if len > m then (minimum chn, len) else p |
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− | s' = foldl' (flip S.delete) s explored |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=96 Problem 96] == |
Devise an algorithm for solving Su Doku puzzles. |
Devise an algorithm for solving Su Doku puzzles. |
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+ | See numerous solutions on the [[Sudoku]] page. |
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− | Solution: |
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<haskell> |
<haskell> |
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+ | import Data.List |
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− | problem_96 = undefined |
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+ | import Char |
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− | </haskell> |
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+ | |||
+ | top3 :: Grid -> Int |
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+ | top3 g = |
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+ | read . take 3 $ (g !! 0) |
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+ | type Grid = [String] |
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− | == [http://projecteuler.net/index.php?section=view&id=97 Problem 97] == |
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+ | type Row = String |
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+ | type Col = String |
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+ | type Cell = String |
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+ | type Pos = Int |
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+ | |||
+ | row :: Grid -> Pos -> Row |
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+ | row [] _ = [] |
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+ | row g p = filter (/='0') (g !! (p `div` 9)) |
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+ | |||
+ | col :: Grid -> Pos -> Col |
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+ | col [] _ = [] |
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+ | col g p = filter (/='0') ((transpose g) !! (p `mod` 9)) |
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+ | |||
+ | cell :: Grid -> Pos -> Cell |
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+ | cell [] _ = [] |
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+ | cell g p = |
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+ | concat rows |
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+ | where |
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+ | r = p `div` 9 `div` 3 * 3 |
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+ | c = p `mod` 9 `div` 3 * 3 |
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+ | rows = |
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+ | map (take 3 . drop c) . map (g !!) $ [r, r+1, r+2] |
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+ | |||
+ | groupsOf _ [] = [] |
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+ | groupsOf n xs = |
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+ | front : groupsOf n back |
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+ | where |
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+ | (front,back) = splitAt n xs |
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+ | |||
+ | extrapolate :: Grid -> [Grid] |
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+ | extrapolate [] = [] |
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+ | extrapolate g = |
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+ | if null zeroes |
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+ | then [] -- no more zeroes, must have solved it |
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+ | else map mkGrid possibilities |
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+ | where |
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+ | flat = concat g |
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+ | numbered = zip [0..] flat |
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+ | zeroes = filter ((=='0') . snd) numbered |
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+ | p = fst . head $ zeroes |
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+ | possibilities = |
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+ | ['1'..'9'] \\ (row g p ++ col g p ++ cell g p) |
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+ | (front,_:back) = splitAt p flat |
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+ | mkGrid new = groupsOf 9 (front ++ [new] ++ back) |
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+ | |||
+ | loop :: [Grid] -> [Grid] |
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+ | loop = concatMap extrapolate |
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+ | |||
+ | solve :: Grid -> Grid |
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+ | solve g = |
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+ | head . |
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+ | last . |
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+ | takeWhile (not . null) . |
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+ | iterate loop $ [g] |
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+ | |||
+ | main = do |
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+ | contents <- readFile "sudoku.txt" |
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+ | let |
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+ | grids :: [Grid] |
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+ | grids = |
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+ | groupsOf 9 . |
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+ | filter ((/='G') . head) . |
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+ | lines $ contents |
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+ | let rgrids=map (concatMap words) grids |
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+ | writeFile "p96.log"$show$ sum $ map (top3 . solve) $ rgrids |
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+ | problem_96 =main |
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+ | </haskell> |
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+ | == [http://projecteuler.net/index.php?section=problems&id=97 Problem 97] == |
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Find the last ten digits of the non-Mersenne prime: 28433 × 2<sup>7830457</sup> + 1. |
Find the last ten digits of the non-Mersenne prime: 28433 × 2<sup>7830457</sup> + 1. |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | problem_97 = |
+ | problem_97 = |
+ | flip mod limit $ 28433 * powMod limit 2 7830457 + 1 |
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+ | where |
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+ | limit=10^10 |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=98 Problem 98] == |
Investigating words, and their anagrams, which can represent square numbers. |
Investigating words, and their anagrams, which can represent square numbers. |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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+ | import Data.List |
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− | problem_98 = undefined |
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+ | import Data.Maybe |
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+ | import Data.Function (on) |
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+ | |||
+ | -- Replace each letter of a word, or digit of a number, with |
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+ | -- the index of where that letter or digit first appears |
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+ | profile :: Ord a => [a] -> [Int] |
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+ | profile x = map (fromJust . flip lookup (indices x)) x |
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+ | where |
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+ | indices = map head . groupBy fstEq . sort . flip zip [0..] |
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+ | |||
+ | -- Check for equality on the first component of a tuple |
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+ | fstEq :: Eq a => (a, b) -> (a, b) -> Bool |
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+ | fstEq = (==) `on` fst |
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+ | |||
+ | -- The histogram of a small list |
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+ | hist :: Ord a => [a] -> [(a, Int)] |
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+ | hist = let item g = (head g, length g) in map item . group . sort |
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+ | |||
+ | -- The list of anagram sets for a word list. |
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+ | anagrams :: Ord a => [[a]] -> [[[a]]] |
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+ | anagrams x = map (map snd) $ filter (not . null . drop 1) $ |
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+ | groupBy fstEq $ sort $ zip (map hist x) x |
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+ | |||
+ | -- Given two finite lists that are a permutation of one |
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+ | -- another, return the permutation function |
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+ | mkPermute :: Ord a => [a] -> [a] -> ([b] -> [b]) |
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+ | mkPermute x y = pairsToPermute $ concat $ |
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+ | zipWith zip (occurs x) (occurs y) |
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+ | where |
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+ | pairsToPermute ps = flip map (map snd $ sort ps) . (!!) |
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+ | occurs = map (map snd) . groupBy fstEq . sort . flip zip [0..] |
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+ | |||
+ | problem_98 :: [String] -> Int |
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+ | problem_98 ws = read $ head |
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+ | [y | was <- sortBy longFirst $ anagrams ws, -- word anagram sets |
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+ | w1:t <- tails was, w2 <- t, |
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+ | let permute = mkPermute w1 w2, |
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+ | nas <- sortBy longFirst $ anagrams $ |
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+ | filter ((== profile w1) . profile) $ |
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+ | dropWhile (flip longerThan w1) $ |
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+ | takeWhile (not . longerThan w1) $ |
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+ | map show $ map (\x -> x * x) [1..], -- number anagram sets |
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+ | x:t <- tails nas, y <- t, |
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+ | permute x == y || permute y == x |
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+ | ] |
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+ | |||
+ | run_problem_98 :: IO Int |
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+ | run_problem_98 = do |
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+ | words_file <- readFile "words.txt" |
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+ | let words = read $ '[' : words_file ++ "]" |
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+ | return $ problem_98 words |
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+ | |||
+ | -- Sort on length of first element, from longest to shortest |
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+ | longFirst :: [[a]] -> [[a]] -> Ordering |
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+ | longFirst = flip compareLen `on` fst |
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+ | |||
+ | -- Is y longer than x? |
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+ | longerThan :: [a] -> [a] -> Bool |
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+ | longerThan x y = compareLen x y == LT |
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+ | |||
+ | -- Compare the lengths of lists, with short-circuiting |
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+ | compareLen :: [a] -> [a] -> Ordering |
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+ | compareLen (_:xs) (_:ys) = compareLen xs ys |
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+ | compareLen (_:_) [] = GT |
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+ | compareLen [] [] = EQ |
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+ | compareLen [] (_:_) = LT |
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</haskell> |
</haskell> |
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+ | (Cf. [[short-circuiting]]) |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=99 Problem 99] == |
Which base/exponent pair in the file has the greatest numerical value? |
Which base/exponent pair in the file has the greatest numerical value? |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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+ | import Data.List |
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− | problem_99 = undefined |
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+ | lognum (b,e) = e * log b |
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+ | logfun x = lognum . read $ "(" ++ x ++ ")" |
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+ | problem_99 = snd . maximum . flip zip [1..] . map logfun . lines |
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+ | main = readFile "base_exp.txt" >>= print . problem_99 |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=100 Problem 100] == |
Finding the number of blue discs for which there is 50% chance of taking two blue. |
Finding the number of blue discs for which there is 50% chance of taking two blue. |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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+ | nextAB a b |
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− | problem_100 = undefined |
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+ | |a+b>10^12 =[a,b] |
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+ | |otherwise=nextAB (3*a+2*b+2) (4*a+3*b+3) |
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+ | problem_100=(+1)$head$nextAB 14 20 |
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</haskell> |
</haskell> |
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− | |||
− | [[Category:Tutorials]] |
||
− | [[Category:Code]] |
Latest revision as of 20:08, 21 February 2010
Problem 91
Find the number of right angle triangles in the quadrant.
Solution:
reduce x y = (quot x d, quot y d)
where d = gcd x y
problem_91 n =
3*n*n + 2* sum others
where
others =[min xc yc|
x1 <- [1..n],
y1 <- [1..n],
let (yi,xi) = reduce x1 y1,
let yc = quot (n-y1) yi,
let xc = quot x1 xi
]
Problem 92
Investigating a square digits number chain with a surprising property.
Solution:
import Data.Array
import Data.Char
import Data.List
makeIncreas 1 minnum = [[a]|a<-[minnum..9]]
makeIncreas digits minnum = [a:b|a<-[minnum ..9],b<-makeIncreas (digits-1) a]
squares :: Array Char Int
squares = array ('0','9') [ (intToDigit x,x^2) | x <- [0..9] ]
next :: Int -> Int
next = sum . map (squares !) . show
factorial n = if n == 0 then 1 else n * factorial (n - 1)
countNum xs=ys
where
ys=product$map (factorial.length)$group xs
yield :: Int -> Int
yield = until (\x -> x == 89 || x == 1) next
problem_92=
sum[div p7 $countNum a|
a<-tail$makeIncreas 7 0,
let k=sum $map (^2) a,
yield k==89
]
where
p7=factorial 7
Problem 93
Using four distinct digits and the rules of arithmetic, find the longest sequence of target numbers.
Solution:
import Data.List
import Control.Monad
import Data.Ord (comparing)
solve [] [x] = [x]
solve ns stack =
pushes ++ ops
where
pushes = do
x <- ns
solve (x `delete` ns) (x:stack)
ops = do
guard (length stack > 1)
x <- opResults (stack!!0) (stack!!1)
solve ns (x : drop 2 stack)
opResults a b =
[a*b,a+b,a-b] ++ (if b /= 0 then [a / b] else [])
results xs = fun 1 ys
where
ys = nub $ sort $ map truncate $
filter (\x -> x > 0 && floor x == ceiling x) $ solve xs []
fun n (x:xs)
|n == x =fun (n+1) xs
|otherwise=n-1
cmp = comparing results
main =
appendFile "p93.log" $ show $
maximumBy cmp $ [[a,b,c,d] |
a <- [1..10],
b <- [a+1..10],
c <- [b+1..10],
d <- [c+1..10]
]
problem_93 = main
Problem 94
Investigating almost equilateral triangles with integral sides and area.
Solution:
import List
findmin d = d:head [[n,m]|m<-[1..10],n<-[1..10],n*n==d*m*m+1]
pow 1 x=x
pow n x =mult x $pow (n-1) x
where
mult [d,a, b] [_,a1, b1]=d:[a*a1+d*b*b1,a*b1+b*a1]
--find it looks like (5-5-6)
f556 =takeWhile (<10^9)
[n2|i<-[1..],
let [_,m,_]=pow i$findmin 12,
let n=div (m-1) 6,
let n1=4*n+1, -- sides
let n2=3*n1+1 -- perimeter
]
--find it looks like (5-6-6)
f665 =takeWhile (<10^9)
[n2|i<-[1..],
let [_,m,_]=pow i$findmin 3,
mod (m-2) 3==0,
let n=div (m-2) 3,
let n1=2*n,
let n2=3*n1+2
]
problem_94=sum f556+sum f665-2
Problem 95
Find the smallest member of the longest amicable chain with no element exceeding one million. Here is a more straightforward solution, without optimization. Yet it solves the problem in a few seconds when compiled with GHC 6.6.1 with the -O2 flag. I like to let the compiler do the optimization, without cluttering my code.
This solution avoids using unboxed arrays, which many consider to be somewhat of an imperitive-style hack. In fact, no memoization at all is required.
import Data.List (foldl1', group)
-- The longest chain of numbers is (n, k), where
-- n is the smallest number in the chain, and k is the length
-- of the chain. We limit the search to chains whose
-- smallest number is no more than m and, optionally, whose
-- largest number is no more than m'.
chain s n n'
| n' == n = s
| n' < n = []
| (< n') 1000000 = []
| n' `elem` s = []
| otherwise = chain(n' : s) n $ eulerTotient n'
findChain n = length$chain [] n $ eulerTotient n
longestChain =
foldl1' cmpChain [(n, findChain n) | n <- [12496..15000]]
where
cmpChain p@(n, k) q@(n', k')
| (k, negate n) < (k', negate n') = q
| otherwise = p
problem_95 = fst $ longestChain
Problem 96
Devise an algorithm for solving Su Doku puzzles.
See numerous solutions on the Sudoku page.
import Data.List
import Char
top3 :: Grid -> Int
top3 g =
read . take 3 $ (g !! 0)
type Grid = [String]
type Row = String
type Col = String
type Cell = String
type Pos = Int
row :: Grid -> Pos -> Row
row [] _ = []
row g p = filter (/='0') (g !! (p `div` 9))
col :: Grid -> Pos -> Col
col [] _ = []
col g p = filter (/='0') ((transpose g) !! (p `mod` 9))
cell :: Grid -> Pos -> Cell
cell [] _ = []
cell g p =
concat rows
where
r = p `div` 9 `div` 3 * 3
c = p `mod` 9 `div` 3 * 3
rows =
map (take 3 . drop c) . map (g !!) $ [r, r+1, r+2]
groupsOf _ [] = []
groupsOf n xs =
front : groupsOf n back
where
(front,back) = splitAt n xs
extrapolate :: Grid -> [Grid]
extrapolate [] = []
extrapolate g =
if null zeroes
then [] -- no more zeroes, must have solved it
else map mkGrid possibilities
where
flat = concat g
numbered = zip [0..] flat
zeroes = filter ((=='0') . snd) numbered
p = fst . head $ zeroes
possibilities =
['1'..'9'] \\ (row g p ++ col g p ++ cell g p)
(front,_:back) = splitAt p flat
mkGrid new = groupsOf 9 (front ++ [new] ++ back)
loop :: [Grid] -> [Grid]
loop = concatMap extrapolate
solve :: Grid -> Grid
solve g =
head .
last .
takeWhile (not . null) .
iterate loop $ [g]
main = do
contents <- readFile "sudoku.txt"
let
grids :: [Grid]
grids =
groupsOf 9 .
filter ((/='G') . head) .
lines $ contents
let rgrids=map (concatMap words) grids
writeFile "p96.log"$show$ sum $ map (top3 . solve) $ rgrids
problem_96 =main
Problem 97
Find the last ten digits of the non-Mersenne prime: 28433 × 27830457 + 1.
Solution:
problem_97 =
flip mod limit $ 28433 * powMod limit 2 7830457 + 1
where
limit=10^10
Problem 98
Investigating words, and their anagrams, which can represent square numbers.
Solution:
import Data.List
import Data.Maybe
import Data.Function (on)
-- Replace each letter of a word, or digit of a number, with
-- the index of where that letter or digit first appears
profile :: Ord a => [a] -> [Int]
profile x = map (fromJust . flip lookup (indices x)) x
where
indices = map head . groupBy fstEq . sort . flip zip [0..]
-- Check for equality on the first component of a tuple
fstEq :: Eq a => (a, b) -> (a, b) -> Bool
fstEq = (==) `on` fst
-- The histogram of a small list
hist :: Ord a => [a] -> [(a, Int)]
hist = let item g = (head g, length g) in map item . group . sort
-- The list of anagram sets for a word list.
anagrams :: Ord a => [[a]] -> [[[a]]]
anagrams x = map (map snd) $ filter (not . null . drop 1) $
groupBy fstEq $ sort $ zip (map hist x) x
-- Given two finite lists that are a permutation of one
-- another, return the permutation function
mkPermute :: Ord a => [a] -> [a] -> ([b] -> [b])
mkPermute x y = pairsToPermute $ concat $
zipWith zip (occurs x) (occurs y)
where
pairsToPermute ps = flip map (map snd $ sort ps) . (!!)
occurs = map (map snd) . groupBy fstEq . sort . flip zip [0..]
problem_98 :: [String] -> Int
problem_98 ws = read $ head
[y | was <- sortBy longFirst $ anagrams ws, -- word anagram sets
w1:t <- tails was, w2 <- t,
let permute = mkPermute w1 w2,
nas <- sortBy longFirst $ anagrams $
filter ((== profile w1) . profile) $
dropWhile (flip longerThan w1) $
takeWhile (not . longerThan w1) $
map show $ map (\x -> x * x) [1..], -- number anagram sets
x:t <- tails nas, y <- t,
permute x == y || permute y == x
]
run_problem_98 :: IO Int
run_problem_98 = do
words_file <- readFile "words.txt"
let words = read $ '[' : words_file ++ "]"
return $ problem_98 words
-- Sort on length of first element, from longest to shortest
longFirst :: [[a]] -> [[a]] -> Ordering
longFirst = flip compareLen `on` fst
-- Is y longer than x?
longerThan :: [a] -> [a] -> Bool
longerThan x y = compareLen x y == LT
-- Compare the lengths of lists, with short-circuiting
compareLen :: [a] -> [a] -> Ordering
compareLen (_:xs) (_:ys) = compareLen xs ys
compareLen (_:_) [] = GT
compareLen [] [] = EQ
compareLen [] (_:_) = LT
(Cf. short-circuiting)
Problem 99
Which base/exponent pair in the file has the greatest numerical value?
Solution:
import Data.List
lognum (b,e) = e * log b
logfun x = lognum . read $ "(" ++ x ++ ")"
problem_99 = snd . maximum . flip zip [1..] . map logfun . lines
main = readFile "base_exp.txt" >>= print . problem_99
Problem 100
Finding the number of blue discs for which there is 50% chance of taking two blue.
Solution:
nextAB a b
|a+b>10^12 =[a,b]
|otherwise=nextAB (3*a+2*b+2) (4*a+3*b+3)
problem_100=(+1)$head$nextAB 14 20