# [Haskell-cafe] Re: Hints for Euler Problem 11

ronguida at mindspring.com ronguida at mindspring.com
Fri Jul 20 13:29:12 EDT 2007

```Thank you for all the hints.

Here's how I ended up solving it:

> module Main
>     where
>
> import Data.List
> import Data.Maybe
>
> gridText = "08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08\n\
> \49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00\n\
> \81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65\n\
> \52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91\n\
> \22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80\n\
> \24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50\n\
> \32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70\n\
> \67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21\n\
> \24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72\n\
> \21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95\n\
> \78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92\n\
> \16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57\n\
> \86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58\n\
> \19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40\n\
> \04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66\n\
> \88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69\n\
> \04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36\n\
> \20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16\n\
> \20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54\n\
> \01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48"

> type Grid a = [[a]]

> grid :: Grid Integer

takeExactly - extract the first "n" elements of a list;
there must be at least "n" elements, n > 0

> takeExactly :: (Monad m) => Int -> [a] -> m [a]
> takeExactly n xs | n > 0 =
>                      let ys = take n xs in
>                        if length ys == n
>                          then return ys
>                          else fail "takeExactly: list is too short"
>                  | otherwise = fail "takeExactly: empty list"

> extractGroups :: Int -> [[a]] -> [[a]]
> extractGroups n = catMaybes . map (takeExactly n)

> gridExtractGroups :: Int -> Grid [a] -> Grid [a]
> gridExtractGroups n = filter (not . null) . map (extractGroups n)

gridTails - generate a list of grids with sequential starting places
Parameters a and b determine the offset bewteen successive grids,
as follows: (note that a, b may not be negative)

Grid x_{0,1.., 0,1..} ->

[ Grid x_{  0,     1 ..,   0,     1 ..}
, Grid x_{  a,   a+1 ..,   b,   b+1 ..}
, Grid x_{2*a, 2*a+1 .., 2*b, 2*b+1 ..}
etc. ]

> gridTails :: (Int, Int) -> Grid a -> [Grid a]
> gridTails (a,b) xs =
>   if not \$ null \$ concat xs
>     then xs : gridTails (a,b) ((drop a) \$ map (drop b) xs)
>     else []

transpose3d - converts x_{i,j,k} to x_{j,k,i}
using sequence is x_{i,j,k} -> x_{j,i,k} -> x_{j,k,i}

> transpose3d :: [Grid a] -> Grid [a]
> transpose3d = map transpose . transpose

reorient - transform a direction vector so that both components are
non-negative

> reorient :: (Int, Int) -> (Grid a -> Grid a, (Int, Int), Grid b -> Grid b)
> reorient (a,b)
>  | a>=0 && b>=0 = (id                   , ( a, b), id)
>  | a< 0 && b>=0 = (reverse              , (-a, b), reverse)
>  | a>=0 && b< 0 = (map reverse          , ( a,-b), map reverse)
>  | a< 0 && b< 0 = (reverse . map reverse, (-a,-b), reverse . map reverse)

makeEls - convert a grid to grid of Equidistant Letter Sequences

> makeEls :: (Int, Int) -> Grid a -> Grid [a]
> makeEls vec = let (reflect2, rvec, reflect1) = reorient vec in
>                 reflect2 . transpose3d . gridTails rvec . reflect1

> getGroups :: Int -> (Int, Int) -> Grid a -> [[a]]
> getGroups n (a,b) = concat . gridExtractGroups n . makeEls (a,b)

> findMaxProduct :: (Ord a, Num a) => Int -> (Int, Int) -> Grid a -> a
> findMaxProduct n (a,b) = maximum . map product . getGroups n (a,b)

> main :: IO()
> main = do
>   print \$ findMaxProduct 4 (1,0) grid
>   print \$ findMaxProduct 4 (0,1) grid
>   print \$ findMaxProduct 4 (1,1) grid
>   print \$ findMaxProduct 4 (1,-1) grid

-- Ron

```