[Haskell-cafe] Hints for Euler Problem 11

[Ronald Guida]

Hi, again.

I started looking at the Euler problems [1].  I had no trouble with
problems 1 through 10, but I'm stuck on problem 11.  I am aware that
the solutions are available ([2]), but I would rather not look just
yet.

In Problem 11, a 20x20 grid of numbers is given, and the problem is to
find the largest product of four numbers along a straight line in the
grid.  The line can be horizontal, vertical, or diagonal.

I figured out how to handle the horizontal and vertical products, but
I'm stuck on how to approach the problem of extracting the diagonals.

Here is what I have so far; it does the horizontal and vertical cases:

 > module Main
 >     where
 >
 > import Data.List
 >
 > 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"
 >
 > readGrid :: (Read a) => String -> [[a]]
 > readGrid = (map ((map read) . words)) . lines
 >
 > grid :: [[Integer]]
 > grid = readGrid gridText
 >
 > makeGroups :: Int -> [a] -> [[a]]
 > makeGroups 0 _ = []
 > makeGroups n xs = let ys = take n xs in
 >                  if n == length ys
 >                    then ys : (makeGroups n $ tail xs)
 >                    else []
 >
 > maxHorizontal :: (Ord a, Num a) => Int -> [[a]] -> a
 > maxHorizontal length = maximum . map product . concat . map 
(makeGroups length)
 >
 > maxVertical :: (Ord a, Num a) => Int -> [[a]] -> a
 > maxVertical length = maxHorizontal length . transpose
 >
 > main :: IO()
 > main = do
 >   print $ maxHorizontal 4 grid
 >   print $ maxVertical 4 grid

To handle the diagonals, my plan is to try to extract each diagonal as
a list of elements and put all the diagonals into a list; then I can
use maxHorizontal.

I came up with this function to try to extract the main diagonal.

 > getDiag :: [[a]] -> [a]
 > getDiag = map (head . head) . iterate (tail . map tail)

The problem is, this function doesn't work unless I have an infinite
grid.

Could anyone provide me with some hints to lead me in the right direction?

Thank you
-- Ron

References:

[1] http://projecteuler.net/index.php?section=view

[2] http://www.haskell.org/haskellwiki/Euler_problems