018-maximum-path-sum-I.lhs

#!/usr/bin/env runhaskell

[Problem 18](http://projecteuler.net/problem=18)

By starting at the top of the triangle below and moving to adjacent numbers on
the row below, the maximum total from top to bottom is 23.

       3
      7 4
     2 4 6
    8 5 9 3

That is, 3 + 7 + 4 + 9 = 23.

Find the maximum total from top to bottom of the triangle below:

                         75
                       95 64
                      17 47 82
                    18 35 87 10
                   20 04 82 47 65
                 19 01 23 75 03 34
                88 02 77 73 07 63 67
              99 65 04 28 06 16 70 92
             41 41 26 56 83 40 80 70 33
           41 48 72 33 47 32 37 16 94 29
          53 71 44 65 25 43 91 52 97 51 14
        70 11 33 28 77 73 17 78 39 68 17 57
       91 71 52 38 17 14 91 43 58 50 27 29 48
     63 66 04 68 89 53 67 30 73 16 69 87 40 31
    04 62 98 27 23 09 70 98 73 93 38 53 60 04 23

NOTE: As there are only 16384 routes, it is possible to solve this problem by
trying every route. However, Problem 67, is the same challenge with a triangle
containing one-hundred rows; it cannot be solved by brute force, and requires a
clever method! ;o)


Code
----

> triangle :: [[Int]]
> triangle = [[75]
>            ,[95,64]
>            ,[17,47,82]
>            ,[18,35,87,10]
>            ,[20,04,82,47,65]
>            ,[19,01,23,75,03,34]
>            ,[88,02,77,73,07,63,67]
>            ,[99,65,04,28,06,16,70,92]
>            ,[41,41,26,56,83,40,80,70,33]
>            ,[41,48,72,33,47,32,37,16,94,29]
>            ,[53,71,44,65,25,43,91,52,97,51,14]
>            ,[70,11,33,28,77,73,17,78,39,68,17,57]
>            ,[91,71,52,38,17,14,91,43,58,50,27,29,48]
>            ,[63,66,04,68,89,53,67,30,73,16,69,87,40,31]
>            ,[04,62,98,27,23,09,70,98,73,93,38,53,60,04,23]]

> step :: [Int] -> [Int] -> [Int]
> step lower higher = let maxs = zipWith max lower (tail lower)
>                     in zipWith (+) maxs higher

> main :: IO ()
> main = let result = head . foldl1 step $ reverse triangle
>        in print result


Answer
------

    1074