Euler project #18 approach

Question

I am looking into an Euler project. Specifically #18.
To sum up, the idea is to find the max path from a triangle:

   3
  7 4
 2 4 6  
8 5 9 3
<         


        

Answer 1

Using your example, the "bottom up" way to approach it is:

Examining the bottom row, the most you can get from each element is

8,5,9,3

Examining the next-to-bottom row, the most you can get from each element is (depending on whether you go left or right from it):

2+max(8,5),4+max(5,9),6+max(9,3) = 10,13,15

So this is great; we've eliminated 2 rows by squishing them together to replace them with one row, reducing the problem to

     3
   7   4
10  13  15

Obviously we can just keep repeating this. Examining the next row up, the most you can get from each element is

7+max(10,13),4+max(13,15) = 20,19

And so from the top, the most you can get is

3+max(20,19) = 23

QED.