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

It's something called dynamic programming.

You have such triangle:

   3
  7 4
 2 4 6  
8 5 9 3

When you move from the bottom to the top, you can calculate the best choices on last iteration. In this case you take the last row 8 5 9 3 and maximize sum in addition to previous line.

Iteration 1: Assume that you are on last-1 step.

You have line 2 4 6, let's iterate on it.

From 2, you can go to either 8 or 5, so 8 is better (maximize your result from 2) so you calculate first sum 8 + 2 = 10.

From 4, you can go to either 5 or 9, so 9 is better (maximize your result from 4) so you calculate second sum 9 + 4 = 13.

From 6, you can go to either 9 or 3, so 9 is better again (maximize your result from 6) so you calculate third sum 9 + 6 = 15.

This is the end of first iteration and you got the line of sums 10 13 15.

Now you've got triangle of lower dimension:

    3
  7  4
10 13 15  

Now go on until you get one value, which is exactly 23.