The problem description is **here**, and **click here** to see all my other Euler solutions in F#.

This is a more difficult version of problem 82, and now you can move in all four directions!

As before, we start by loading the input data into a 2D array:

and initialize another matrix of the same size to hold the minimum sum leading to each cell:

What we have done differently this time though, is to choose initial defaults:

- for the top-left corner (0, 0) the minimum sum is itself
- for all other cells at (
*row, col*) we’ll always traverse right and then down- if
*row*= 0, then this equates to only move right until we reach*col* - if
*col*= 0, then this equates to only move down until we reach*row*

- if

e.g.

Next, as we did for the problem 82, we’ll add a couple of helper to help us traverse both matrices:

and we’ll find the new minimum sum to each cell by taking the minimum from the four directions:

Unlike before, we can’t isolate to optimizing one column at a time, and instead we’ll need to traverse the whole matrix. For better* cache locality*, we’ll traverse row first and recursively optimize the whole matrix until no more optimization is possible:

Finally, find the final value for the bottom-right corner of the sum matrix:

This solution runs in 73ms on my laptop.

The source code for this solution is here.