Choosing the maximum adjacent value is not guaranteed to give the maximum sum from top to bottom (as demonstrated here).
Here's a more obvious example:
2 1 1
1 1 8 9
Your approach will choose 1 + 1 + 2 + 1, but the best approach is 1 + 1 + 1 + 9.
If you want to solve this problem the "lazy way", you can use a brute force solver and try every path. As the problem hint suggests, this will work here, but not on problem 67.
There's a more efficient solution, though (as the problem suggested). As a hint, think about the problem going backwards.