No, there is only 1 test for each problem (all cases at once).

find a regular pattern

The easiest way in my opinion is to consider a DP on width of currently covered rows and number of currently painted balls (figure out why width works and why you don't need left + right). This would work even for a n × m grid.

However, some solutions are based on counting arguments, and some people just noticed a pattern :)

If you remove one edge, you get a tree. Then you can compute shortest path efficiently via LCA (segment tree) as d(u, v). Suppose the edge you removed was (a, b). Then the shortest distance in the graph is min(d(u, v), d(u, a) + w(a, b) + d(b, v), d(u, b) + w(a, b) + d(a, v)).

We can turn it into a set of difference constraints rather than sum constraints (e.g. replace b_i with  - c_i). Then look at using Bellman-Ford for determining whether a set of difference constraints can be satisfied.

-_- why necro bump this when you can make your own blog post