When Width Is Fixed

For fixed small width $w$, broken profile DP has $2^w$ states per column. With small $w$ (like $3$ or $4$), you can often discover a formula (like $a_n = f(a_{n-1}, a_{n-2})$) by running the DP on paper. Once you have the formula, code it directly without the profile machinery.

But for variable width or complex tile shapes, you need the full DP. The profile approach is the general tool. In contests, if $w$ is small and fixed, try small cases by hand. If $w$ varies or is large (up to $10$), implement broken profile DP.