Burst Balloons
Recognize the pattern
Brute force idea
If you approach Burst Balloons in the most literal way possible, you get this: Try every order of balloon bursting. Factorial because each burst changes what's adjacent. It is a fair place to begin because it matches the surface of the question, yet it does not capture the deeper structure that makes the problem simpler.
Better approach
The deeper shift in Burst Balloons is this: Interval DP: think of adding balloons instead of removing them. dp[i][j] = max coins from bursting all balloons between i and j (exclusive). For each possible last balloon k in (i,j), dp[i][j] = max(dp[i][k] + dp[k][j] + nums[i]*nums[k]*nums[j]). Once you hold onto the right piece of information from moment to moment, the problem feels less like trial and error and more like following a shape that was there all along.
Key invariant
At the center of Burst Balloons is one steady idea: By thinking of k as the LAST balloon burst in the interval (i,j), the left and right subproblems become independent — the boundaries i and j are guaranteed to still exist when k is burst. When you keep that truth intact, each local choice supports the larger solution instead of fighting it.
Watch out for
A common way to get lost in Burst Balloons is this: Thinking about which balloon to burst FIRST — that makes subproblems dependent. Reversing the perspective to 'which is burst LAST' is the key insight that makes intervals independent. Most mistakes here are not about syntax; they come from losing track of what your state, pointer, or structure is supposed to mean.