Everyone of us will have this choice to make — to solve the complex parts first or the most common ones?

Usually there are a few complex parts, and a lot of simpler ones. Let's assume they are 20% complex parts and 80% common ones. Both are needed.

So, if you need most of the system working then do the 80% first. That way those are out of the way and you can focus on the hard parts. Or, you can finish the hard parts first, when your mind is fresh.

The deciding factor is always the situation and what you want from it. For example, if you need a system built fast, finishing all the easier parts will allow you to use at least 80% of the system while you build the last complicated bits.

However, if the needle will move only once the complex parts is solved, then that's the obvious first.