Homework Help: Equivalent Diagram- How does counting paths let us eyeball?

All the youtube links are time stamped



1. The problem statement, all variables and given/known data





2. Relevant equations


3. The attempt at a solution
I wrote it out the operator expressions each by hand and got the same result but I don't understand how he is able to just eyeball it. I'm trying to develop some intuition.

a. To start, why is he counting distinct signal paths paths? What is the motivation to see why this information may be useful in letting us eyeball which block diagrams are teh same. He mentioned earlier in the video that this could be useful but I didn't understand.

b.

He says something about "making a sum" and seeing "how many of them have the same sum" What does that mean? He then looks for the paths with the biggest delay. I don't see why this info would help us other than to maybe to disprove two block diagrams are different.
^ See time stamped youtube vid

At the very end, he is able to conclude that the answer is 3 but I don't see how we have proof

 

I have used X₁ to denote X after one delay

So for example then, what does 4X₂; 4X₁ mean? Aren't there 9 paths because you have 4, 4, and 1? What is 4X2 for example? So it seems like based on your notation that would mean 4 times X after 2 delays in part a). Where does this 4 times X after two delays come from?

Is this his method of eyebaling?

 

So the reason we know it is equivalent by eyeballing is that they all sum to 4X1; 4X2?