I think it was pretty good. Its not really for people who don't know math, but for people (like me) who know some math but have never looked into the paradox.

I think it was pretty good. Its not really for people who don't know math, but for people (like me) who know some math but have never looked into the paradox. The key here is to explain what pieces means and the whole cloud idea. With that, I sorta get it (I think so anyway, I'd need to understand the proof better to see if my intuitive feeling is correct). That being said, I notice a problem immediately. Why do we need 4 pieces? From the explanation, its pretty much giving a sort of "algorithm" for creating the 4 (or 5) pieces. Essentially, start somewhere, and go around and around, and put each "atom" into one of 4 sets. I.e. the first "atom" goes into the first bucket, the second atom into the second bucket, etc... Then, take bucket 1 and 2, and translate them (really just to avoid intersection) and then rotate set 2 so that the atoms are "half way" between those of set 1, and do the same rotation for bucket 4. So... why not just call buckets 1 and 3 "bucket A" and buckets 3 and 4 "bucket B" and just do the translation? I know enough to know that my "algorithm" doesn't work, but its what the intuition here suggests. So, there is still some gap in my understanding. I'm curious how far an intuitive explanation can go. I'll take a look at the link someone else gave below, but does anyone have an intuitive answer to my question?