Visa-free travel on inbound and outbound flights

Will I qualify for additional free travel if I exit China after a free stay only to return a few days later?

Specifically, I would like to enjoy free stays on both the inbound and return legs of my trip, such as:

- INBOUND: USA direct to Beijing (3 days or less), then direct to Hong Kong
- RETURN: Hong Kong direct to Shanghai (6 days or less), then direct to USA

Finally, would the policy be any different if...

1: inbound and return trips include transit in the same city? (e.g. 3-day stays in Beijing both directions)

2: return occurs on the same day as transit? (e.g. leave Beijing for Hong Kong, then immediately return through Shanghai) Perhaps a better way to ask this question is whether a minimum stay is required in the destination country (e.g. Hong Kong) before a traveler may return to China and enjoy an additional free transit.