72 hour free China Visa - 3rd country

Do you have to be in the 3rd country longer than China as have been told by various UK travel agencies and visa agencies that travelers have been rejected upon arrival in China with less days spent in 3rd country e.g., London-Shanghai (2 days)-Singapore (1 day)-Hong Kong (1 day)-London, and that they have to get the paid visa instead. Please clarify how the 72 hour or 144 hr free visa works. Thanks.