Does this travel qualify for visa free?

My daughter and her 4 children traveled from USA, our home country, to Taipei on July 12, 2023 for educational reasons. We have return tickets out of Taipei confirmed for August 15, 2023. If we added a trip to Beijing on the end (from 8/11 to 8/14) would our tickets - USA to Taiwan to Beijing to Taiwan to USA meet the visa free qualifications?