This is a challenging problem from the c.100AD Chinese mathematical work, *Jiǔ zhāng suàn shù* (*The Nine Chapters on the Mathematical Art*) found at the MAA Convergence website.

“Now a good horse and an inferior horse set out from Chang’an to Qi. Qi is 3000* li *from Chang’an. The good horse travels 193 *li* on the first day and daily increases by 13 *li*; the inferior horse travels 97 *li* on the first day and daily decreases by ½ *li*. The good horse reaches Qi first and turns back to meet the inferior horse. Tell: how many days until they meet and how far has each traveled?”

The solution involves common fractions, which the Chinese were already adept at using by 100 BC.

See Horses to Qi for a solution.