On what date did the Day of Atonement fall each year?

Answer: 10th day of the 7th month

The tenth day of the seventh month was the Day of Atonement, when atonement was made to cleanse Israel from their sins.

“This is to be a permanent statute for you: On the tenth day of the seventh month, you shall humble yourselves and not do any work—whether the native or the foreigner who resides among you—”— Leviticus 16:29 (BSB)