לרפואת ר' מרדכי בן שרה תוך שח"י
The Gemara in Rosh Hashana [21a] relates that Levi came to Bavel and saw them eating on what they thought was the 11th of Tishrei but he was coming from Eretz Yisrael [where they determined the beginning of the new months] and knew that it was actually the 10th and thus Yom Kippur.
Asked Tosfos - How could he allow them to eat on what he knew was Yom Kippur??
Answers Tosfos: The law is that Kiddush Hachodesh is valid even if done in error, so even though they were mistaken, their sanctification is valid.
WONDER OF WONDERS!!! That law is talking about the BEIS DIN establishing a new month by mistake. But here the Beis Din wasn't making the mistake - the PEOPLE were??
So Tosfos' answer is [seemingly] not an answer and the question resurfaces!! I am left scratching my head!!!
I believe that in this shiur, one possible approach is offered.... [there are at least two more].