Literally: to give one's tongue to the cat
Origins: At first it was,“jeter sa langue aux chiens” literally to throw your tongue to the dogs. Then slowly evolved to become in the 19th century, “donner sa langue au chat” literally give your tongue to the cat.
Meaning: to give up trying to guess something, e.g. the answer to a riddle
Synonyms: laisser tomber ; s’avouer vaincu.