Donner sa langue au chat

Literally: To give one's tongue to the cat

Example: “Je ne trouve pas la réponse à ta devinette, je donne ma langue au chat.” — I can't figure out the answer to your riddle; I give up.

Origin: Older still than the cat: Madame de Sévigné has jeter sa langue aux chiens — to throw one's tongue to the dogs, dogs being what you gave your leftovers to, so surrendering your tongue to them meant giving up an organ of speech that had stopped earning its keep, since you were never going to say the answer you'd given up guessing. George Sand's generation instead confided a secret dans l'oreille du chat — into the cat's ear — on the reasoning that a cat, unable to talk, could be trusted with anything (barring, of course, Puss in Boots). The dog and the cat merged into the modern verb only in 1860, with Edmond de Goncourt's Charles Demailly an early instance of donner sa langue au chat in print.