Guess what? Come on, guess! Guess again! Give up? Are you giving your tongue to the cat?
The French expression donner sa langue au chat - literally to give one's tongue to the cat - is roughly equivalent to the English throwing in the towel. It simply means that you don't know the solution to a problem, or the answer to a guessing game, you don't want to guess any longer, and you'd like the other person to reveal the answer. In other words, "je donne ma langue au chat" is a way of saying: Well, I don’t know. Just tell me.
We find records of the expression jeter sa langue aux chiens - throwing dogs one's tongue - with Mme de Sévigné in her 17th century letters. The original idiom had the same meaning of wanting to stop guessing the answer to a question. When you couldn’t guess any more, your tongue was useless, so you threw it away to the dogs like so much leftover food.
But how did throw it to the dogs become give it to the cat?
By the wonderful magic of unintentional idiom blend - also called malaphor!
In the 19th century, we find the expression mettre quelque chose dans l'oreille du chat - putting something in the cat's ear (Georges Sand-) which meant confiding a secret that would never be disclosed. The cat was then considered an animal knowing a multitude of secrets, without ever being able to divulge them.
Both idioms might have later blended into donner sa langue au chat to indicate that you're throwing out your useless tongue while giving it to the cat who certainly knows the answer to the question. One might add that it would be worth getting it back at this point. Or let the cat spill the beans.
Les problèmes d'algèbre sont tellement difficiles: je finis toujours par donner ma langue au chat.
Algebra problems are so difficult. I always give up in the end.
Tu ne trouves pas? Tu donnes ta langue au chat?
You can't figure it out? Shall I give you the answer?