It is generally thought that a person cannot accept an offer of which they are unaware, because inorder to create a binding contract, the parties must reach an agreement. If their wishes merely happen to coincide, that may be very convenient for both, but it does not constitute a contract and cannot legally bind them.
In the American case of Williams v Carwardine (1833), a lady who was aware of a reward for information did give the information requested by the defendant not because of the reward but due to the believe that she had only a short time to live and her conscience will be free if she gave the information. The court held that she was entitled to the reward as she was aware of the offer and complied with its terms. By contrast, a second US case of Fitch v Snedaker (1868) saw the court hold the position that a person who gives information without knowledge of the offer of a reward cannot claim the reward.
It will also appear that if the offeree knew of the offer in the past but has completely forgotten about it, they are treated as never having known about it. This position was illustrated in the case of R V Clarke (1927).