62. 

This time, Alice met both brothers together. She asked the first one, "Are you the owner of the rattle?" - And received the answer: "Yes." Then Alice asked the second brother, "Is this your rattle?" The second one answered, and Alice gave the rattle to one of them.

To which of the brothers did Alice hand over the rattle: the first or the second?

## G. FROM THE MOUTH OF THE BANDERSNATCH

Of all the adventures Alice had with the two brothers Tralala and Trulala, the one I want to tell you about now was the most extraordinary and was remembered by Alice in the smallest detail.

It all began like this. One day, Shaltay-Boltay met Alice and, pulling her aside, said:

- Child, I want to tell you a terrible secret. Although most people don't even suspect it, Tralala and Trulala have a third brother, and his name is Trululala. He lives far, far away from here, but sometimes he comes to our lands. He looks as much like Tralala and Trulala as they look like each other.

Shaltay-Boltay's message greatly alarmed Alice. After all! The mere fact that a third brother existed made all her previous conclusions invalid. Even the day of the week could no longer be determined with absolute certainty. And most importantly - how could she now return the rattle to its rightful owner?

Alice fell into deep thought and then asked Shaltay-Boltay a sensible question:

- On which days of the week does Trululala lie?
- Trululala always lies, - Shaltay-Boltay replied.

Alice left silently. Her heart was uneasy. "Maybe there is no third brother at all?" she tried to comfort herself. "Maybe it's all Shaltay-Boltay's invention? After all, everyone would agree that his story sounds very strange." But no matter how hard Alice tried, she couldn't shake off the troubling thought: "What if it's all true?"

What happened next is told in different ways (there are four versions of events, and I will not hide any of them from you). I ask you to accept two assumptions: 1) if there is indeed someone besides Tralala and Trulala who is indistinguishable from them in appearance, then his name is Trululala; 2) if such an individual exists, then he always lies. I must note that the second assumption is not necessary for solving the first puzzle, but is necessary for solving the next two puzzles.
62. The key to the solution lies in that part of the problem statement where it is clear that, upon receiving the second brother's answer, Alice knew to whom she needed to give the rattle. If the second brother had answered "yes," then one of the brothers would have been telling the truth while the other was lying, and Alice would not have been able to determine who the owner of the rattle was. But since the problem states that Alice did give the rattle, the second brother did not answer "yes." Therefore, either both brothers were lying or both were telling the truth. From this, we conclude that both were telling the truth, and the meeting with Alice took place on Sunday. Therefore, Alice gave the rattle to the first brother.