61. 

A few days later, while wandering through the forest, Alice came across one of the brothers sitting under a tree. She asked him the same question and heard in response: “The true owner of the rattle speaks the truth today.”

Alice paused to think. She wanted to assess how likely it was that the brother who said this was the true owner of the rattle.

- I know what you're thinking,- said Shaltai-Boltai, who had appeared nearby.-The odds are high! Thirteen out of fourteen.

How did Shaltai-Boltai come up with these numbers?
61. Shaltai-Boltai correctly assessed the chances. Suppose the brother was lying. This would mean that the rattle owner does not tell the truth on the day of the meeting. On the day of the meeting, he lies and, therefore, must be the very brother whom Alice met. Now suppose that the brother Alice met is telling the truth. This would mean that the rattle owner does tell the truth on the day of the meeting. If the meeting takes place on a weekday, the rattle belongs to the brother Alice met, and if on a Sunday, then (since both brothers tell the truth on Sundays) the rattle can belong to either of them.

In summary, if the meeting takes place on a weekday, the rattle belongs to the one Alice was talking to. If the meeting takes place on a Sunday, the chances that he is the owner of the rattle are $6^{1} / 2$ out of 7, or 13 out of 14.