Which Alice?

The Twelfth Question
believed that both he and the Red Queen were asleep at that time.
Was the Red Queen asleep or awake at the time?"
Alice thought this over, and at first believed the situation was
impossible. But she suddenly realized that the situation was not
impossible and was able to figure out the answer.
What is the answer?
AWAKE OR ASLEEP? "I am like the Red King," said the Red
Queen. "I also believe only false things when I am asleep, and I
believe only true things when I am awake. Now, the night before
last at eleven o'clock, the Red King believed I was asleep. At the
same time, I either believed that he was asleep or I believed that he
was awake. Which did I believe?"
Alice had to think rather hard to get this one, but she finally did.
What is the solution?
HOW MANY RATTLES? "That last puzzle made my head
ache," said the White Queen. "Let's get back to arithmetic puzzles.
Now, you know Tweedledum and Tweedledee?"
"Oh, yes, indeed!" replied Alice.
"Very well. Tweedledum and Tweedledee once had a wager."
"What was the wager about?" asked Alice.
"It was about the monstrous Crow. Tweedledum believed it
would come back the next day, and Tweedledee believed it
wouldn't. So they decided to have a wager about it."
"How much did they wager?" asked Alice.
"Well," replied the Queen, "you know they both collect rattles?"
"I know that Tweedledum had a rattle," replied Alice, "one that
he claimed that Tweedledee had spoiled, but I didn't know that
Tweedledee also had a rattle."
"They both had rattles," replied the Queen. "Indeed each had
several rattles. The wager was for one rattle."
"How funny!" Alice laughed. "How many rattles did each have?"
69