Knuck tells Marv, Alice, and Blorb that his botday is one of these days.

He then whispers something to Alice, and then something else to Blorb.

“I’ve told Alice the month of my botday, and Blorb the numerical day of my botday,” Knuck explains.

“I don’t know when Knuck’s botday is, but I know that Blorb doesn’t know either,” says Alice.

“At first I didn't know when Knuck's botday is, but now I do,” says Blorb.

“Now I know when Knuck’s botday is, too!”, exclaims Alice.

When is Knuck’s botday? Answer: July 16

Solution:

“I don’t know when Knuck’s botday is, but I know that Blorb doesn’t know either,” says Alice.

Alice knows that Blorb knows the numerical day, but not the month of the botday. 

Let's suppose that the robot told Alice her botday is in May. Alice wouldn't know whether it's on the 15th, 16th, or 19th. But Alice would know there's a chance the botday is May 19. She also knows if that was the case, Blorb would immediately know the botday since May is the only month where the botday might fall on the 19th. By saying she knows Blorb doesn't know the botday, though, Alice is saying she's confident the date is not May 19. Alice can only know that from the robot telling her a month other than May. 

Following the same logic, the robot couldn't have told Alice her botday is in June.

From Alice's statement, we know that Knuck’s botday is in either July or August.

“At first I didn't know when Knuck's botday is, but now I do,” says Blorb.

If the botday was July 14 or August 14, the robot would have told Blorb only that her botday is on the 14th of some month. It would be impossible for Blorb to know the botday only from that information. Since he does know her botday, it can't be on July 14 or August 14. Every remaining date's numerical day is unique. Since Blorb knows the botday, we only know that the robot told him one of these days.

“Now I know when Knuck’s botday is, too!”, exclaims Alice.

Alice knows only the month of the botday. If the month was August, she would have no way of knowing whether the date was August 15 or August 17. Hence it must be July 16.

Problem 1

Alice, Blorb, Carol, and Dan are playing a hat game with this box of hats:

First, they each put a hat from the box on their head without looking at it and without peeking at what hats remain in the box.

Once they each have a hat, they stand in a ring so that each player can see all of the other players' hats. Then, each player gets one chance to guess their own hat color.

• First, Alice says, “I'm certain my hat is blue.”

• Then, Blorb says, “I'm certain my hat is yellow.”

Indicate which hat colors Carol and Dan must have.

Carol:
A. Red
B. Blue
C. Yellow

Dan:
A. Red
B. Blue
C. Yellow

Problem 2

Flushed with victory, Alice, Blorb, Carol, and Dan decide to try the game again, to see if it gets any harder. 

They play with the same rules and the same hats.

• First, Alice says, “I don't know my hat color.”

• Then, Blorb says, “I don't know my hat color.”

Carol sees that Alice and Blorb are wearing blue hats, and Dan is wearing a yellow hat.

What should Carol say at this point?

A. “I don't know my hat color.”
B. “I'm certain my hat is red.”
C. “I'm certain my hat is blue.”
D. “I'm certain my hat is yellow.”