Logic
# Deterministic Games

Harry and Ron told me separately how many times they watched the movie *Titanic*. I told them "you guys both watched it, but one of you watched it once more than the other.". Then they had the following conversation:

Ron: I have no idea if you watched *Titanic* more than I did.

Harry: Me neither. Do you know now?

Ron: Yes, indeed!

Harry: Really? Then so do I!

What is the possible number of times Ron watched *Titanic*? (Select *every* answer that is possible.)

(**Note:** At no point did Harry and Ron watch *Titanic* together.)

Ted rolls two tetrahedral fair dice. Each die has its four faces labelled: 1, 2, 3, and 4. He tells Marshall the sum of the outcomes, and tells Lily the product. Ted asks them to guess the numbers he got. Then Marshall and Lily have the following conversation:

Marshall: Do you know the answer?

Lily: No, do you?

Marshall: At first I didn't, but I do now!

Lily: I still don't get it.

Marshall: We heard different numbers.

Lily: Oh!

If the two numbers that Ted got were $a$ and $b,$ what is $a^2+b^2?$

**Note:** The order of the two numbers does not matter.

A warden has three prisoners: a normal, a one-eyed, and a blind man.

He shows five hats and says, truthfully, "I have three white hats and two red hats."

He then orders the two men who can see to close their eyes and hides two of the five hats. He then distributes the remaining three hats to the three prisoners and orders them to wear them. Now, the warden orders the prisoners to open their eyes and tells the rule of the game: whoever knows the color of the hat he himself is wearing gets a free pack of cigarettes everyday for the rest of their prison life.

He first asks the normal man, who answers, "I don't know." Then he asks the one-eyed man, who answers, "I don't know." The warden also asks the blind man, who answers, "Yes, I definitely know the color of my hat." Surprisingly, the blind man actually got it right. Which of the following could be the hat colors of the (normal, one-eyed, blind) men?

(A) (white, red, red)

(B) (white, white, white)

(C) (red, white, red)

(D) (white, white, red)

Harry and Ron told me how many times they watched the movie *Titanic*. I told them "you guys both watched it, but one of you watched it once more than the other". Then they had the following conversation:

Ron: I have no idea if I watched *Titanic* more than you.

Harry: Me neither. Do you know now?

Ron: Nope. Do you?

Harry: Not quite yet.

Ron: OK, now I get it!

Harry: Really? Then so do I!

What is the sum of the possible number of times Ron watched *Titanic*?

A group of three thieves (including the boss) obtained three bars of gold. Initially, the boss is holding all three bars. Every morning, the three thieves have a conference, at which the boss must make a proposition. He suggests how to distribute the three bars, i.e. how many bars each person gets to have. (This means redistributing **all** three bars, not just those held by the boss.) If at least half (including the boss himself) agree, then the gold is distributed as the boss suggested. If less than half agree, then the boss is killed. At most how many bars of gold can the boss constantly keep?

**Note:** The other two thieves are very selfish. They agree to every plan that lets themselves have more than before, and disagree to every proposition that takes gold from them (or leaves their gold amount the same).