Impossible puzzles

I’ve been on holiday, visiting friends and family, and one of my friends gave me three impossible puzzles. I thought I would pass them on.

The impossible puzzle

X and Y are two different whole numbers greater than 1. Their sum is not greater than 100, and Y is greater than X. S and P are two mathematicians (and consequently perfect logicians); S knows the sum X + Y and P knows the product X × Y. Both S and P know all the information in this paragraph.

The following conversation occurs (both participants are telling the truth):

• S says “P does not know X and Y.”
• P says “Now I know X and Y.”
• S says “Now I also know X and Y.”

What are X and Y?

The Seemingly Impossible “Guess The Number Logic Puzzle“

I got this from Mind your decisions by Presh Talwalkar.

Alice and Bob are on a game show. Each is secretly told a whole, positive number. They are told the two numbers are consecutive, but neither knows the other person’s number. For example, if Alice is told 20, she does not know if Bob was told 19 or 21. And if Bob is told 21, he does not know if Alice was told 20 or 22. The point of the game is to guess the other person’s number. The game works as follows.

• Alice and Bob cannot communicate with each other, and they are not allowed to plan their strategy either.
• The two are in a room where a clock rings every minute.
• After the clock rings, either player can call out a guess of the other player’s number, or they can stay silent.
• The game continues until Alice or Bob makes a guess. After the first guess is made, the game ends.
• Alice and Bob win $1 million each if the guess is correct, and they lose and get nothing if the guess is incorrect.

How should Alice and Bob play this game to have the best chance of winning? Each knows the other person is perfect at logical reasoning.

Pirates, and being thrown overboard

Some pirates caught a yacht on the high seas. The 8 people on the yacht were all logicians.

The pirate king said “Tomorrow I will put a red dot or a black dot on your forehead. You cannot see the dot, but every one else can. I will place you in a circle, so you can see every one else, and I will ask each of you in turn, if you have a red dot or a black dot. If you get it right – you live, if you get it wrong you walk the plank, and feed the sharks. Now go to the cell and plan your strategy, arr Jim lad”.

What strategy should they use to save most people.