Counterfeit Coin [Puzzle]

There are 12 coins, one of which is a counterfeit. The false coin differs in weight from the true ones, but you don’t know whether it’s heavier or lighter. Find the counterfeit coin using three weighings in a pan balance.

[From Futility Closet]


Two Prizes [Puzzle]

Suppose I offer you two prizes: prize A and prize B. You are to make a statement – any statement you like.

If the statement is clearly true, I will give you a prize. You will win either A or B, I am not saying which one.

And if your statement is false, you won’t get any prize.

It’s clear from this that you want to make a statement that is clearly true, because otherwise you won’t win any prize. You can say something like “One plus one equals two.” This is clearly true, and you will win either A or B.

However, let’s say you really would like to win prize A. The question is: what statement can you make that will ensure that you will win prize A?

[From Forever Undecided: A Puzzle Guide to Gödel]

Wise Men with Hats [Puzzles]

The first one is an old classic hat puzzle; I’ve put it here as a warm-up for the second one which is a tad more challenging.

Puzzle #1 Three wise men are told to stand in a straight line, one in front of the other. A hat is put on each of their heads. They are told that each of these hats was selected from a group of five hats: two black hats and three white hats. The first man, standing at the front of the line, can’t see either of the men behind him or their hats. The second man, in the middle, can see only the first man and his hat. The last man, at the rear, can see both other men and their hats.

None of the men can see the hat on his own head. They are asked to deduce its color. Some time goes by as the wise men ponder the puzzle in silence. Finally the first one, at the front of the line, makes an announcement: “My hat is white.”

He is correct. How did he come to this conclusion?

Puzzle #2 One hundred persons will be lined up single file, facing north. Each person will be assigned either a red hat or a blue hat. No one can see the color of his or her own hat. However, each person is able to see the color of the hat worn by every person in front of him or her. That is, for example, the last person in line can see the color of the hat on 99 persons in front of him or her; and the first person, who is at the front of the line, cannot see the color of any hat.

Beginning with the last person in line, and then moving to the 99th person, the 98th, etc., each will be asked to name the color of his or her own hat. If the color is correctly named, the person lives; if incorrectly named, the person is shot dead on the spot. Everyone in line is able to hear every response as well as hear the gunshot; also, everyone in line is able to remember all that needs to be remembered and is able to compute all that needs to be computed.

Before being lined up, the 100 persons are allowed to discuss strategy, with an eye toward developing a plan that will allow as many of them as possible to name the correct color of his or her own hat (and thus survive). They know all of the preceding information in this problem. Once lined up, each person is allowed only to say “Red” or “Blue” when his or her turn arrives, beginning with the last person in line.

Your assignment: Develop a plan that allows as many people as possible to survive. [Source: TienrneyLab ]

I’ll provide answers in the Comments section next week.

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