Department of
Philosophy

Puzzles and Paradoxes

There Are Two Errors in the
the Title of This Page

Many philosophers enjoy amusing puzzles and paradoxes when not doing more serious work; we hope you will too. The puzzles here have solutions: with careful analysis you can figure them out but the answer may not be what you first expect. The paradoxes provide an amusing way to raise deep philosophical problems; you may enjoy thinking about them.

Puzzles

On the Island of Knights and Knaves: On the island of Knights and Knaves, every inhabitant is either a knight or a knave. Knights always tell the truth. Knaves never tell the truth; any sentence uttered by a knave is false. A stranger came to the island and encountered three inhabitants, A, B, and C. He asked A, "Are you a knight, or a knave?" A mumbled an answer that the stranger could not understand. The stranger then asked B, "What did he say?" B replied, "A said that there is exactly one knight among us." Then C burst out, "Don't believe B, he is lying!" What are B and C?

One day I went to the island of knights and knaves and encountered an inhabitant who said, "Either I am a knave or else two plus two equals five." What should you conclude? 
(Knights and Knaves puzzles by Raymond Smullyan)

The Surprise Test: One day the professor came into the class and announced, "Next week I will give you a surprise test. It will be a surprise, because you won't be able to figure out on which day it will occur until the class meets on the day of the test. It could happen on Monday, Tuesday, Wednesday, Thursday, or Friday, but I won't tell you which day."

The class were clever. They reasoned as follows: "She can't give the test on Friday, because then it wouldn't be a surprise; we'd know after class on Thursday that the test hadn't yet occurred, and hence we'd figure out that it would have to be on Friday. So we know the test can't be on Friday. But then it can't be on Thursday either, because if it were, we would know after class on Wednesday that it would have to be on Thursday, since it wouldn't have happened yet, and we have already shown that it can't be on Friday." Reasoning in this manner, the students concluded that the test could not occur on Wednesday either, nor on Tuesday, nor on Monday. Having concluded that a surprise test was impossible, the students didn't study. They were very disappointed and very surprised on Wednesday when they got a test. Where did the students' reasoning go wrong?

The Problem of the Light Switch: I have an ordinary light switch connected to a light. When the switch is closed, the light is on. When the switch is open, the light is off. At two minutes to noon, the light is on. At one minute to noon I flip the switch, turning the light off. At half a minute to noon I flip it again, turning the light on. At a quarter of a minute before noon I flip it again, turning the light off. I continue in this way, cutting the time between flippings of the switch in half each time. Now this will be an infinite series of flips. The switch flippings will occur closer and closer to noon, but will all be completed before noon. Will the switch be on or off at noon?

The Super Bullet: The Acme Arms Company has invented a Super Bullet: a Super Bullet penetrates anything it hits. But the Adamantine Armor company has invented a Super Strong Armor Plate: nothing that hits a Super Strong Armor Plate penetrates it. The army is planning to shoot a Super Bullet at a Super Strong Armor Plate. What will happen?

The monkey: Hanging over a pulley there is a rope with a weight at one end; at the other end clings a monkey of equal weight. The rope weighs 4 ounces per foot. The sum of the ages of the monkey and its mother is eight years, and the weight of the monkey is as many pounds as its mother is years old. The mother is twice as old as the monkey was when the mother was half as old as the monkey will be when the monkey is three times as old as its mother was when she was three times as old as the monkey was. The weight of the rope and weight is half again as much as the difference between the weight of the weight and the weight plus the weight of the monkey. How long is the rope?

Paradoxes

The problem of the Two Sentences: Consider the first of the following sentences -- is it true or false?

1) The sentence numbered 2) is true.

2) The sentence numbered 1) is false.

Have you found two errors in the main heading of this page yet? (Title adapted from Robert M. Martin)

The Ship of Theseus: Theseus has just bought himself a brand new wooden ship. He is a maintenance fanatic. Whenever some part of his ship gets a mark on it, he removes the part and replaces it with a new duplicate. Over many years, Theseus has eventually replaced every single original part with new wood. He is proud of how well he has kept up the condition of his ship.

Xanthippe buys used ship parts for refurbishing and resale. Theseus has been a source of high quality material. Xanthippe has bought from him each part of his ship as he replaced them. As it happens, Xanthippe has not sold any of them. She realizes that the now has the parts of a whole wooden ship, and they all have just minor cosmetic flaws. Xanthippe cleans of the marks on the wood and re-assembles the parts into a ship, placing them back in their exact original arrangement.

Does Theseus still own the ship that he purchased? If so, how does the ship that he purchased differ from the ship that Zanthippe now has? If not, when did Theseus stop owning the ship that he purchased?

(This version of the classic story was written by Earl Conee.)

The Malfunctioning Transporter: You step into the Transporter on the Star Ship Enterprise to be transported down to the surface of the planet Duplo. However, while you are being transported, an unfortunate energy surge occurs, with surprising results. Two copies of you materialize on the planet's surface instead of one. Each is a molecule-for-molecule duplicate of you as you were when the transporting process began. Each of the duplicates claims to be you, but they can't both be right. For if each is identical to you, they are identical to each other, because things identical to the same thing are identical to each other. But these two duplicates are obviously not identical: they are two distinct people. On the other hand, it is very difficult to think of any basis for saying that one of them is you and the other isn't, for any relation to your earlier self that the one has the other has as well. Of course, you could say that neither one is you, but why? Each of them has as much claim to be you as you have after any other transporter journey. What has happened to you?

Answers to the puzzles are available on the web. Answers to the paradoxes are not so easy to come by, but you could always take a philosophy course or two and see what you can figure out.