The World of Paradox

This is a collection of some of the well known paradoxes and interesting logic puzzles.

There are two boxes on the table: one opaque and one transparent. The transparent box has a dollar bill in it. The opaque box is empty at the moment. You have two choices: take the opaque box only or take both boxes.

One hour later, both boxes are removed. A computer called the decision prediction machine predicts the choice you have made. From experiments, the machine has 99% chance of predicting you decision correctly.

If the prediction machine predicts that you will take the opaque box only, a thousand dollars will be put into the opaque box. On the other hand, if it predicts that you will take both boxes, the opaque box will be left empty.

The boxes are returned to the table and you pick the box(es). Note that you have 99% chance of getting $1000 by picking the opaque box only. On the other hand, you always get $1 more by taking both boxes, regardless of the contents in the opaque box. What choice would you make?

This is a puzzle about a man condemned to be hanged. The man was sentenced on Saturday. "The hanging will take place at noon," said the judge to the prisoner, "on one of the seven days of next week. But you will not know which day it is until you are informed on the morning of the day of the hanging."

The judge was known to be a man who always kept his word. The prisoner, accompanied by his lawyer, went back to his cell. As soon as the two men were alone the lawyer broke into a grin. "Don’t you see?" he exclaimed. "The judge’s sentenced cannot possibly be carried out."

"I don’t understand," said the prisoner. "Let me explain. They obviously cannot hang you next Saturday. Saturday is the last day of the week. On Friday afternoon, you will still be alive and know with certainty that the hanging will be on Saturday. You would know this before you were told on Saturday morning. That would violate the judge’s decree."

"True," said the prisoner. "Therefore, Saturday is ruled out," continued the lawyer. "This leaves Friday as the last day they can hang you. But they can’t hang you on Friday because by Thursday afternoon, only two days would remain : Friday and Saturday. Since Saturday is not possible, the hanging would have to be on Friday. Your knowledge of this fact would again violate the judge’s decree. So Friday is out. This leaves Thursday as the last possible day. But Thursday is out because if you are alive on Wednesday afternoon, you will know that Thursday will be the day."

"I get it," said the prisoner, who was beginning to feel much better. "In exactly the same way I can rule out Wednesday, Tuesday and Monday. That leaves only tomorrow. But they can’t hang me tomorrow because I know it today!"

The prisoner is thus convinced, by what appears to be unimpeachable logic, that he cannot be hanged without violating the judge’s decree. Then, on Thursday morning, the hangman arrives. Clearly, the prisoner did not expect him. What is more surprising, the conditions stated by the judge is now fulfilled.

What went wrong with the reasoning provided by the lawyer?

Once upon a time, there was a small country called Mamajorca, where very smart women rule the land.

One morning, the Queen decided to put an end to the problem of male infidelity in Mamajorca. She summoned all the women heads of households before her and read the following statement :

"There are (one or more) unfaithful husbands in our country. Although none of you knew before this gathering whether your own husband was faithful, each of you know which of the other husbands are unfaithful. I forbid you to discuss the matter of your husband’s fidelity with anyone. However, should you discover that your husband is unfaithful, you maust shoot him on the midnight of the day you found out about it."

Thirty-nine silent night went by, and on the fortieth night, shots were heard.

Was any innocent husband shot? Did any unfaithful husband escape? Why were there shots on the fortieth night, but not before?

This problem is very similar to Newcomb’s paradox.

Two prisoners were arrested for committing a crime. However, the police does not have enough information to convict them.

The prisoners were put into separate cells. The police officer went to talk to each of them separately. "Each of you has two choices: either keep mum or give us enough information to convict your accomplice," the police officer said.

"If both of you keep mum, we do not have sufficient evidence, so we will sentenced you to one month’s jail for petty crime. If both of you give us the information we need, you will each serve six month’s jail for the crime you committed. However, if one keeps mum and the other provides the information, the one who keeps mum will serve nine month’s jail for the crime and obstructing justice. The one who helped us will be rewarded and will be freed immediately."

Both prisoners know their choices and that the police officer told both of them exactly the same thing. What would their choices be?