# Famous Math and Logic Paradoxes

November 9, 2016 1 Comment

Math and Logic are full of paradoxes.

**1. Achilles and the Tortoise** The Paradox of Achilles and the Tortoise was described by the Greek philosopher **Zeno of Elea** in the 5th century BC. The great hero Achilles challenges a tortoise to a footrace. He agrees to give the tortoise a head start of 100m. When the race begins, Achilles starts running, so that by the time he has reached the 100m mark, the tortoise has only walked 10m. But by the time Achilles has reached the 110m mark, the tortoise has walked another 1m. By the time he has reached the 111m mark, the tortoise has walked another 0.1m, then 0.01m, then 0.001m, and so on. The tortoise always moves forwards while Achilles always plays catch up. Why is Achilles always behind the tortoise?

**2. Bermuda Triangle Paradox. **Why the sum of the interior angles of the Bermuda triangle is not 180 degrees?

**3. Simpson Paradox. **The average score for dance of boys and girls in class A are 16 and 21, respectively. The average score of boys and girls in class B are 15 and 20, respectively. Twenty percent of class A students are girls. Forty percent of class B students are girls. Which class has a higher average score?

**4. Braess paradox.** The diagram shows a road network. All cars drive in one direction from A to F. The numbers represent the maximum flow rate in vehicles per hour. Engineers want to construct a new road with a flow rate of 100 vehicles per hour. Drivers randomly choose the road at crossroads. What new road decreases the capacity of the network (the number of vehicles at point F)?

**5. Barber Paradox **In a city, the barber is the ‘one who shaves all those, and those only, who do not shave themselves.’

Who shaves the barber?

**6. The Two Envelope Paradox.** One envelope has twice as much money as the second one. Gerry does not know which envelope contains the larger amount. He takes one of the envelopes, counts the money, and is offered the chance to switch the envelope. He thinks “If the amount of money in the chosen envelope is X dollars, then the other envelope contains either 2X of 0.5X dollars, with equal probability of 0.5. The expected value of switching is 0.5 (2X) + 0.5 (0.5X) = 1.25X. This is greater than the value in the initially chosen envelope. It is better to switch.” What is your advice?

**7. Potato Paradox.** I have 100kg of potatoes, which are 99 percent water. I dry them until they are 98 percent water. How much do they weigh now?

**8. Leonard Euler’s Paradox. **Why the average of all of the numbers is not a zero?

1, -1, 2, -2, 3, -3, . . .

9. **Friendship Paradox. **Your friends have more friends than you. Why?

**10. Uninteresting Number Paradox. **How many uninteresting numbers exist?

**11. Gabriel’ Horn Paradox.** The shape obtained from rotating the equation about x-axis resembles a trumpet. If we need an infinite volume of paint to paint the infinite horn, how much paint does the horn can contain inside itself?

**12. Pop Quiz Paradox.** A teacher announces that there will be a quiz one day during the next week. The teacher gives the definition that they would not when they come in to the class that the quiz was going to be given that day. The brightest student says that the quiz cannot be on Friday because they will know the day. With the same technique, she eliminates Thursday, Wednesday, Tuesday, and Monday. “You cannot give us a pop quiz next week” she says. When does the teacher give the pop quiz? I know the paradox from Charles Carter Wald. Probably, Martin Gardner described it for the first time in The Colossal Book of Mathematics.

**Answers**

10. Uninteresting Number Paradox

In the Two Envelope Problem, you have both too much, and not enough, information to answer the question. Too much, because the solution you gave applies only if Gerry doesn’t look in his envelope and count the money. But he did. And too little, because once he determines what his value X is, the answer no longer depends on the probability he would choose the higher or lower value, but on the probability that the set would be constructed with half, or twice, that amount in the other envelope.

For example, say two sets of envelopes were prepared, one set with $10 and $20, and the other with $20 and $40. Then they are shuffled, and only one set is given to Gerry. If he opens it and finds $20, the expected value in the other envelope is indeed $25. But if he finds $10 to $40, the other envelope is certain to contain $20.

The point of the paradox, is that Gerry’s solution assumes that the distribution of possible envelopes is uniform over all values from 0 to infinity, which is not possible. This can be easier to understand if you add a third envelope to my set (with $40 and $80), then a fourth, etc. Gerry’s solution will always be correct, except for the lowest and highest possible values.