Famous Math and Logic Paradoxes

Math and Logic are full of paradoxes.tortoise

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?

bermudatriangle2. 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?

roadnetwork_24.  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)?Beard grooming.

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?

aplusclickhorn11. 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.



1. Achilles and the Tortoise

2. Bermuda Triangle Paradox

3. Simpson Paradox

4.  Braess paradox

5. Barber Paradox

6. The Two Envelope Paradox

7. Potato Paradox

8.  Leonard Euler’s Paradox

9. Friendship Paradox

10. Uninteresting Number Paradox

11. Gabriel’s Horn Paradox

12. Pop Quiz Paradox


Logical Reasoning in Pattern Recognition

Bongard problem is a kind of puzzle invented by the Soviet computer scientist Michael Bongard (1924–1971) in the mid-1960s. He died in 1971 during a hiking expedition in the Pamir Mountains. The tests played an important role in the disciplines of cognitive psychology and cognitive science. Human logical reasoning has a great advantage over computer intelligence.

Be smart.   Train your brain!

Here several problems similar to the original Bongard problems go:

What is the main difference between the pictures on the left page and on the right page?

A. z5461

B. z5462

C. z5468

Try to solve the problem yourself before looking for the answers  in the A+Click Brainteaser Problems.



Harry Foundalis collected hundreds of Bongard problems.


The logic puzzle almost everyone gets wrong

This is a very famous logic problem:

“Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person?

  • A: Yes
  • B: No
  • C: Cannot be determined”


According to the Keith E. Stanovich in Scientific American more than 80 percent of people choose C, which is not the correct answer. The correct answer is A. Why?

This is another example of a similar problem:

There are only two handshakes in a meeting: John shakes hands with a person, this person shakes hands with Anna. Does a man shake hands with a woman in the meeting?


How to Solve a Problem – Read the Question

Try to solve the following problem suggested by Leslie Green:


What is the difference between the number of letters in UPPER CASE and lower case in the text?

Most of the people and me too answered the question fast: None.

If you give the same answer, read the question again.

The trivial advice is often useful: “Try to answer exactly to the question.”

Do you get another answer? What is it?

1932Another example:  A shopkeeper of a Dairy stands six feet tall and wears size 13 sneakers. What does he weigh?



Everyday Geek Questions

What happens once in a minute, twice in a week and once in a year?


I will be X years old in year X2. How old am I?


How many planes of symmetry does a cube have?


“1” costs $10. “20” costs $20. How much does “100” cost?


Crazy Puzzle

What number and why?Aplusclick crazy puzzle

Wittygon – Game and Math Lesson

aplusclickWittygonAplusclick has just launched Wittiygon. It is a mix of Math practice and a logic game.

icon-173-tile Aplusclick Logic Puzzle Wittygon is a free educational game for Very Intelligent People (VIP) in grades one through twelve and smart adults.

The game is free and ad-free.

 What is the story about?

In the Wittygon planetary system, the aim is to move a planet to a nearest neighbour in order to exchange positions. Match 4 or 5 planets in a row to win stars.  You win additional points if you correctly answer IQ Bonus Questions. The questions will be automatically adapted to your skills and motivation. The collection of the questions includes more than 400 logic questions, which don’t ask for special mathematical knowledge.

Where can you play?

Online: www.wittygon.com ,   iPhone or iPad,    Android:

online aplusclickAppsStore AplusclickGooglePlay

Enjoy the challenge – become clever!