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


Project Starters

Challenging puzzlers to create a positive attitude to real projects.

Betting Math

Leslie Green’s Casino Problem

A bunch of Physicists, fresh from College, go to their local casino with plans to profit with their advanced gaming strategiesCasino Roulette - 3d render for Roulette. They have chosen a Roulette game where there are 36 numbers from 1 to 36, alternately colored black and red. There is only one green slot, which is usual in Europe but unusual in the USA. When the ball lands in the green slot, 0, all bets are lost to the House.

All participants place 223 bets. Each bet pays out at 1/1 odds, meaning you get your bet back plus an equal amount of winnings.

John bets only on RED, and only immediately after a run of 3 reds.

Sally bets only on EVEN, but only after the sequence ODD, ODD, EVEN, ODD.

Peter looks for a run of 8 events, either all ODD, all EVEN, all RED or all BLACK. He then bets against this run continuing.

At the end of the 223 bets, who is the most likely to have won?

Leslie Green’s Sports Betting Problem

736The all-male under-10 football team, the Hunky Heroes, are playing the all-girl under-12 football team, the Girly Girls. I am facilitating the betting, although this may not be strictly legal in all jurisdictions.

Buddy and his friends have put a total of $200 on the favorites, the Hunky Heroes, to win. If they win I have to pay out a total of $3 for every $2 placed.

Samantha and her friends have put a total of $50 on the Girly Girls to win. If they win I will pay out a total of $6 for each dollar placed. (Some people would call this 5/1 odds, where you win $5 for each dollar placed AND you get your $1 bet back).

Some of the parents have individually placed a total of $100 on a draw. In this case I pay out a total of $3 for each dollar placed.

Who has the greatest chance of winning?

Casino Statistics

In a casino on a Sunday,  90% of the visitors lost $200 each, # 9% of the visitors lost $1,000 each, and # the rest won $10,000 each. # If the profit of the casino is $340,000, how many people visited the casino?

The answer

Bankrupt Business

Jane and Gerry visit a casino. In one game, they have a 1/5 probability of winning $100 and 1/2 probability of losing $50. They have also a chance of no win / no loss. What is the most likely amount of money they will win (or lose) at the end of 100 games?

The answer

Gambling is bad for your health and budget!

The Importance of Recreational Math

“The body of recreational mathematics that Mr. Gardner tended to and augmented is a valuable resource for mankind.” –

Aplusclick project tries to destroy the border between the Common Core and the Recreational Math.

The man who counted

The man who counted

“In 1932 Malba Tahan published what would became one of the most successful books ever written in Brazil – O Homem que Calculava – The Man Who Counted.”

Beautiful Math Problems

What are characteristics of the most beautiful math problems?


They are practical, give an impression that the problem cannot be solved, and finish by an unexpected (surprise) solution.

It is not about the beautiful math equations or mathematical beauty. It is mostly about recreational math, brain teasers, and thinking outside the box.

Find a short list of my favourite beautiful problems:

1. Morozkin’s problem:


Vladimir Arnold (1937-2010), one of the greatest 20th century Russian mathematicians told the following story:

“Our schoolteacher I. V. Morozkin gave us the following problem: 

Two old women started at sunrise and each walked at a constant (different) velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. 

At what time was the sunrise on this day?”



2. Martin Gardner’s favorite problem

“Three sailors come across a pile of coconuts. The first sailor takes half of them plus half

a coconut. The second sailor takes half of what is left, plus half a coconut. The third sailor also takes half of what remains, plus half a coconut. Left over is exactly one coconut, which they toss to a monkey. 

How many coconuts were in the original pile?”


3. Lucas problem

François Édouard Anatole Lucas (1842 – 1891) was a French mathematician.


Every day at noon, a ship leave Le Havre for New York and another ship leaves New York for Le Havre. The trip lasts 7 days and 7 nights. 

How many ships will a ship leaving Le Havre today meet at sea?


4. Euler bridge problem


In a city Konigsberg, there were seven bridges. 
There was a tradition to walk and cross over each of the seven bridges only once. 
If a person starts and finishes at the same point, can he acomplish this task?


5. Secretary problem


An entrepreneur wants to hire the best person for a position. 
He makes a decision immediately after the interview. 
Once rejected, an applicant cannot be recalled. 
He interviews N randomly chosen people out of 100 applicants, rejects them and records the best score S. 
After that, he continues to interview others and stops when the person has a sc

ore better than S. 

What number N do you recommend to the cruel man?


6. Monty Hall


A venture capitalist will invest in only one of three start-up companies: A, B, or C. 
I will make a lot of money if I invest in the same company, and will lose all of my money if I choose another company. 
I decide to invest in company A and I inform the venture capitalist. 
He assures me that he does not invest in company C. 

What company do you recommend for me to make the investment?


7. The Legend of Carthage


The Legend of Carthage: Queen Dido and her followers arrived in North Africa. 
The locals told them that they could have the coastal area that an ox hide would cover. 
She cut the hide into a series of thin strips, jointed them t

ogether, and formed a coastal shape. 
The ox-hide enclosed area was known as Carthage. 

If you had a 10 km long strip, which shape (rectangle, triangle, semi-circle, or semi-ellipse) would you choose to maximise the enclosed area?


8. Lewis Carroll’s Coaches

twotrainsA coach leaves London for York and another at the same moment leaves York for London. They go at uniform rates, one faster than the other. After meeting and passing, one requires sixteen hours and the other nine hours to complete the journey. What total time does each coach require for the whole journey?


You are welcome to expand the list by submitting your input at


Funny Questions and Unexpected Answers at Tech Job Interview

Brain Teaser Interview Questions And Answers on Amazon

Brain Teaser Interview Questions And Answers on iTunes