A man walks into a casino and spots an interesting new game. He is presented with two buttons, either of which he can push. One of the buttons will win him a certain amount of money and the other will win him twice the amount, although he doesn’t know what the amount is. So he pushes one of the buttons at random and is surprised and very pleased to learn he just won $1 million. Suddenly, the machine presents him with two new buttons. He can push one to keep his original choice, keeping the $1 million. Or he can switch his choice to the button he didn’t push. What would you do and why?

A father tells his children, a boy and a girl, to play in the backyard without getting dirty, but they play in the mud anyway. Both children get mud on their foreheads. The father says, “at least one of you has mud on his or her forehead, which I asked you not to do. I’m going to take away desert tonight, but I’ll give you a chance to get your desert back.” He tells them he will ask the following question twice: do you know if you have mud on your forehead? Each child must answer honestly and simultaneously each time the question is asked. Each child can see the other’s forehead but not his or her own. If both children can correctly tell whether or not they have mud on their foreheads, they can have desert that night. What should the children do?

The solution is here.

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A nobleman is throwing a party in 24 hours, and he wants to serve wine from 1000 casks in his cellar. Unfortunately, an assassin has poisoned one of these casks. The poison takes 23 hours to take effect, and it shows no symptoms before causing death. The nobleman has servants who are “willing” to taste the bottles.

What is the least number of servants the nobleman needs to correctly identify the poisoned bottle, and what is his strategy?

The solution is here

The physicist Enrico Fermi used to pose seemingly complex puzzles which could be solved with some simple logic. Here are some of my own “Fermi puzzles” from Biology.

Fermi Puzzle 1: The fertilized human egg cell divides and differentiates (on average) into ~ 3.72 x 10¹³ cells, which form the tissues and organs in an adult. On average, the body replaces itself with a largely new set of cells every 7 to 10 years, with some tissues (gut lining, skin) being renewed even more rapidly. Given this data, how many cells does an adult body produce (on average), per second? Recall that the number of seconds in a year is approximately 3.15 x 10⁷.

Solution: Given the data, the body produces 3.72 x 10¹³/(7-10) cells per year or 3.7-5.3 x 10¹² cells in one year. 1 year = 3.15 x 10⁷ seconds. Hence the body produces, on average, 1.2-1.7 x 10⁵cells per second. And it does so with astonishing fidelity.

Fermi Puzzle 2: The human genome consists of about 6.4 billion nucleotides (A,C,G,T).

a. If the size of a nucleotide is approximately 0.34 x 10^-9 meters, what would be the length of the DNA in one cell if it were stretched out in a straight line?

b. What is the total length L of DNA in an average body (use the data given in problem 1)? The distance from the earth to the sun is called an Astronomical Unit = 1.5 x 10¹¹ meters. How many round trips to the sun does L represent?

Solution:

a. Length of DNA in one cell = (length of 1 bp) x (number of bp per cell) = (0.34 x 10^-9 meters) x (6 x 10⁹) ~ 2.04 meters.

b. L = Total length of DNA in an adult human = (Length of DNA in one cell) x (number of cells in the body) = (0.34 × 10^-9 m)(6 × 10⁹)(3.72 x 10¹³) ~ 7.6 x 10¹³ meters

d = 1 Astronomical Unit (AU) = Distance from EARTH to SUN = 1.5 x 10¹¹ meters. Hence L/(2d) ~ 253.

This means that the length of DNA in one human is the equivalent of over 250 round trips between the earth and the sun.

Fermi Puzzle 3: Worldwide, about 7 million people have died from Covid-19 as of April 2025. If these people stood one behind the other with a 1-foot gap between them, how far would the line stretch?

Solution: Assume that the “thickness of a person” is about 1.5 feet. The 6.58 million people line would stretch 7 *(1.5+1)* 10⁶ feet = 17,500,000 feet. 1 mile = 5280 feet – so this is equivalent to ~ 3314 miles. As a comparison, the driving distance from New York to San Francisco is 2900 miles.

Fermi Puzzle 4: The current world population is about 7.8 billion people. Assume that 100 humans left Africa to populate the rest of the world about 60,000 years ago and the generation time is 30 years.

a. Estimate the average growth rate of the population (% increase per generation) from this data.

b. Estimate how many humans have lived on the earth in the past 60,000 years.

Solution: The number of generations since the “Out of Africa” event is 60,000/30 = 2000.

a. In this time the population went from 100 to 7.8 x 10⁹. If the growth rate is “a” per generation, then in 2000 generations 100 people would become: 100 (1+a)²⁰⁰⁰ = 7.8 x 10⁹. Hence, a = 0.0091 or approximately 1% every generation. In fact, the growth rate in the distant past was much lower and the rate now is much higher.

b. The number of humans who have lived on the earth in the past 60,000 years is:

100 [ 1 + e^a + e^2a + ...e^2000a] = 100 *(e^2001a-1)/(e^a-1) ~ 8.6 x 10¹¹.

This is an overestimate (can you see why?)

Fermi Puzzle 5: Every human has two parents who have two parents etc. Going back 2000 year, how many ancestors do any 20 people selected at random have? How does this number compare to the result of problem 4b? How do you resolve the apparent paradox? If you choose two people at random, assuming complete mixing, how far many generations back would you have to go before they would share a common ancestor with high probability?

Solution: 2000 years is approximately 2000/30 = 66.7 generations. In this time 20 people have 20 x 2⁶⁷ ancestors. Now 2¹⁰ ~ 10³ so 20 x 2^66.7 = 20 x (2¹⁰)^6.67 ~ 20 x (10³)^6.67 = 2 x 10²¹. From Problem 4b, the number of people who have ever lived on the earth is ~ 10¹². So, these 20 random people have a factor of 2x10⁹ more ancestors than the number of people who have ever lived on the earth! This does not make sense. WHAT IS GOING ON? The only explanation is that these ancestors cannot all be different people. Any random set of people share a huge number of ancestors. We are all related in our ancestry.

How can we estimate how far back (how many generations) we need to go for two random people to have a common ancestor? A simple estimate is obtained as follows: The number of ancestors for one person in n generations is ~ 2ⁿ. When this equals the number of people who have ever been alive, then this person must share an ancestor with at least one other person. This happens when 2ⁿ = 10¹² which gives n =39.9. You share a common ancestor with any other random human if you go back 40 generations ~ 1200 years.

Fermi Puzzle 6: Bacterial generation times range from 1 minute to over 30 hours. Let us consider e coli, which double every 20 minutes in ideal conditions. The mass (wet) of e coli is 10^-12 gm. How long would it take a single e coli to make a colony with the same mass as the earth (~ 6 x 10²⁷ gm).

E-coli double once every 20 minutes. After 20 x n minutes we will have 2ⁿ e coli. The ratio of the mass of the earth to that of an e-coli is 6 x 10²⁷/10^-12 = 6 x 10³⁹. For the colony to equal the mass of the earth would require n generations where 6 x 10³⁹ = 2ⁿ.

This gives, n = 132.14 generations. Since each generation is 20 minutes, this would take, 20 x 132.14 = 44 hours (less than 2 days).

By how much would sea levels rise if the icepack on Antarctica melted? The icepack on Antarctica is over 2 km thick on average. The land area of Antarctica is about 5.4 million square miles and the surface area of the earth is about 196.9 million square miles. This means that Antarctica’s surface area is roughly 2.7 % that of the earth. About 70% of the earth’s surface is ocean. Hence, if all the icepack on Antarctica melted, and the water was added to the oceans, it would raise the ocean levels by: 2000 x 0.027 / 0.7 meters = 77 meters. Even if we cover the melt water over the whole of the earth’s surface, the water level increase would be 2000 x 0.027 meters = 54 meters. This means that if the icepack on Antarctica melted, almost all coastal towns on earth would be under water.

A man needs to go through a train tunnel to reach the other side. He starts running at 10 mph through the tunnel in an effort to reach his destination as soon as possible. When he is ¼ th of the way through the tunnel, he hears the whistle of the train behind him. Assuming the tunnel is not big enough for him and the train, he has to get out of the tunnel in order to survive. We know the following: Given 1. If he runs back, he will make it out of the tunnel by a whisker. Given 2. If he continues running forward, he will still make it out through the other end by a whisker. What is the speed of the train?

Solution is here

Five mathematically minded pirates discover a chest containing 100 gold coins. They sit down and devise a distribution strategy. The pirates are ranked Pirate 1 to Pirate 5, where Pirate 1 is the highest ranked. The highest ranked pirate gets to propose a plan on how to distribute the gold and then all the pirates vote on it. If at least half of the pirates agree to his plan, the gold is split according to the proposal. If not, the proposing pirate is thrown off the ship and the process will continue with the remaining pirates until a proposal is accepted. The first priority of the pirates is to stay alive and second is to maximize the gold they get. Pirate 1 devises a plan which he knows will be accepted for sure and will maximize his gold. What was the plan he proposed?

Solution is here

You walk up a mountain starting at 6 AM and get to the top at 6 PM. You spend the night at the top and on the next day, you start walking from the top at 6 AM and get to the bottom at 6 PM along the same path. You do not necessarily travel at the same speed on the two days. Show that there must be at least one clock time of day when, while going up and coming down, you were at the same height.

Solution is here

You go to the bank and take out some money (say X dollars and Y cents). By mistake, the teller gives you dollars instead of cents and cents instead of dollars, i.e. you got Y dollars and X cents. On the way home, you buy a newspaper for 20 cents. When you get home, you find you have twice as much money in your wallet as what you had originally intended to withdraw. What are X and Y?

Solution is here

Five pirates are marooned on an island and spend the day collecting coconuts, intending to divide them up the next morning. One of the pirates decides to get more than his share. So, he gets up at night and finding that the coconuts they had collected do not divide by five unless one coconut is removed. So he gives one coconut to a monkey, removes 1/5 of the remaining hides them. Each pirate does this in turn, giving one to the monkey and hiding 1/5 of the rest. When they wake up, everyone notices the diminished pile but since they all stole, no one says anything. So, they give one to the monkey and take 1/5 th of the rest each. What is the least number of coconuts for which this is possible?

Solution is here

This is a problem which is extensively discussed online with videos and text. Here we present the general solution.

Here is the Problem: N people are lined up facing in the same direction. With their eyes closed, hats are placed on their heads. The hats are of two colors, either white or black. Each person can see the hats on the heads of people ahead of him. Starting from the last person in the line, they are asked to predict the color of the hat on their heads. What is the maximum number of correct answers possible if they use some rule?

Solution is here

A wants to send B a secure message. He puts it in a locked box and sends it via the postal service. The postal service is reliable and will get the box to its destination without the lock being opened and the message read. However, B does not have the key to open the box. How will he get the message?

Solution is here

Once upon a time, in a far away kingdom, there lived a king who was a mathematician. He wanted to find a suitable friend with whom to discuss mathematics. He invited the best mathematicians in his kingdom to come and solve some riddles that he posed to them. The first one of these was to determine which of the above two boxes has the gold.

Can you solve it?

Solution is here

An Arab sheikh wants to will all his fortune to only the wisest among his two sons.He makes them a proposition. His two sons will ride their camels in a race, and the son whose camel crosses the finish line last will inherit it all. The two brothers wander aimlessly for days, neither willing to cross the finish line. In desperation, they ask a Wise Dervish for advice. He tells them something. The brothers leap onto the camels and charge toward the finish line. What did the Dervish tell them?

Solution is here

There is a wonderful book by Raymond D. Smullyan that everyone who likes puzzles should read.

Its name is: What Is the Name of This Book?: The Riddle of Dracula and Other Logical Puzzles.

Here are some puzzles from this book. All the puzzles below are about the “Island of Knights and Knaves.” Knights always tell the truth and Knaves always lie.

Puzzle 1: A stranger from out of town meets three inhabitants A, B, C standing in a garden. He asks A: “Are you a Knight or a Knave?” A mumbles something the stranger could not hear so he asks B: “What did he say?” B said: “He said he was a Knave.” C said: “Don’t believe B, he is lying !” What are B and C (Knights or Knaves)?

Puzzle 2: Now the stranger meets two people A and B

A says: “At least one of us is a Knave.” What are A and B?

Solutions are here

And here is one more that I will leave you to solve on your own:

Puzzle 3: The stranger from out of town meets three other inhabitants A, B, C. He asks A: “How many Knight are among you?” A mumbles something the stranger could not hear so he asks B: “What did he say?” B said: “He said there is one Knight among us.” C said: “Don’t believe B, he is lying !” What are B and C (Knights or Knaves)?

An old farmer had 19 horses that he willed to his three children as follows: The oldest child gets half the horses, the middle child gets one fourth of the horses and the youngest child gets one fifth of the horses. When the old man died, the children did not know how to divide the horses without killing them. They asked for help from a wise neighboring farmer, who came over the next day riding on his horse. How did he solve their problem?

Solution is here

Ramsey theory is a branch of mathematics that looks for ordered subsets in random systems when they get large enough. The simplest problem in Ramsey theory is the following: What is the smallest number of people to guarantee a clique of size 3? A clique in this context is defined as a set of people who are either friends or strangers. To make the problem simpler, we will give the answer. A clique of size 3 is guaranteed among six people. The problem we will pose is to prove that this is true.

Prove that in a group of six people, there must be three people who are friends (know each other) or strangers (do not know each other).

Solution is here

Two people, let us call them A and B, decide to have a duel. They agree to meet at a designated spot sometime within the hour between noon to 1 PM. Each would arrive at a random time and stay for 5 minutes. If the other arrived within these 5 minutes, they would fight. Otherwise, they would leave, honor served. What is the probability that they will fight?

Solution is here

In a room with N > 1 people, show that there are at least two people who know the same number of people in the room. The assumption is that knowing someone is reciprocal so if A knows B, then B knows A.

Solution is here:

Prove that if people in a room shake hands with whoever they know, the number of people who shake hands with an odd number of people is even. You are given that there are at least two people who know each other.

Solution is here