Some novel "Fermi Puzzles"

And their solutions

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).