How to Guess Smartly Using Fermi Problems?
How to look out for the smartest guy?
Nowadays many interviewers, whether recruiting for jobs or university admissions have started asking 'out of the box' questions to check the level of smartness in the prospective employees or students. It is a difficult task for the recruitment committee or the admission committee to select the right candidate out of thousands of applicants with good credentials. Usually bright candidates fare well in the written tests which usually comprise of aptitude test, test of English language and some subject-specific questionnaire. Hence the interviews of the short-listed candidates hold a good weightage and to a great extent become the deciding factor whether the person will get through or not.
In this hub, I would give examples of such questions, which are the favourites of the interviewers/examiners and the ones which can differentiate a great thinker from an ordinary one, distinguish a potential troubleshooter and decision maker from any other average person.
Example question 1: You are on a yatch sailing the Pacific Ocean. Your navigator announces that you are over the deepest point, the Mariana Trench. Just then, a clumsy guest accidantally drops a six kilo cannonball over the side. How long will it take for the cannonball to reach the bottom of the ocean? [Source: Reader's Digest June 1997]
Example question 2: Assuming you're not in a big lecture hall and the professor shuts the door at the start of class, how long does it take for you and your classmates to deplete the oxygen enough to feel it? [Source: Physics Buzz]
Let's identify the common features of the above two problems:
1. Seems impossible to answer at the beginning, many people might just answer qualitatively, such as 'long time', 'several hours' without mentally working out the solution. But a good acceptable answer should be quantitative.
2. Certain information in the problems will not be given, so brave estimation and approximation strategies should be adopted. For example the depth of Mariana Trench and the speed with which a cannonball falls under water are not mentioned, so some people might answer, 'insufficient data, so cannot be solved'.
3. These kinds of problems can be solved in multiple ways and can also come out with different answers, so the exactness of the answers are not important, the process of thinking and coming up with the answer is important. Any sensible guesstimation will be close to the correct answer within the 'order of magnitude'.
4. These are multi-step problems and can be best solved by breaking the problem into simple manageable steps. Even if approximation is too high or too low at different stages of solving the problems, the law of averages balance the estimations and generate a close-to-the-true answer.
5. These questions can be successfully solved when a few risks are taken and new knowledge is constructed based on previously existing knowledge.
These kind of problems are called Fermi problems, named after Nobel prize-winning physicist Ernico Fermi who used such problems to teach thinking skills to his students. Making intelligent guesses, estimating a rough figure, approximating, making sensible assumptions are extremely important aspects of problem solving.
In the classrooms or at various stages of learning, we usually focus on deriving the exact answer from the given data, hardly do we practice taking risks and estimating.
How to steer your thinking for the solutions?
Solution to example 1: Let's break the problem into further simpler questions and approximate and make assumptions where necessary.
How deep is the deepest point of ocean, Mariana Trench?
Could be the same as the highest point on Earth, the Mt. Everest which is 8900 metres. Any answer between 8 to 11 km could be thought of approximately.
How fast can the cannonball travel under water?
Any heavy object usually takes about one second to reach the bottom of a 3 m deep swimming pool, so it's speed is approximately 3 metres per second
How long does the cannonball take to reach the bottom?
8900/3 = 2967 seconds which is close to 3600 seconds, that means close to one hour or about 50 minutes. (1 hr = 60 x 60 seconds) (2967/60 = 49 minutes)
ABOUT 50 MINUTES
Isn't it fun and challenging?
Hope by now you've got a hang of how to solve these problems, your answer could be different and even the method of solving could be different and that is the beauty of these problems. The interviewer will not get bored by asking the same quiz to a lot of candidates and would rather enjoy the various ways of thinking and estimation strategies which they use to get to the solution.
But remember, some answer is definitely better than no answer. Not being able to answer will prove that you tend to give up when a crisis situation comes or you hesitate to try new ways when challenges arise.
Solution to example 2: Try to recall what you had learnt in your science classes and the basic information you have about oxygen. The solution is worked out in the table below:
What is the volume of your classroom?
Estimate the volume of any classroom you can think of, mentally calculate the length, width and height and find the volume (length x breadth x height)
How much of oxygen is available in the closed door classroom?
Volume of oxygen is 21% of the air in the classroom, so calculate 21% of the calculated volume above
How many classmates (+ one teacher) are present in a lecture hall?
Think of a number of students in a classroom depending on the average student-teacher ratio in your place
What volume of oxygen is consumed by each person in a day?
Approximately 600 litres intake per day per person
What volume of oxygen is consumed by each person in a minute?
600/24 = 25 litres/60 = 0.4167 litres per minute
Now find out how much time is required for all the people inside the room to consume the available oxygen. You will probably start feeling uncomfortable before the entire oxygen is depleted, hence you can also estimate a little less time than the time required to consume all of the oxygen as your FINAL ANSWER.
Impress your prospective boss or professor!
Instead of giving any vague answer, if you start your way forward and express the process which led to your solution, you have already impressed your future boss/professor. Because as you see, volume of a classroom can vary and number of students can vary from person to person, the approximate 21% and 600 litres are recalled from basic scientific knowledge and the rest is weaved from these existing knowledge and assumptions.
More examples of Fermi questions:
- Fermi problem - Wikipedia, the free encyclopedia
Visit to find the original Fermi question, actually asked by Ernico Fermi to his students. The question was: How many piano-tuners are there in Chicago?
Try to crack these questions using reasonable estimation and approximation strategies.
Create your own Fermi problem!
Be creative, think of more such problems and challenge your peers and siblings, collaborative solving of such problems will sharpen your thinking skills and prepare you for such mind-bending problems whether in interviews or in any other problem solving situations.
For more examples of such Fermi problems, visit the following links:
Books which will give additional ideas:
I personally love reading the following books, which I would recommend all of you to read. These will help broaden your perspective and connects different real world issues in coherent chains. For example, how sale of Hush Puppies in New York city can reduce crime rate in the city?
1. Tipping Point by Malcolm Gladwell
2. What the Dog Saw by Malcolm Gladwell
3. Freakonomics by Steven Levitt and Stephen J. Dubner
4. Super Freakonomics by Steven Levitt and Stephen J. Dubner
I am sure after reading these books, your analytical skills will improve and you will enjoy perceiving the real world scenarios from different angles.
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