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Are you a smart guesser or one that usually takes too much wild guesses?

Last year during the summer holiday, I attended an interview for a job as a telemarketer. I had very much rehearsed my speech and thought of all possible questions that the interviewer may ask me and the appropriate answers. But guess what? The first and only question, the smart lady asked me was this:

Make as if you are watching Captain Jack Sparrow on TV, sailing the pacific in his ship. One of his mates accidentally drops a cannonball over the deepest point on earth, the Mariana Trench. How long will it take for the cannonball to reach the floor of the ocean?

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Before reading on – please try to solve the problem yourself.

Did you make a wild guess because of the “lack of information”? Did you get muddled up in the equations of motion of Newton and the laws of gravity trying to come up with an “exact answer”? Or are you among the smart ones and you have managed to hit the problem on the head with these two important points:

1. How deep is the Mariana Trench?
2. How fast might a cannonball fall through water?

Afterwards, the interviewer told me that most of the candidates simply took up wild guesses and rarely was someone willing to ponder over the problem and come up with an approximation.

What does this has to do with you or me? Everything! In the real world, we frequently need to make decisions when we do not have access to full information. From what clothes to wear or how to raise kids, creative people must think for themselves. There may not be time or money to make success of your decisions. Your best guess will often be the best you can do in most circumstances.

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The above riddle can be solved with what scientists call a Fermi problem, named after Nobel Prize-winning Physicist Enrico Fermi, who used problems such as this to teach his students how to think for themselves. A Fermi problem does not contain all the information you need to solve it precisely.

The classic Fermi problem, generally attributed to Fermi, is How many piano tuners are there in Chicago? A typical solution to this problem would involve multiplying together a series of estimates that would yield the correct answer if the estimates were correct. For example, we might make the following assumptions:

1. There are approximately 5,000,000 people living in Chicago.
2. On average, there are two persons in each household in Chicago.
3. Roughly one household in twenty has a piano that is tuned regularly.
4. Pianos that are tuned regularly are tuned on average about once per year.
5. It takes a piano tuner about two hours to tune a piano, including travel time.
6. Each piano tuner works eight hours in a day, five days in a week, and 50 weeks in a year.

From these assumptions we can compute that the number of piano tunings in a single year in Chicago is :

(5,000,000 persons in Chicago) / (2 persons/household) × (1 piano/20 households) × (1 piano tuning per piano per year) = 125,000 piano tunings per year in Chicago.

And we can similarly calculate that the average piano tuner performs :

(50 weeks/year)×(5 days/week)×(8 hours/day)×(1 piano tuning per 2 hours per piano tuner) = 1000 piano tunings per year per piano tuner.

Dividing gives :

(125,000 piano tuning per year in Chicago) / (1000 piano tunings per year per piano tuner) = 125 piano tuners in Chicago.

This turns out to be reasonably close to the actual number in the yellow pages.

Why was guesswork so accurate? The law of averages is partly responsible. At any point, your assumptions may be too high or too low. But because of the law of averages, your mistakes will frequently balance out.

One of my favourite “guess-estimators” is inventor Stan Mason, who developed microwave cookware specially designed to position food in the best spot for cooking. To do this, Mason needed to know where the microwave’s “hot spots” were – the place where the rays hit the food with the highest intensity. To find out, he put shelves of unpopped popcorn kernels in the microwave and watched to see which kernels popped first. He discovered a pattern in the oven’s hottest rays, they weren’t in the corners or at the centre, but in the shape of a mushroom cloud. Then he designed cooking dishes to fit the pattern. He had come up with a resourceful way to approximate the answer rather than using scientifically sophisticated testing equipment.

Fermi would have approved.

By the way, the Mariana Trench is about eleven kilometers deep, and a cannonball drops at a rate of 3 metres per second. So it will take the cannonball about 1 hour and 1 minute to reach the bottom of the trench. Could this be guessed?

If you know that Earth’s highest point, Mount Everest, is about 10 000 metres high(actually 8900 metres high), you might reasonably conclude that its lowest point is close to the same distance. Then you might imagine that a heavy object will take one second to fall through a 3-metre deep swimming pool. These estimates will bring you close enough to the correct answer.

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And if you were smart enough to read the post carefully, you would have spotted another little problem and maybe would have come up with the answer.

Did I get the job?

I dumped the interview and didn’t get the job. But as life is, I learnt something fundamental that day, which continues to serve me up to now and I sincerely hope it will serve you too.

related reading

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• Count the words from the world shortest stories.
• You’ll Never Guess What’s In McDonald’s Chicken McNuggets!
• Tap a tune with the space bar and it guesses it!!
• Guess-the-Google Game

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