Growing up as a singularly unathletic child, my favourite form of recreation was usually through books. And a favourite amongst the books were Agatha Christie's murder mysteries featuring Hercule Poirot. Poirot fascinated me. I guess there was the element of vicariously living through the act of evil being punished by good. (Which probably attracts us to all mystery writers).
But another aspect that made Poirot more appealing than the more energetic specimens like Sherlock Holmes was his reliance on "ze little grey cells". The power of analytical reasoning practiced through the simple mechanism of question and answer being used to solve fiendishly difficult murders. How romantic an idea!
I recently came across a intriguing set of problems, which require rigorous exercise of the grey cells. Called Fermi problems, these are just plain old estimation problems. Typical examples go like "estimate the number of piano tuners in New York City", "estimate the number of Mustangs in the US". It requires one to start out with some basic facts and figures and then get to the answer through a number of logical reasoning steps. For the piano tuner question, it is usually good to start off with some estimate of the population of New York City. Starting off with ridiculous numbers (like 1 million or 100 million) will definitely lead you to the wrong answer. So, what the estimation exercise really calls for is some general knowledge with some ability to think and reason logically.
The solution to the piano tuner problem typically goes as follows:
- Number of people in NYC -> Number of households
- Number of households -> Estimating the numbers with a piano
- Number of pianos -> Some tuning frequency -> Demand for number of pianos that need tuning in a month
- Assuming a certain number of pianos that can be tuned in a day and a certain number of working days, you get to the number of likely tuners
The exercise definitely teaches some ability to make logical connections. The other thing this type of thinking teaches is parsimony of assumptions. One could make the problem more complex by assuming a different population for NYC's different buroughs, different estimates for the proportion of households with pianos for Manhattan vs the Bronx and so on. In practice however, these assumptions only introduce false precision to the answer. Just because you have thought through the solution in an enormous degree of detail doesn't necessarily make it right.
Some typical example of Fermi problems can be found at this link. Enjoy the experience. And I would love to hear some of the feelings that strike you as you try and solve these problems. Some of the "a-ha" moments for me were around the parsimony of assumptions, needing to find the point of greatest uncertainty and then fix it with the least cost, so as to narrow down the range of answers.
The exercise overall taught me a fair bit about about modeling, the way we approach modeling problems, ranges of uncertainty and how we deal with them, parsimony of assumptions.