How Many Licks?

Or, How to Estimate Damn Near Anything


By Aaron Santos

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How many licks to the center of a Tootsie Pop? How many people are having sex at this moment? How long would it take a monkey on a typewriter to produce the plays of Shakespeare? For all those questions that keep you up at night, here’s the way to answer them. And the beauty of it is that it’s all approximate!

Using Enrico Fermi’s theory of approximation, Santos brings the world of numbers into perspective. For puzzle junkies and trivia fanatics, these 70 word puzzles will show the reader how to take a bit of information, add what they already know, and extrapolate an answer.

Santos has done the impossible: make math and the multiple possibilities of numbers fun and informative. Can you really cry a river? Is it possible to dig your way out of jail with just a teaspoon and before your life sentence is up?

Taking an academic subject and using it as the prism to view everyday off-the-wall questions as math problems to be solved is a natural step for the lovers of sudoku, cryptograms, word puzzles, and other thought-provoking games.


For Anna
I had no idea taking the Fung Wah bus ride would be so lucky.



You're on the bus and running a little late for work. In your hurry, you forgot your watch and your cell phone is at the bottom of your briefcase, so you ask the woman next to you what time it is. She glances at her watch, which reads 8:33:46, and replies "Eight-thirty."
Did she lie? Why didn't she say, "Eight thirty-three and forty-six seconds A.M.?" Maybe she doesn't keep her watch set exactly. Maybe she, too, was in a rush and didn't want to waste precious time talking to you. Maybe she realized that by the time she finished her sentence, it would be 8:33:48.
For various reasons, people rarely ever use such precision when dealing with numbers. It probably didn't matter to you whether the time was 8:33:46 or 8:31:27, and reading off the extra digits wastes your time and hers. Whether it's saying "eight-thirty" or adding a "pinch" of nutmeg, we all approximate things. Unless you work at NASA, sacrificing a little bit of precision to save time is a good trade-off because, for the most part, we all have more important things to do. But approximation is more than just rounding to the nearest minute. It's also a valuable intellectual tool that helps us conceptualize numbers.
Approximation works as an idea filter that can be used to weed out bad ideas when making a decision. Whether you're a business-man deciding which product to pursue, a congressman voting on whether or not to build a fence around the Mexican border, or a physicist trying to detect the Higgs Boson, approximation is the first technique you should use as a feasibility test.
Let's say you're a government official working on a missile defense plan. Do you pursue it immediately, potentially wasting billions of taxpayer dollars only later to find out whether or not it ever had a chance of stopping a nuclear attack? Or should you first make a rough estimate of the plan's likelihood of success, potentially saving time and money while you come up with a legitimate plan? If you estimate the plan has a 90% chance of success, you definitely pursue it. Maybe even if you estimate it has only a 10% chance, you still pursue it. But, if your estimate concludes its chances of realization are less than the chance of winning the lottery, it would be foolish to ever consider it. Even without an advanced engineering degree, estimations like these are possible and necessary for making some of the critical decisions we face every day. Whether it be missile defense or simply when to cross the street, we make estimates all the time, and having a sound mathematical footing only improves our accuracy.
In addition to providing an idea filter, getting in the habit of formally approximating things also boosts one's numerical prowess, specifically one's ability to conceptualize very big (and very small) numbers. Although you may not yet have an appreciation for the difference between a billion and a trillion, experience using these numbers builds an understanding of them quickly. One rapidly develops "numerical landmarks" that act as a conceptual guide. For example, at the time of this writing, a billion is about one-seventh the world population and a trillion is about one-eleventh the national debt. With simple arithmetic and a little bit of practice, you can estimate just about anything (no matter how large or small) and grow comfortable understanding big numbers in the process.


There are many approximation techniques, but one of the most powerful is the Fermi approximation. Its power stems from the fact that it's both easy and quick to use and can be done with little background information. Though there isn't a well-defined procedure for Fermi approximations, the general steps are to first make basic assumptions that seem reasonable and then to use those assumptions to calculate what you want to know.
Let's say I wanted to know how many leaves are on a tree. I could first estimate that each branch has about 30 leaves. For a given tree, 30 leaves is a reasonable number to have on a branch. I could then assume there are 10 branches on each tree. Some trees have more, others less, but 10 branches is a reasonable number to have on a tree. I then know that if each of the 10 branches has 30 leaves then there are (30 leaves per branch) x (10 branches per tree) = 300 leaves per tree. This example is simple enough, but for more complicated examples it helps to have some basic guidelines.
Enrico Fermi (1901-1954) was an Italian physicist and Nobel laureate known mostly for his work on quantum theory. He was renowned for solving seemingly impossible problems using order of magnitude estimates. Half of the particles in the universe—fermions—are named after him.


Maybe you're calculating how much it's going to cost for all the bricks needed to build a new schoolhouse. You don't know how many bricks are in a schoolhouse, so you can't start there. Maybe you start off knowing that each brick is about half a foot long. You also know the school house is to be about 100 feet long. From this you calculate that 200 bricks are needed to make that length. Next you guess the building is to be about 20 feet high. All of these are reasonable estimates. Maybe you're off by a few percent in each one, but if you were to start by guessing the number of bricks in the whole building you could be off by a factor of 10, 100, or even more. Start with approximations that you're pretty sure about and then calculate anything you're not sure about.


In the example using leaves, I know that there are a certain number of "leaves per branch" and a certain number of "branches per tree." While you may have forgotten how you canceled units in chemistry class, it's simple enough to remember if you can see that it's just like multiplying and dividing numbers. If I take "branches per tree" and multiply by "leaves per branch," the "branches" cancel, giving "leaves per tree."
It's simple factoring. One divided by one is one, 37 divided by 37 is one, branches divided by branches is one, thingy divided by thingy is one. Sometimes you need a string of units canceling to get the answer you seek. For example, in the schoolhouse calculation it may be that you use (X dollars per brick) x (Y bricks per wall) x (Z walls per schoolhouse) to get a final answer of X*Y*Z dollars per schoolhouse.


Sometimes you won't be able to start with something that's a reasonable guess. Maybe you want to know the number of teachers living in the Los Angeles city limits, but have no idea where to start. Well, you know there's certainly greater than 100 teachers, as that would fill up only two decent-sized schools. You also know that it should be less than 10% of the total city population (which in this case would be about 1 million people). Here we have an upper bound and a lower bound. Though you might not know what the answer is, you certainly know what the answer is not, and you can then use other information and the process of elimination to chip away until you find a better estimate.


There's no shame in using the Web to look up numbers you don't know. There are a variety of websites (Google, Wikipedia, etc.) that are useful for this purpose. Throughout the book, I will often look up things (physical constants, governmental budgets, etc.) that aren't interesting to calculate and don't aid in learning the process of approximation. Except where instructional, I won't calculate anything that can be found with a simple Google search, and neither should you.


This is perhaps the most important rule. We all have personal biases. Especially when there's some uncertainty involved, it's important to make sure that your preconceptions about the way things "should" work do not affect the outcome of calculations you do. I've often caught myself trying to force a calculation to turn out the way I want. At times like this, it's important to use conservative estimates and the utmost skepticism when deriving a result.


The old cliché "Practice makes perfect," is certainly true when it comes to approximations. The more you do it, the better you get at it and the easier it will be to visualize numbers. Over time, you'll build landmarks in number space that'll help you navigate. For example, when you hear things like a million and a billion, you should remember that a million is about the number of seconds in a week and a half, whereas a billion seconds is a little less than 32 years. Having facts like these to refer to greatly increases one's understanding of large numbers.


One thing that makes approximations fast and particularly easy is that you can put the numbers in a simpler form. For example, since we're only approximating, the number 397 could just as easily be rounded to 400 since the difference isn't significant given the degree of precision we're working with. Below are some tricks that I'll use to make the math simpler.


Just like our fellow bus passenger, we're not interested in knowing the time exactly, only what's significant to us. For the most part, this book will deal with two significant figures. Anything that's 187 becomes 190, and anything that's 7,432 becomes 7,400. It makes the math easier and saves time.


Many people prefer seeing numbers written out or hearing words like a million or a billion because "they make more sense." But can we really make more sense out of these? For example, most us would be happy to have a million or a billion dollars. With one million dollars we could live on a thousand dollars a day for a little less than three years before it ran out. How long would a billion dollars last? Six years? Thirty years? How about 2,740 years? That's right, if we had a billion dollars back in the time of Jesus, we'd still be spending it today because one billion is one thousand times greater than one million.
Our conceptions of very big numbers (and very small numbers) aren't particularly good, but exponential notation makes them effortless to think about. In exponential notation, numbers are written as a decimal number with a single non-zero digit before the decimal point times 10 to some power. The power tells us how many spaces we have to move the decimal point if we tried to write the whole number out. When you run out of places to move the decimal point, you add zeroes to the end of the number.
For example, 4.1 x 1015 could be written out by moving the decimal point one place to the right to get 41 and then adding 14 more zeroes so that you'd have a 41 followed by fourteen zeroes. The number 6.234 x 1053 could be written out by moving the decimal point 3 places to the right and then writing out 50 more zeroes after the 6234. It's easy to see that reading 6.234 x 1053 is much more comprehensible than reading
For negative powers, you move the decimal point to the left so that the written-out number is now a zero followed by a decimal point followed by the "magnitude of the power minus one" zeroes followed by the number in front. For example, 3.23 x 10-3 is a decimal point followed by 3-1=2 zeroes followed by 323, i.e., 0.00323.
Exponential notation reduces writing space and makes it easier to read and visualize numbers. It's worth noting that in some cases it's impractical (if not impossible) to write numbers as anything but exponential notation. The largest number in the book is 3.8 x 105,767,416. If I were to try to write this out with the actual number of zeroes, it alone would produce a 5,000-page book!


The nearest "order of magnitude" means the nearest power of 10. For example, 0.00001, 1, and 1,000,000 are all integer powers of 10. Written in exponential notation, I would have 10-5, 100, and 106, respectively. Using a property of exponentials, doing calculations with these numbers becomes very easy because you can multiply by adding the exponents. For example: 102 x 105 = 107 and 102 x 10-5 = 10-3.


The problems in the book are roughly organized from easy to difficult. The first several problems serve as an introduction to the Fermi approximation technique and lay out the format for the rest of the book. These problems are less intimidating than later problems, and several have answers that can be looked up. In this way, they demonstrate the power of the Fermi approximation by showing how accurate it can be.
Problems in the middle of the book are more likely to require multiple steps and may deal with mathematical and physical concepts that the reader is less familiar with.
Finally, the last set of problems deals with scientific concepts and algebraic equations that may be quite daunting for the unfamiliar reader. These are the problems that you do for honor as well as enjoyment.
Each problem consists of six parts:
• First, A BACKGROUND STORY describes the question we're considering and gives my own take on the problem. In some instances, the background story contains numbers that will help solve the problem.
ASK YOURSELF THIS . . .—This is a list of questions that provide some useful ideas you might want to consider before trying to solve the problem.
HELPFUL HINTS—I address each of the questions asked above and list the numbers I use to calculate the answer. If the problem requires use of physics equations, a brief tutorial is given on the physics that underlies the problem. (Feel free to skip this section if you're feeling studly and want to solve the problem on your own!)
CONSTRUCT A FORMULA—I describe in words the operations you can do to get the result. This is a good place to see if your units have canceled the right way.
MESSY MATH—The numbers are plugged in.
ANSWERS—Finally, I show the results and try to provide some context.


One disclaimer before we dive in: This is not a book of answers. I make no guarantees that the answers are right. This is a book about methods, about how to use numbers to calculate things. Don't buy this book if you're only looking to find out some interesting factoids1. I won't be happy if you buy it and don't learn something from it2. If this book accomplishes its task, then by the end you should be able to approximate answers to questions that you come up with. In fact, you'll probably even be able to do a better job on some of the things I've estimated in the book. After all, there's more than one way to skin a math problem. Ultimately, if you end up doing a better job than what I've written, then I'll feel like my work here is done.

Fine Tuning
To illustrate the power of the methods described in the previous section, let's start with Fermi's classic example: HOW MANY PIANO TUNERS ARE IN CHICAGO? Seem impossible? At first glance, such questions may seem hopeless to answer without looking in the Chicago Yellow Pages. As you'll soon see, it's not nearly as difficult as it seems.
We have to start somewhere, so let's start with what we know. How many people live in Chicago? Okay, we probably don't know that, but maybe we know how many people live in Boston or Detroit or any other major city. Most major cities have around 3.0 x 106 (3 million) people living in them. Some have more, others less, but they're usually around that size. If it's not exact, it should be close. Even in the worst-case scenario, it will certainly be between 500,000 (a medium-sized city) and 50 million (more than the largest city in the world). We have to make basic assumptions like these, but we shouldn't sweat how precise they are since they're just approximations that give us roughly the right number. If the calculation needs to be really precise, you can always go online and look up the exact result (2.8 million people, according to Wikipedia at the time of this writing), but if you like results fast and dirty, this is the way to go.
Moving on, we'll next guess that 30 percent of residents are homeowners3. We'll assume that 5 percent of these homeowners have pianos and that their pianos are probably tuned once every six months. We don't consider the number of musical halls, performance areas, etc. because they're probably much fewer than the number of nonprofessionals who own pianos. Lastly, a piano tuner would probably tune about two pianos a day, or equivalently 730 pianos a year.


a. How many people are in Chicago?
b. What percentage of them are homeowners?
c. What percentage own pianos?


There are about 3.0 x 106 people in Chicago and other major cities.


Here, we construct a formula that can be used to find the number of pianos tuners. We do this by combining the variables listed above in such a way that their units cancel leaving us with units of "piano tuners." In order to simplify our formula, we can do a few simple calculations first.
1) For example, if we multiply the fraction4 of residents who are homeowners (0.3 homeowners per resident) times the number of total residents (3.0 x 106 residents) we get the number of homeowners (9.0 x 105 homeowners).
2) If we multiply this again by the fraction of homeowners who own pianos (0.05 piano owners per homeowner), we find there are about 4.5 x 104 pianos.
3) These pianos need to be tuned twice each year, meaning that there are about 9.0 x 104 tunings per year.
4) Finally, we can obtain the number of piano tuners by dividing the number of tunings per year by the number of tunings per year per tuner (730 tunings per year per tuner):
(total # of tunings per year)
(# of tunings per year per tuner)



This is a reasonable estimate. The Chicago Yellow Pages has 43 listings under piano tuners and not every tuner pays for Yellow Pages listings, so we're off by less than a factor of 3.


On Sale
Aug 25, 2009
Page Count
176 pages
Running Press

Aaron Santos

About the Author

Aaron Santos received a Ph.D. in Physics from Boston University in 2007, and is currently a visiting assistant professor in the Physics and Astronomy Department at Oberlin College. He is the author of How Many Licks?: Or How to Estimate Damn Near Anything, and enjoys writing his Diary of Numbers blog ( He lives in Oberlin, Ohio.

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