How much money is in a Brinks truck?

A lot, certainly, assuming it's full. But is it a million? A hundred million? Somewhere in between? Most of us, when presented with such a question, throw up our hands.

Sanjoy Mahajan sharpens his pencil.

To say that Mr. Mahajan, a lecturer at the Massachusetts Institute of Technology, is a good guesser doesn't do him justice. He can start with seemingly zero information and, after some furious scribbling and rapid-fire explanations, come up with an answer that's close to the mark.

Take the Brinks truck. He begins by estimating its size, figuring that it's big enough inside for a person to stand up or lie down. It probably has nooks and crannies, but he assumes, for the sake of simplicity, that it does not. As for the money, he figures the truck is filled with stacks of twenties, because thousand-dollar bills are rare, and a truck filled with ones seems silly. The engine, he assumes, is similar to that of a pickup, which can haul a ton or two.

Eventually, after estimating the weight and density of the bills, he arrives at a number: $20-million. How close is that? Well, the Brinks people tend to be tight-lipped about such things (a spokeswoman politely declined to provide a figure), but the largest heist on record is said to be $18-million. So he's in the ballpark.

And ballpark is all he's aiming for. Trying to be too exact can be paralyzing; or, as he likes to say, rigor leads to rigor mortis. That's the message of his new book, *Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving*, in which he lays out his principles for back-of-the-envelope calculations, including divide and conquer, take out the big part, and trust your gut.

The book isn't light reading (binomial coefficients, anyone?), but the core ideas don't require Algebra II: The world is messy, so do the best you can. You know more than you think you do. Use whatever tools are available to do the job.

It's as much an attitude as a technique. And it's an attitude that isn't the norm in mathematics classes, where down-to-the-decimal accuracy is prized. For MIT students, who are use to getting the right answers, the transition can be tough. "There was a lot of nervousness at the beginning of the class," says Sean Clarke, a graduate student in biological engineering. A few weeks in, though, students were coming up with their own problems—and the class gets consistently high marks in student evaluations.

Mr. Mahajan, 41, is bespectacled and boyish despite a smattering of gray hair. A physicist by training, he's associate director of MIT's Teaching and Learning Laboratory and sort of floats between departments. His affinity for math extends to childhood. When he was a toddler, he informed his parents, correctly, that a heating coil on the ceiling was a hexagon. In first grade, he told his teacher he wanted to be a mathematician when he grew up. The teacher cheerfully announced to the class: "Sanjoy wants to be a magician!"

They were both right, in a sense: some of the calculations he pulls off have a hint of Houdini. For instance, he can start with two paper cones, to find the relation between drag force and velocity, and—believe it or not—arrive at the cost of a round-trip plane ticket from New York to Los Angeles. He works out the problem in a blur of equations, remarking that a gram of gasoline and a gram of fat contain the same amount of energy, that drag force is proportional to velocity squared, and so on. The number he arrives at ($700) isn't the cheapest deal out there, but it's roughly right.

The airfare example is well-rehearsed. I decided to see how he'd cope with an unfamiliar quandary. How much, I asked him, is the annual state budget of Delaware? He didn't know the state's population, but he knew that California has about 40 million people and, creatively applying Zipf's law, a statistical observation from which it can be asserted that the largest city is twice the size of the second-largest, he determined that Delaware has about a million people.

It's actually 885,122. So far, so good.

He then assumed that everyone makes $50,000 a year. Some make more, no doubt, and some don't make anything, but this seemed reasonable. He further assumed that the state income tax is 5 percent, the same as in his home state, Massachusetts. He wasn't sure that Delaware has an income tax (it does) but figured that, even if it didn't, revenues from sales taxes would probably be equivalent.

Final answer: $2.5-billion. The actual number for the 2010 fiscal year is $3-billion. For comparison purposes, the budget of neighboring Pennsylvania is $29-billion.

Not bad at all.

A few years ago, Mr. Mahajan became a friend of Jeff Schmidt, a former editor of *Physics Today*, who sued that publication after he was fired and got an undisclosed settlement. When a reporter asked Mr. Mahajan to estimate the size of the settlement, he came up with $500,000—assuming that, with back pay and damages, Mr. Schmidt would have asked for around a million and settled for half.

The lawyers for the company that owns *Physics Today* accused Mr. Schmidt of revealing the figure to Mr. Mahajan—which Mr. Schmidt said wasn't true. "They didn't know he was one of the world's experts in estimation," he told me, adding that getting to know Mr. Mahajan was "almost worth getting fired."

I attended one of Mr. Mahajan's classes recently. Afterward I asked him for advice about getting back to the airport. He suggested that I walk across the campus, take the subway, get off three stops later, and wait for a shuttle. Once I did that, printed my boarding pass, and made it through security, Mr. Mahajan estimated, I'd arrive at the gate at 4 p.m.

When I did make it to my gate, I checked the time: 3:54. Close enough.

## Comments

1. 11186108 - May 04, 2010 at 02:27 pm

Wonderful!

"The Back of the Envelope" by Jon Bentley dealt with exactly the same aspects of computation. It was published in his November 1984 column in Communications of the ACM and republished in his Programming Pearls book in 1986 (Addison-Wesley)

2. edwcarney - May 04, 2010 at 05:22 pm

I remember that column of Jon Bentley's so well. Actually it was March, 1984. (I looked it up. I don't remember it that well.)

The full citation is Bentley, J. (1984). The back of the envelope. Communications of the ACM, 27(3), 180-184.

3. willismg - May 05, 2010 at 11:00 pm

Let's not forget to pay homage to the great Enrico Fermi who is famous for estimating how many piano tuners are in NYC (I believe).

4. arrive2__net - May 06, 2010 at 03:53 am

When you are working with experts in the very early stage of some big project you often need exactly that kind of accurate guess-timation of what to expect in order to get the project off the ground. It is is good to see the good professor is developing, teaching, refining that kind integration of true expertise and street savvy realism. That's teaching real world critical thinking. Learning those type of techniques really sharpens the mind and enables the student to realize what is really possible in estimation. Good article.

Bernard Schuster

Arrive2.net

5. 22000394 - May 06, 2010 at 08:20 am

We have an article in the Chronicle of *HIGHER* Education that essentially apologizes to its readership for suggesting the need for 10th grade mathematics? No wonder the country's understanding of science is so poor.

6. dnewton137 - May 06, 2010 at 08:57 am

Making good estimates is one of the most important tools used by all physicists. (It's not about "10th grade mathematics." It's about getting a preliminary handle on scientific questions.) That's why I routinely assigned estimation problems to my students. The results often revealed unexpected insights into their thinking processes. An example:

A student at my Ivy League university once came to complain about receiving no points for his answer to that week's estimation question, which was "Estimate the number of grains of sand on all the beaches of Earth." (That's an interesting number, incidentally. It's less than Avogrado's Number!) I asked him to take me through his argument, and he whipped out his calculator and eventually produced a number like 98.37155. Does that seem plausible, I asked. Certainly, he said, that's what my calculator says, so it must be right. Further questioning revealed that he spent several weeks each summer on the Atlantic beaches. So, I asked, "Ever notice anything about sand?" Long pause, blank look, and "Whaddya mean?"

That was one observation behind my general deep conviction that the world is full of people who see everything and observe nothing. That phenomenon is sometimes apparent among the Chronicle's commentators.

7. willismg - May 06, 2010 at 09:16 am

What #8 said...

8. dthornton9 - May 06, 2010 at 10:09 am

And met a clerk at the cookie store the other day who stared at me blankly when after being told a dozen cookies was $3.95 - I responded, "So, basically 30 cents a cookie?" She had no clue either about estimating math, or real math - much less Algebra II. After we completed the transaction, I asked her if she was in college - "yep," she said..."a sophomore at XXX." .....I worry.

9. ianative - May 06, 2010 at 10:37 am

I'm working on an action research project right now that deals with helping students link theory and practice to produce understanding. The student who thought there were 98 grains of sand could probably recite theory out the wazoo and he could certainly do calculations... but to what end? As disheartening as it sometimes is, we need to remember that for every fact our students learn they should be able to answer the "So what?" question that follows.

10. actlibrary - May 06, 2010 at 10:50 am

Although I agree with the main point made by #8, I think #7 was talking about the inclusion and tone of the following statement in the article: ""The book isn't light reading (binomial coefficients, anyone?), but the core ideas don't require Algebra II:"

11. 11232247 - May 06, 2010 at 11:15 am

If a bat and a ball together cost $1.10 and the bat costs $1 more than the ball, how much does the ball cost? (*Answer below)

Comment: The key hurdle to making accurate estimations (i.e. prudent decisions)is making sure our initial assumptions are more or less correct. In the above word problem, a majority of college level respondents will reflexivly (and incorrectly answer) that the ball costs 10 cents. The lesson in all of this is that it is oftentimes possible to be off in our estimates if we are not careful in properly assessing our assumptions.

*The ball costs 5 cents (the bat costs $1.05) or (.5 + $1.05 = $1.10)

12. dnewton137 - May 06, 2010 at 01:21 pm

Another (true)story to go alongside dthornton9's. Long ago I was checking out of a Washington hotel. It wasn't a very fancy hotel so the clerk was figuring my bill on a calculator. At the time Washington had a 10% room tax. She pecked away, then threw the calculator down and strode off, saying "The battery's dead. I'll have to go get our accountant to calculate the room tax."

13. arrive2__net - May 06, 2010 at 03:31 pm

What I think is important about what the prof teaches is not following a procedural formula, like 3.95/12=32.9. (Like the article said, "Trying to be too exact can be paralyzing", cookie store clerk. ) What is significant about what the prof teaches is that it is possible to develop a meaningful estimate even where the connection between whatever evidence you can dig-up, and the solution is obscure or barely discernable. He also teaches them how to think about the problem to develop an original solution. In the article, the methodology for estimating the number of bills was based first on another set of assumptions about and estimate of volume and weight (without being paralyzed), guessing or assuming the average denomination of the bills, securing the relevant count inference data, and evaluating the accuracy of the estimate based on a old newspaper story about a robbery. Once someone tells you how it was done ... it may seem obvious, but the difficult part is before someone tells you how it was done. Most of the time, in school, you are told what is important, given the already thought out evidence ... the situation is structured for you, and you are merely required to put it together. If you can pass a pre-structured multiple-choice test, you are good to go. However, in real life you are often required to structure the problem yourself, to search for whatever evidence you can connect to the problem (evaluating its reliability), and to develop your own methodology for hooking up the evidence into a formula. Precedent, or even "the school solution", may just be a distraction, it may be inadequate and inferior to what you can develop, now, yourself. In real life developing estimation solutions is often competitive. Is your solution merely satisfactory of does another contractor have a better heuristic? Also realizing that problems like that may be solvable, before you settle, give up, or move on to something else ... that is important for students to learn as well. The demands for estimation solutions in, for example, "Research and Development" go way beyond 10th grade, believe it or not. Maybe college should prepare you to develop solutions on your own. Also, modern students are often embarassed if they don't get the exact solution, exactly as the teacher expects, so they have a lot of trouble even thinking about "estimation".

Bernard Schuster

Arrive2.net

14. arrive2__net - May 06, 2010 at 03:36 pm

Make that 395/12=32.9.

15. joeefr - May 06, 2010 at 07:14 pm

Another lovely example that was running around the web last year was the phone company people who couldn't distinguish between 0.001 cents and 0.001 dollars. My homework problem "Explaining fractions to the phone company" (http://www.physics.umd.edu/perg/abp/TPProbs/Problems/G/G38.htm) has a link to the audio clip. I give estimation problems regularly to my university level physics students -- one on every homework and one on every exam. You can find them in our problem collections (links at http://www.physics.umd.edu/perg/.)

16. sanjoymahajan - May 06, 2010 at 09:37 pm

Perhaps one reason I got interested in this area is the following experience. Our advisor was leaving for another university, and we graduate students were packing up our computer lab -- the first step being to back up all the hard drives onto DAT tapes. I went to the computing center, asked how much a DAT tape was, and was told $6.50.

After thinking it over, I decided that we'd need more than one tape anyway, so I may as well buy a box (of 10). When I asked for its price, the cashier pulled out a pocket calculator and started keying away. Oh, I thought, it's that random 8.25% (or whatever it is) sales tax. When the cashier eventually reported, "That'll be 65 dollars," I knew I'd chosen a fruitful area as my career path.

p.s. Comments #1 and #2 reminded me of another "Back of the envelope" series, by the physicist Edward M. Purcell. They ran for several years in the American Journal of Physics starting in January 1983 (Am J Phys 51(1):11).

17. daughtht - May 06, 2010 at 11:32 pm

Over 100 so-called "Fermi problems" may be found at http://www.physics.odu.edu/~weinstei/wag.html, with links to other sites.

These invitations to estimate are named in honor of Enrico Fermi, who was known for asking such posers as how many piano tuners there are in Chicago.

18. drstarsky - May 07, 2010 at 02:22 pm

Another book dealing with the same subject is "Guesstimation" by Weinstein and Adam.

19. sahara - June 02, 2010 at 09:39 am

I got my MBA degree when he was about 12, and we were using these approaches to problem solving then...it's not new...