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.