How Many Licks? or How to Estimate Damn Near Anything by Aaron Santos

One of the odd things about the C-list celebrity life of a semi-pro blogger is that I get a bunch of requests to review books on physics-related topics. Some of these take the form of a book showing up out of the blue, others are preceded by a polite request from the author. Aaron Santos's How Many Licks? is one of the latter, which helps bump it up the list of things to do...

This is a short little book-- 176 pages total-- built around the idea of Fermi Problems, the order-of-magnitude estimates that Enrico Fermi was famous for. The idea is that, with a little basic knowledge and some really simple math, you can come up with ballpark estimates of all sorts of things-- the canonical Fermi problem example is "How many piano tuners are there in Chicago?"

Santos starts off with Chicago piano tuners, and adds 68 more, from classics like "How many hairs are on your head?" to less obvious questions like "Is more silicon used in computer chips or breast implants in a given year?" Each problem has the same structure: A statement of the number to be estimated, a set of "Ask Yourself This" questions to guide the solution, some "Helpful Hints" derived from Wikipedia and other common sources, leading to the construction of a formula, the plugging in of numbers, and the final answer. All these sections are written in a breezy and accessible style, and even the most demanding questions (the last half-dozen require some real knowledge of physics) are written in a style that is clear and easy to follow.

Of course, as a sometime book reviewer, I am required to have some quibbles...

One minor point has to do with possible uses of the book. While it's entertaining to read on its own, one could also imagine using it as a supplemental text in an introductory physical science course, or maybe a general education course aimed at non-scientists. In that case, it would be better to have the calculations on a different page than the set-up work-- as it is, it's a little too easy to just go with the flow without really thinking through the problem yourself.

The other, somewhat more serious quibble, is that the presentation he uses is smooth and convincing enough that it gives the impression that there's only one "right" path toward the answer. In reality, you can estimate a lot of these things in more than one way, and at times I found myself wanting to argue with his chosen method.

For example, take problem #24, "How long a line could you draw with one pen before it ran out of ink?" In the book, Santos proceeds by trying to estimate the useful lifetime of a pen, and the length of lines drawn in a day of pen usage. He arrives comes up with an estimate of about 3000 m for one pen.

You could come at this question another way though, that seems a little more concrete to me: The little tube of ink in a typical ballpoint pen is around 10cm in length, and 1mm in diameter, meaning that the volume of ink in one pen is around 100 mm3. The line drawn by the medium-point pens I favor is somewhat less than 1mm wide-- call it 0.2 mm. If you estimate the depth of the ink in a smooth line as 0.001 mm (one micron, which is probably on the low side, but then I'm probably underestimating the ink volume a little), this means that 100mm3 of ink, you could drawn a line that is 0.2 mm x 0.001 mm x 500000mm, or 500 m in length.

Which of these methods is "right"? They're both equally valid ways of attacking the problem, and they give answers that are pretty close-- separated by a factor of just 6. Unless somebody with a lot of free time and a whole bunch of pens would like to do the experiment, there's no obvious reason to prefer one or the other-- it comes down to whether you prefer to guess at the length of a day's worth of text, or to guess at the effective thickness of a line of ink.

So, my one complaint with the structure of the book is that I would've liked to see some discussion of alternative methods one could use for some of these problems. I think that would give a better sense of the mental flexibility that is the real goal of this Fermi estimation business.

All in all, though, this is a very entertaining book for anyone who enjoys mathematical puzzles. And it may be that only having a single method presented will spur readers to argue with the text (as I did with the pen thing), and develop their own estimates for some of these quantities.

Santos is currently a post-doc at Michigan, having gotten his Ph.D. in 2007. If he decides to pursue an academic job, this book suggests that he'll have no problem demonstrating competence in teaching. It's a well-done book and a fun read, and I'll definitely be thinking of ways I might use it in class...

(I am aware that there is another book, Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin, that uses Fermi problems as a way to approach all sorts of science. I have not read it (not having been sent a free copy...), so I can't say how this book compares to that one.)

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And if you're interested in a set of purely physics Fermi problems, you should look up back issues of The American Journal of Physics--beginning in 1983, and continuing for several years, Edward Purcell wrote a "back of the envelope" column featuring 3 such problems, with solutions appearing the following month. If Santos's and Weinstein's books indicate a general interest in the subject, AmJPhys would be wise to issue these in book form.

Well, 500 meters is way off, and 3000 meters is in the middle of the range for actual ink cartridges.

The Bic web site gives specs on their ink cartridges. The typical skinny ballpoint cartridge writes a line of 1600 meters. And that's about the minimum. A fat ballpoint cartridge writes for 4800 meters.

The experiment wouldn't be hard to set up, and it wouldn't take but one person to change pens or reload a drum of paper. Basically you would modify one of those old seismographs with a drum of paper. You know how long the paper is (or can calculate it fairly easily), so you just measure time until the pen runs dry. If you really need to automate it, rig some sort of conductivity sensor or photogate/image recognition to sense the start and end of the pen's marking. If one needs to, build a housing for a while package's worth and measure the start and finish of all the lines, then do some statistics.

It sure seems to make business sense that pen makers would be able to reliably estimate how much a given pen will write as one of the factors in pricing their product.

There was an episode of "How It's Made" that shows how the pens are tested. The pen is made to sweep back and forth over the moving paper so it has to write from different angles. Then they check to make sure the length of the line meets the specifications.