How to catch a bus. Step 1: Stand there and wait.

Step 2: Repeat Step 1 ad infinitum.

That's right, don't get your baggies in a twist 'cause you think you could get there faster by walking, because most likely you won't.

That's the conclusion of a paper just published in the January 23, 2008 issue of New Scientist.

The New Scientist News Service offered the following account of the work.

Lazy option is best when waiting for the bus

Ever lose patience waiting for a bus and decided to walk instead? Next time, stick around, it's nearly always the best strategy.

Scott Kominers, a mathematician at Harvard University, and his colleagues derived a formula for the optimal time that you should wait for a tardy bus at each stop en route before giving up and walking on. "Many mathematicians probably ponder this on their way to work, but never get round to working it out," he says.

The team found that the solution was surprisingly simple. When both options seem reasonably attractive, the formula advises you to choose the "lazy" option: wait at the first stop, no matter how frustrating.

The formula does break down in extreme cases, Kominers says, when the time interval between buses is longer than an hour, for example, and your destination is only a kilometre away.

If you do choose to walk, you should make your decision before you start waiting, he says. You will still reach your destination later than the bus you'd have caught, but it will be much less frustrating than waiting for a while and then watching the bus shoot by. "It certainly has changed the way I travel," Kominers says.

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Okay, now your brain's in gear and ready for the abstract of the New Scientist paper; it follows.

Walk versus Wait: The Lazy Mathematician Wins

In this recreational mathematics note, we address a simple, yet instructive question:

Justin has to travel a distance of d miles along a bus route. Along this route, there are n bus stops i, each spaced at a distance of d_i from the starting point. At each bus stop, Justin is faced with a choice: to walk or to wait. If he walks on, he can still catch a bus at the next bus stop — but if a bus passes him while he walks, he is almost assured a longer wait.

We model Justin's decision constraint and completely solve the model in a special case. The answer is intuitive: the optimal strategy is the laziest.

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For those who'd like to do the math themselves, a link to the full paper in PDF format appears in the upper right hand corner of this page.