A Random Place at the Table
Have you ever encountered a problem that seemed to require a lengthy calculation only to realize that, with a simple shortcut, you can solve it in your head?
A paradigmatic problem of this kind is that of the two bicyclists and the fly:
Two bicyclists start 20 miles apart and head toward each other, each going at a steady rate of 10 mph. At the same time, a fly starts flying at a steady 15 mph from the handlebar of the one bicycle to the handlebar of the other one, and then turns around and flies to the first one again, continuing in this manner back and forth until the bicyclists meet. What total distance did the fly cover?
The “obvious” and standard way to find the answer is to add up the distance that the fly covers on the first leg of the trip, then on the second leg, then on the third, and so on. Each distance gets smaller and smaller, giving an infinite series of distances, which can be summed.
There is, however, a simpler way. Notice that the bicycles meet exactly one hour after their start, so that the fly had exactly one hour of flying time. Since the fly’s speed was a constant 15 mph, the answer must be 15 miles.
Legend has it that a version of this question was put to John von Neumann, the incandescent genius who laid the mathematical foundations of quantum mechanics and game theory. Von Neumann’s calculating ability was so stratospheric that many of his peers, brilliant themselves, thought of him as “the smartest man in the world.” As expected, von Neumann solved it in an instant. His questioner said in disappointment, “Oh, you’ve heard the trick before!”
“What trick?” von Neumann reportedly said. “All I did was sum the geometric series.”
Here’s our problem this month. You don’t have to be a von Neumann to solve it!
A clubhouse for a cult has a table with 10 chairs. There are five people who sit around it, occupying a chair each, completely at random. They call their original chairs the “anointed” chairs. After half of the meeting is done, all five people get up and move clockwise to different places, each one moving the same number of places as the others, so that their positions relative to one another are not changed. They have to do so, however, in a way that maximizes anointed chairs. Show that no matter what their original positions were, they can always find new places such that at least three of the anointed chairs are used for the second sitting.
The obvious — and long — way to solve this is to consider all possible positions around the table that five people can occupy (how many?) and then show for each that there is some number of places the cultists can move so that three or more chairs are reused. This would probably require a computer program. But there is a simpler way. Can you find it?
Finding such “short circuit” solutions can be thrilling and, for some, addictive. Here’s a bonus problem for those who can’t get enough:
A ball is thrown up in the air from a certain height. It rises to a maximum height and falls back to the initial height. Does it take longer to go up from its starting to its maximum height or longer to fall down from the top of its flight back to its initial height?
In high school physics, when students are asked to solve such problems, they are told to neglect air resistance. Then you can apply the standard equations of projectile motion and find that both times are the same. But what if you do not ignore the turbulent drag of the air on the ball, which is proportional to the square of the ball’s velocity? Do the times still remain the same or does that change? Factoring in the drag makes the problem much harder if you try to do it the usual way. But you can still answer the above question simply if you short circuit it!
I invite readers to share their favorite “short circuit problem.” Please share your insights with all of us.
Editor’s note: The reader who submits the most interesting, creative or insightful solution (as judged by the columnist) in the comments section will receive a Quanta Magazine T-shirt. (Update: The solution is now available here.) And if you’d like to suggest a favorite puzzle for a future Insights column, submit it as a comment below, clearly marked “NEW PUZZLE SUGGESTION” (it will not appear online, so solutions to the puzzle above should be submitted separately).
Note that we will hold comments for the first day or two to allow for independent contributions.