The Problem of the Month
These problems are my attempt to give high school students, like you, a glimpse into "real mathematics" by creating a math puzzle which will challenge you to think in ways that you have never thought before. Typically, in your high school math classes, you are given a problem to solve and a fixed step-by-step procedure for solving that type of problem — follow the steps, and you are certain to get an answer. However, "real mathematics" is not like that. When mathematicians work on problems, they often start off with no idea about how to solve the problem. There are lots of math questions that no one yet knows how to solve. I invite you to stretch your mind and take my mathematics challenge. I look forward to see what you come up with.
— Joseph DiMuro, Associate Professor for the Department of Math and Computer Science
We’re in the Dark
You have a flashlight you want to use; the flashlight requires two batteries to run. You have eight batteries, but only four of them are good. And you have no idea which ones are good and which ones are bad. You have no way to test the batteries, except by trying pairs of batteries in the flashlight and seeing if the flashlight turns on. (Of course, the flashlight will only turn on if both batteries you put into the flashlight are good.)
Come up with a strategy that guarantees that you turn on the flashlight in as few attempts as possible. That is: if your strategy guarantees success in at most n attempts, then you want n to be as small as possible.
If you figure that one out, here are a couple of more complicated variations for you to try:
- Five good batteries, eight bad batteries, and the flashlight needs two good batteries to run.
- Five good batteries, two bad batteries, and the flashlight needs three good batteries to run.
Send your answers to these problems to firstname.lastname@example.org by November 30, 2017.