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 seeing what you come up with.
— Joseph DiMuro, Associate Professor for the Department of Math and Computer Science
Who Buys Lunch?
Alice, Bob and Charlie frequently have lunch together, and at the end of their meals, they like to randomly choose one person to pay for lunch. The problem is, the only randomizing device they have is a coin. And it's a bit tricky to use a two-sided coin to pick one person out of three.
The three friends use the following method: they flip the coin twice. Alice buys lunch on heads-heads, Bob buys lunch on heads-tails and Charlie buys lunch on tails-heads. If the coin lands tails-tails, then they start over. This method works well for them, since each person has an equal chance of buying lunch.
Now, one day, the group invited Dennis and Eleanor to join them for lunch. As usual, they decided to pick one person at random to buy lunch, but they still only have a coin to use. What to do?
Your challenge: find a method that the five friends can use to pick someone to buy lunch, using only a fair coin. Each person must have an equal chance of buying lunch, and you need to find a method that requires as few coin flips (on average) as possible.
Send your answers to firstname.lastname@example.org by February 28th, 2018. Look out for the next problem of the month on March 1, 2018.