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, Assistant Professor for the Department of Math and Computer Science.
Let's say I offered you a chance to win a small cash prize. I give you four standard dice, each with numbers from 1 to 6. Your goal is to roll as high a total on the dice as you can; whatever total you get, that's the number of dollars you win.
Here's how the game works. You roll the four dice, and then you must choose one or more of the dice to "freeze" ("frozen dice" are set aside, and are not rolled again for the rest of the game). Then you re-roll all unfrozen dice (if any), and again you must freeze one or more of the dice you rolled. Continue until all four dice are frozen; then the game ends.
Here's a sample game. You roll 6-2-2-1; you freeze the 6 and re-roll the other three dice. You roll 5-4-2; you freeze the 5 and the 4 and re-roll the last die. You roll a 1; bad luck, but you must freeze that last die. The dice are now showing 6-5-4-1, for a total of 16. You win $16.
Assume that your goal is to maximize the amount of money you won on average. Which dice should you freeze if your first roll is 5-5-5-3? What if your first roll is 5-5-4-4?
Joseph DiMuro (firstname.lastname@example.org) by September 30, 2016.
I will review your submission and whether you are correct or incorrect, I will be in touch with you to provide personal feedback, guidance, and background information regarding the problem.