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
The Annoying ATM
You live in a small town. This town has only one ATM, and it's an annoying one to use; it only allows deposits of exactly $10 and withdrawals of exactly $17. If you want to deposit or withdraw any other amount of money, you need to use multiple transactions. For instance: if you want to deposit exactly $3, you can do that by withdrawing $17 first, then depositing $10 twice.
Note: your bank does not allow overdrafts. If you have less than $17 in your account, then you can't make a withdrawal of exactly $17, so you can't withdraw any money at all.
You want to keep enough money in your account so that, through multiple transactions, you can withdraw any whole number of dollars you want, from $1 up to your entire account. And you want to be able to do this even if you step up to the ATM with no money in your pocket. What is the least amount of money you can keep in your account and still be able to do this?
Send your answers to firstname.lastname@example.org by October 31, 2017.