Abstract: When two numbers are added in base 10, what's the chance that a carry occurs? Questions like this motivates the study of maximizing/minimizing the number of solutions to a linear equation, such as x+y=z, with the variables lying in a set of given size N. Some of these questions are easy (i.e. challenge problems for high school students), but some others are hard (i.e. open). I will discuss problems from both categories. Based on joint work with P. Diaconis and K. Soundararajan.