Math Memos

# The Multiples of Nine Problem

I recently discovered this problem, and I really like it for a number of reasons. First, it requires a little bit of vocabulary in order to get started. Students will have to know what a multiple is, they will have to know what digits are—and more specifically, how digits can differ from numbers—and they’ll have to understand the difference between even and odd numbers. I also like how nonintimidating it looks at first glance. “How hard could it be to find a multiple of 9 that has only even digits? I shouldn’t have to count up very far.” Because the problem doesn’t look lengthy or challenging, it comes as a surprise when the correct answer is actually the 32nd multiple of nine. I anticipate a lot of students writing out 9, 18, 27, 36, 45, 54, etc, and then getting frustrated or giving up when they don’t get to the answer fairly quickly.

The cookie problem does not require any advanced mathematics techniques, but it is quite challenging nonetheless.  Thus, it is a “problem” for my students, but it has few barriers to entry and is approachable for everyone in my classes.  Without giving any parameters for a solution, students can come up with a variety of ways to solve the problem.  In the end, those various representations present a verdant opportunity for discussing algebra, algebraic thinking, diagrams, as well as mean and median and central tendency, and arithmetic series.  So, altogether, it is a simple but rich problem.