New Observations and Data on M&m Sequences

A mean and median sequence (known as an M&m sequence) begins with any three real numbers. Successive terms are determined in such a way that the nth term satisfies the condition that the mean of the first n terms equals the median of the first n-1 terms. No examples of M&m sequences are known that fail to eventually become stable. it was conjectured by Harris Shultz in a 2005 College Math Journal article that all M&m sequences eventually become stable. At that time, some but not all M&m sequences were shown to be eventually stable. Using a C++ program to generate large quantities of M&m sequences, we collect data, look for patterns, and make some new observations about the sequences. We also present a few tricks for tackling the conjecture that all M&m sequences stabilize. In particular, we are able to show that some new families of M&m sequences are eventually stable. Finally, we present a few interesting related questions.