July 2, 2020

Mean Median Surprise

Start with three numbers, say 5, 17, and 23. Their median (middle value) is 17. Find a fourth number so that the arithmetic mean of all four is 17. This number must be 23 (4 ✕ 17 – 5 – 17 – 23).

Repeat the process. The median of 5, 17, 23, and 23 is halfway between 17 and 23 (20). Find a fifth number so that the mean of all five numbers is 20. This number is 32 (5 ✕ 20 – 5 – 17 – 23 – 23).

Repeat the process. The median of 5, 17, 23, 23, and 32 is 23. Find a sixth number so that the mean of all six is 23. The sixth number must be 38.

Continuing the process, you get the sequence 5, 17, 23, 23, 32, 38, 23, 23, 23, 23,…. The sequence reaches a constant value!

The same thing happens if you start with the numbers 6, 46, and 78. You get the sequence 6, 46, 78, 54, 66, 74, 96, 108, 102, 110, 96, 100, 195, 213, 96, 96, 96,….

It also happens with 13, 41, and 53. You get the sequence 13, 41, 53, 57, 71, 83, 67, 71, 102, 112, 89, 93, 71, 71, 71,….

"To our surprise, the same thing happened to every sequence we examined, with whatever three numbers we started," Harris S. Schultz and Ray C. Shiflett reported in an article in the May 2005 College Mathematics Journal. Schultz and Shiflet dubbed these strings M&m sequences for "mean and median."

In these sequences, "we calculate the median of the list of the first k values and choose the k + 1 value so that the mean of the first k + 1 values equals this median," the mathematicians noted.

An M&m sequence is considered stable if it eventually reaches a constant value. The length of the sequence is the number of terms it takes to get to the repeating value for the first time. For example, the sequence starting with 6, 46, and 78 has a stable value of 96 and its length is 15.

The starting numbers don't have to be integers. The numbers 5, 5.5, and 33.9, for example, yield a sequence of length 73 and stable value –4.65625.

Schultz and Shiflet conjectured that every M&m sequence is stable. In investigating the problem, the mathematicians proved a variety of results that fall short of the ultimate goal but provide useful insights into what's going on.

It's obvious that any sequence that starts with three identical numbers is constant. It's also easy to show that if two of the values are the same, the M&m sequence has length 5. Various other intriguing patterns also emerged.

Schultz and Shiflet proved some stability results, particularly for sequences that start with 0, x, and x + 1, where x is greater than or equal to 1.

"Our hope is that readers will be motivated to study and explore these M&m sequences," Schultz and Shiflet wrote. The question remains: Is every M&m sequence stable?

Originally posted May 30, 2005

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