ILL 1987 Problem 65

The runs of a decimal number are its increasing or decreasing blocks of digits. Thus $024379$ has three runs : $024, 43$ , and $379$ . Determine the average number of runs for a decimal number in the set $\{d_1d_2 \cdots d_n | d_k \neq d_{k+1}, k = 1, 2,\cdots, n - 1\}$ , where $n \geq 2.$ Attachments: