2024 DMC Mock 10 Problems/Problem 15

Problem

Shreyan and Kaylee are on opposite vertices of a cube. Each turn, they both randomly move across an adjacent edge to another vertex with equal probability. What is the expected number of turns that occur until the two meet in the middle of an edge for the first time?

$\textbf{(A)}\ 13\qquad\textbf{(B)}\ 13.5\qquad\textbf{(C)}\ 14\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 26$

Solution

Consider the following states: let $A$ be the state where both are directly opposite of each other and $B$ be the state where they are one edge away from each other. Notice that on every turn, they move an even number of edges, so since they initially are three edges away from each other, they will always be an odd number of edges away.

We can deduce from casework that: $A=\frac{2}{3}(B+1)+\frac{1}{3}(A+1)$ $B=\frac{2}{3}(B+1)+\frac{2}{9}(A+1)+\frac{1}{9}(1)$ After simplifying both equations: $2A-2B=3$ $2A-3B=-9$ Clearly $B=12$ , so $A=\boxed{\textbf{(B)}\ 13.5}$ .

~ eevee9406

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2024 DMC Mock 10 Problems/Problem 15

Problem

Solution