Rolling until two rolls are close

HardProbability Expected Value~10m

A fair $6$ -sided die is rolled repeatedly. What is the expected number of rolls until two consecutive rolls differ by at most $1$ (i.e. $|{\text{current}} - {\text{previous}}| \le 1$ )?

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Let $e_v$ be expected extra rolls given last face $v$ ; a next roll within distance $1$ stops it. Use symmetry $1\leftrightarrow6, 2\leftrightarrow5, 3\leftrightarrow4$ .
Solve the $3$ -variable system, then $\mathbb{E}=1+\tfrac16\sum_v e_v$ .

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3.278 (± 0.01)

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Asked at: Optiver, Akuna Capital

Rolling until two rolls are close

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