Rolling until two rolls differ by two

MediumProbability Expected Value~8m

A fair $6$ -sided die is rolled repeatedly. What is the expected number of rolls until two consecutive rolls differ by exactly $2$ (i.e. $|{\text{current}} - {\text{previous}}| = 2$ )?

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Let $e_v$ be expected extra rolls given last face $v$ ; how many faces are exactly $2$ away from $v$ ? (One for $1,2,5,6$ ; two for $3,4$ .)
Solve the small symmetric system ( $e_1=e_2=5$ , $e_3=4$ ), then $\mathbb{E}=1+\tfrac16\sum e_v$ .

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5.6667 (± 0.01)

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Asked at: Optiver, Akuna Capital

Rolling until two rolls differ by two

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