Cuts until one orb remains

HardProbability Expected Value~9m

Seven orbs, labelled $1$ (attached to the ceiling) down to $7$ , hang in a vertical chain, each joined to its neighbor by a link ( $6$ links total). Each step, a uniformly random remaining link is cut, and every orb below that link falls and shatters. What is the expected number of cuts until only orb $1$ remains? Round to two decimal places.

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Cutting link $j$ (from the top) leaves $j-1$ links - uniform on $\{0,\dots,m-1\}$ .
$e_m = 1 + \tfrac1m\sum_{i<m} e_i$ solves to $e_m = H_m$ ; take $H_6$ .

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2.45 (± 0.01)

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Cuts until one orb remains

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