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For the purposes of this question, we will assume that they both landed successfully. Ingenuity was deployed properly. Then, Perseverance drove away, and then Perserverance say hit a large rock and failed. Could Ingenuity survive on its own? Or does it rely on Perserverance?

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No. Perseverance is the communications relay for Ingenuity.

Ingenuity uses essentially ZigBee to communicate, the same protocol that possibly your smart bulbs use. ZigBee has a range of about 300m line-of-sight without obstructions which is just a tiny bit too short for communicating with Earth.

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    $\begingroup$ Indeed. Note that my answer is a bit tongue-in-cheek, since of course Perseverance also (usually) does not communicate with Earth (although it can in a pinch) but with one of the Martian relays. They're not quite as far away as Earth, but still orders of magnitude farther than 1000m. $\endgroup$ Commented May 23, 2022 at 21:24
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    $\begingroup$ Imagine a whole planet free of EM noise-generating devices! $\endgroup$
    – fraxinus
    Commented May 24, 2022 at 9:01
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    $\begingroup$ All the unfortunate sub-surface Martian homes finding their smart bulbs going haywire :( $\endgroup$
    – dbmag9
    Commented May 24, 2022 at 11:39
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    $\begingroup$ As we say on explainxkcd.com: {{citation needed}} $\endgroup$
    – Barmar
    Commented May 24, 2022 at 14:24

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