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I have a ceiling fan with 4 states. The states always go in the order of off, high, medium, low, off. A cord pull will change the state.

  • One pull of a cord to turn the fan to the off state from the low state.
  • Two pulls of a cord to turn the fan to the off state from the medium state.
  • Three pulls of a cord to turn the fan to the off state from the high state.
  • One pull of a cord to turn the fan to the high state from the off state.
  • Two pulls of a cord to turn the fan to the medium state from the off state.
  • Three pulls of a cord to turn the fan to the low state from the off state.

The fan is in an unknown state that is not off. What is the minimum number of cord pulls to guarantee that the fan will be in the off state when the starting state is unknown?

Please provide a solution or prove how it is not possible.

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  • $\begingroup$ In 3 pulls you cross all 4 states $\endgroup$
    – mousetail
    May 29, 2023 at 12:39
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    $\begingroup$ Community members have provided edits to your question to make it more readable, by fixing grammar and formatting as well as removing parts which add no useful information and thus simply take up space. Why do you want to make your question harder for answerers to parse? We really don't care that you copy-pasted something into Chat GPT when that did not end up providing a solution at all. What matters now is that fellow humans can understand your writing. $\endgroup$
    – bobble
    May 29, 2023 at 14:52
  • $\begingroup$ The post was presented as it was to ChatGPT as stated in text. Creating a different format does not help in any way, shape, or form. $\endgroup$
    – Vandalous1
    May 29, 2023 at 14:58
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    $\begingroup$ Why does it matter that the question was “presented to ChatGPT”, let alone how it was presented? $\endgroup$
    – Sneftel
    May 29, 2023 at 17:16
  • $\begingroup$ So, a post to "puzzling" had a format that was "hard to read." And you thought that was a bad thing and so you "fixed" it. Makes me wonder if the concept is clear. $\endgroup$
    – Boba Fit
    May 30, 2023 at 18:22

2 Answers 2

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It's not possible. Every pull of the cord simply moves it to the next state; mathematically speaking, pulling the cord is a bijection. Before pulling the cord, the fan can be in three states (high, medium, low); after pulling the cord, it can still be in three states (medium, low, off).

For a puzzle like this to be solvable, you'll need to have an operation which has the same output for more than one initial state.

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  • $\begingroup$ Glorfindel, I thought this might be surjective, as low to off is equal in count to off to high. (1) This allows us to have to relational states then, "off" and "on" with "on" having a subset of 3 states (high, medium, and low). I know this doesn't change the issue, but does it allow for at least two of the states to be solved for then? $\endgroup$
    – Vandalous1
    May 29, 2023 at 13:42
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    $\begingroup$ No - another way to look at it is that it's reversible - given a final state and the performed operations, you can always deduce was the initial state was. $\endgroup$
    – Glorfindel
    May 29, 2023 at 13:54
  • $\begingroup$ It might be easier to think of the group that is formed rather than the bijection. Effectively there is a pointer that gets rotated one step for each cord pull. Or, possibly, depending on where your largest experience lies, it might be easier to count cord pulls mod-4. With only four cord pulls to get you back to the start, there is a lot of overlap between the various views. $\endgroup$
    – Boba Fit
    May 30, 2023 at 18:30
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There is no fixed number that can guarantee it will end in the off state. Your only available options are (0,) 1, 2, or 3 (because it's mod 4). But because the current state is unknown, choosing any one of those numbers will necessarily be the incorrect number of pulls for the other two unknown states.

From a real-world point of view, though ... just pull the cord once and wait to hear the change in sound. If it gets noticeably quieter, then you've just transitioned from high to off.

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