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my name is Fblthp, and I am completely lost.

I don't know the location of my home planet, and given the barriers in language and culture, I can't tell you its name in any meaningful manner either.

The one thing I can tell is that in our base-10 mathematics, (we count both fingers in all 5 of our hands) the gravitational acceleration on the planet's surface was just under 10 length units per time units squared.

Can you help me find my home planet, or at least narrow down my choices?

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  • $\begingroup$ Would the knowledge tag be relevant? $\endgroup$ – mbjb Jun 20 '20 at 3:54
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    $\begingroup$ Dear stackexchange network: you don't have to delete the "Hello, " from the beginning of every single post. That said, if you could please send me an air-horn armed alert whenever someone posts a question with this exact tag combination, I'd be very, very much obliged. $\endgroup$ – Bass Jun 20 '20 at 3:58
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    $\begingroup$ @mestackoverflow If you have a very good reason, why we should have knowledge of all possible other planets' properties and measuring systems, then definitely yes. $\endgroup$ – Bass Jun 20 '20 at 3:59
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    $\begingroup$ Hello Fablthp, StackExchange is not a homework delivering service, please show us what you've tried to find your home planet so far. $\endgroup$ – Hasan Jun 20 '20 at 11:48
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    $\begingroup$ Cool puzzle but the intended answer was a stretch. $\endgroup$ – 6005 Jun 20 '20 at 15:22
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Initially, it seems that the face-value interpretation of the question can have no answer. The tag, though, poses a definite problem, which enables us to think about the problem slightly more in depth. My first thought was as follows:

Historically, the metre was defined as one ten-millionth of a quarter of the circumference of the earth, although our calculations were slightly off. Similarly, the second was historically defined as one $86,400$th of a solar day which is slightly longer than the sidereal day (the period of the rotation of the earth - different by one day every year because after the earth has completed a revolution it is back in the same position but having completed an 'extra' rotation).

Thus if (which is a big if, given the barriers in language and culture), fblthp's countrypeople have arrived at their definitions of length and time units in similar ways to us, it implies that their length and time units are in the same relation to the circumference and period of rotation of their planet as our own. This does not however, narrow down the possibilities very much as the length of the day of a planet is highly variable. Specifically, if this were the case, it would mean that if we let $G$ be the gravitational constant, $M$ the mass of the planet, $r_p$ its radius, $g$ the gravity at its surface, $C$ the circumference of the earth, $C_p$ the circumference of their planet, $D$ the length of our day, $D_p$ the length of their day, $m_p$ their unit of length, and $s_p$ their unit of time, we have:

$$g=\frac{GM}{r_p^2}=\frac{10m_p}{s_p^2}=\frac{10\left(\frac{C_p}{C}\right)}{\left(\frac{D_p}{D}\right)^2}\frac{m}{s^2}$$ $$\therefore D_p=D\sqrt{\frac{20\pi r_p^3}{GMC}}$$
Which is an expression for the length of their day in terms of the radius of their planet and its mass. Unfortunately, there's very little physical connection (as far as I'm aware) between the period of rotation of a planet and its radius and mass, which enables us to deduce very little.

However, an even bleaker prospect opens up for fblthp if we step back even further historically:

The original historical connection between the definition of metre and second was that the metre was defined as roughly the length of a pendulum that swings from one side to the other in one second. This doesn't add any new information though, as it's due to the fact that $g\approx\pi^2$, since the period of a pendulum of length $L$ is roughly:
$$T\approx2\pi\sqrt{\frac{L}{g}}$$ Put another way, no matter what planet you are on, and no matter how you have defined your time unit, if you define your length unit (like we roughly did) as the length of a pendulum with a period of $2$ time units, then $g$ the surface gravity on that planet will be just under $10$ (i.e. $\approx\pi^2$ length units per time units squared). Because of this, if fblthp's society has arrived at their length and time units similarly to how we did, then knowing the surface gravity of the planet in their units conveys precisely zero information about the planet itself.

The only problem with this reasoning of course, is that there is no a priori reason that fblthp's society should define their units the same way as us. However, the OP has indicated that this is the intended solution to the puzzle.

As a more general reflection on this type of problem:

Thinking about how different societies could communicate about physical constants is interesting. Of course, the only such constants whose numerical values could be communicated straightforwardly are the dimensionless constants, such as the fine-structure constant. All dimensional quantities have to be described in proportion to some other agreed quantity (e.g. here, also see here), which of course becomes a problem of chicken-and-egg. Without doing something like telling us how many Planck lengths are in the radius of their earth, we cannot learn anything specific about their units.

This would make the situation no longer a puzzle though. For what it's worth, the trick of recognizing the historical connection between metre and second was a good one, and therefore gripes aside I say 'good puzzle!'.

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    $\begingroup$ The second spoiler block is exactly what I was aiming at. Poor Fblthp. (I completely agree with all the gripes too.) $\endgroup$ – Bass Jun 20 '20 at 13:04
  • $\begingroup$ @Bass Fair enough! I've edited the answer accordingly. Even though the answer is non-unique (and I think the sentence about barriers in language and culture is a bit misleading since we need to assume they approached measuring length the same non-obvious way we did), the puzzle did make me think. Initially I thought it would be something to do with Planck lengths, but the ultimate punch line of the puzzle is interesting in its own right. Nice puzzle! $\endgroup$ – Anon Jun 20 '20 at 13:55
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    $\begingroup$ It's true that it's not entirely coincidence that we earthlings use units with that property, as I was already aware, but it's absurd to suggest that there's any kind of inevitability about it. E.g., the standard units of length here on earth have often been feet and inches, and even in metric the "cgs system", in which the centimetre is the primary unit of length despite its name, was commonly used for a while. $\endgroup$ – Gareth McCaughan♦ Jun 20 '20 at 14:06
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    $\begingroup$ For a method somewhat better than Planck lengths, note the SI second is defined relative to a property of a caesium-133 atom, because this is a fixed time effect we can measure very precisely. So if both parties have good enough physics comprehension and measurement, that could be a Rosetta stone. Or for something easier, if maybe less precise with our technology, the brightest frequencies and wavelengths of hydrogen emission. $\endgroup$ – aschepler Jun 20 '20 at 19:38
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This seems too obvious so maybe I'm missing some steganographic hint or something, but

it seems like the answer is "No, of course not", because the length and time units could be anything at all so the actual gravitational acceleration could be anything at all.

Perhaps the fact that

what Fblthp says happens to apply to us here on earth with the SI system

is meant to be relevant somehow, but

I don't see how (again, unless there's some secret hint here indicating that Fblthp is actually an amnesiac earthling or something of the kind).

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    $\begingroup$ This seems like an anti-answer. My feeling is that, given the user, there must be something more to this question that goes beyond the surface reading. $\endgroup$ – hexomino Jun 20 '20 at 11:22
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    $\begingroup$ I have something of the same feeling. I dithered between making this an answer and making it a comment, but eventually decided that, damn it. it is an answer, even if I think it's likely that it'll turn out to be wrong. $\endgroup$ – Gareth McCaughan♦ Jun 20 '20 at 14:02
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    $\begingroup$ It turns out that something else was intended. I'm quite familiar with the something-else in question, but I don't think it really works as intended for this question... $\endgroup$ – Gareth McCaughan♦ Jun 20 '20 at 14:07
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    $\begingroup$ @hexomino I don't agree it is a non-answer. If you take a random alien and ask about their gravity expressen in their units, the chances that it is 10 is pretty much zero. From a bayesian point of view, the fact that it is 10 for Fblthp makes it highly more probable he is form Earth rather than any other planet. If you add that he can communicate with us and does, I think it is a sure bet he is from Earth. (PS: that is, even if he believes he has 5 hands) $\endgroup$ – Florian F Jun 20 '20 at 22:20
  • $\begingroup$ @FlorianF Alternatively, you could argue that the fact that it is around $\pi^2$ for fblthp makes it much more probable that they have used the same idea to define their metre, which then gives you no information about the planet whatsoever. What 'more probable' means in the context of something infinitely unlikely is another question altogether... $\endgroup$ – Anon Jun 20 '20 at 22:37

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