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I found the answer to this question to be surprising, but obvious in retrospect; I recommend trying to figure out the question before looking up the answer.  (via Michael Lugo)
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Alok Tiwari's profile photoJohn Baez's profile photoAkshay S Dinesh's profile photoRobert King's profile photo
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Mars and Venus are the obvious guesses, so since you say you were surprised, it has to be...

Earth!
 
(I actually guessed what Quora claims is the right answer, based on the psychological method I described - but then I guessed the actual correct answer.)
 
Venus I thought, but that is the obvious answer... so probably wrong ?
 
What number is closest to 0?
 
This is a paragraph of irrelevant waffle so that my actual answer appears after the Read More, in case you don't want to read it. I repeat: this is a paragraph of irrelevant waffle so that my actual answer appears after the Read More, in case you don't want to read it. I hope that's clear.

Here's an argument for Mercury (though it sounds from the above as though I'm wrong). Assume that Mercury and Venus are at random places in their orbits. Then for each position of Mercury, there's quite a reasonable chance that it's closer than Venus. (I'm assuming for simplicity that the orbits are circular.) I now can't be bothered to do the calculation, but given that Terry was surprised, I go for Mercury over Venus. I'm assuming that the Earth answer is not allowed.
 
PS I wrote that before looking at the answer ...
 
Mars. I haven't looked up the answer nor read the above comments. I can only see two of +Timothy Gowers' anyway.

I say Mars, 'cos the inner planets spend as much time on the other side of the sun as on Earths side. And Mars moves slower, meaning when it is closer to Earth, it spend more time with Earth. I think this line of logic rationalisation doesn't apply to the gas giants because they're already too far.

Mercury could be considered a fixed distance, wave function from the Earth, it's centre at the Sun, but then so can Venus. So that line of reasoning doesn't work for me.

Now let me see how wrong I am! :)
 
I more or less guessed the answer by thinking about the maximum distance between planets when they are at opposite side of the Sun to the Earth. But it was a guess, because I don't remember the orbital periods of planets other than the Earth :-) and they have a significant impact on the answer.
 
Umm, I can't read the article. It's asking me to sign up. I will wait for the revelation of the answer here.

I see people hinting that it might be Earth. I'm going to say the way the question was put, "Which planet spends the greatest share of time closest to Earth?" implies that it has to be another planet other than Earth.
 
+Satyr Icon You can click on the option to read the first answer without signing up. :-)
 
Thanks, I was able to see the first answer.

The displayed charts were promising, at first, but I realised they were incomplete charts. Surely Mars too has to end up on the other side of the Sun, I thought. Where are those bars?

Oh well, not the first time I have been wrong.
 
One way is to calculate the average distance from any given point to a circle, both of which are in the same plane (in principle this is not equivalent to our problem here since the planets are moving with specific periods).  It turns out that for Mercury and Venus their average distances to Earth are very close: 1.036 AU versus 1.13 AU (AU=astronomical unit = distance between Earth and the Sun).  So I am not sure how could one get to the correct answer based merely on qualitative reasoning.

For Mars, its average distance to Earth calculated this way is 1.69 AU.
 
Here is a general version of the statistical simplification by +Timothy Gowers. Given  x0, x1, x2 three points sampled from a uniform distribution on concentric circles of radius r0, r1, r2, what is the probability that |x1 - x0| > |x1 - x2|.  Plotting that as a function of r0 and r2 should be interesting. No time to do this between compiles though. :)
 
+Satyr Icon - obviously the 'official' answer wasn't Earth!  That's the true answer to the question as stated: of all planets, Earth spends the greatest share of its time closest to Earth.   But of course that's too boring for people to care about it. 

Quora is actually a very good site, worth joining.  The analyses of this problem there are quite detailed and nice.
 
And, Mercury revolves around the Sun in 88 days. If that shouldn't make a difference, what will?
 
I saw +Timothy Gowers answer. Didn't read it, but couldn't help seeing Venus and Mercury mentioned. So I thought, I'll go for a lateral answer. This time it wasn't right, and it wasn't anywhere near as lateral as +John Baez's answer either. So either way no marks for me.
 
I guessed Mercury was the official correct answer.  But I wish they'd allowed the Sun into the competition, too!
 
+Satyr Icon Apologies for spoiling your fun slightly. I've edited my answer so that my guess is no longer visible. The new Google Plus design is annoying but it has its advantages ...
 
Shame on you for sharing content that you have to "log in" to view.... 
 
You had to "log in" to join this conversation here!   I think all this "logging in" stuff sucks, but at least Quora, unlike Google, doesn't track your every motion on the internet.
 
What a wonderful post. Three answers with three different approaches. 
 
To add to +John Baez 's latest comment, one should note +Terence Tao 's suggestion above: recommend trying to figure out the question before looking up the answer… . (I didn't even click the link yet.)
 
+Timothy Gowers +Philippe Beaudoin Your (and others') argument is basically what Mark Eichenlaub answered. Excellent! +Satyr Icon Yes, Mars does end up on the opposite side of the Sun! The semi-major axis for Mars's orbit is about 1.5 AU, which is why the distribution for Mars goes out to about 2.5 AU, which is the farthest apart the Earth and Mars can get. +Kwan Lowe if you know how JPL's HORIZONS system works, you recognize that my response and Jesse's responses are actually the same approach ;)

It should be possible to do this problem analytically, taking into account the different planes and eccentricities of the orbits of Mercury, Venus, Earth, and Mars, in terms of elliptic integrals. But since I said "in terms of elliptic integrals", I did not want to do the problem analytically!
 
The obvious answer is Mars. We just have to define closest as easiest to reach. 
 
+Leo Stein This was my first exposure to HORIZONS (and thank you for the gentle correction :D ).  
 
i hate you for advertising service which forces me to register
 
I also deduced Mercury from the word 'surprised'.  I'm not qualified to guess what my instantaneous answer would have been otherwise.
 
I thought I would find an answer in percentage terms, where the time each planet passes closer to the Earth, would be expressed thus: Mars = X%. Venus = Y%. Mercrio = Z%.
I think to find the answers, you need to align the four planets from the sun, measuring the time until the next alignment, and find the respective percentage of time for each planet.
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