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My friend had an interview at Cambridge. He was asked the following question, and was stumped:

I fly to Chicago. The plane trip is $8$ hours. I look at the time and then set my watch back $6$ hours. Knowing that the Earth rotates $360^\circ$ in $24$ hours, what is the radius of the Earth?

Rodrigo de Azevedo
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burn_burn_55
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    Without knowing the speed of the plane, that $8$ hours seem to be a useless information –  Sep 17 '16 at 08:53
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    Can you clarify what you mean by this "my watch back 6 hours"? – John Mayne Sep 17 '16 at 08:55
  • But it's only an estimation. You may just take an approximate number of the plane speed. – Cave Johnson Sep 17 '16 at 08:55
  • @Edward I guess it means that the person traversed 6 time zones? – Bobson Dugnutt Sep 17 '16 at 08:57
  • Edward - I mean take the original time (in England) and take 6 hours off. For example 18:00 - 12:00 because of a 6 hour time difference – burn_burn_55 Sep 17 '16 at 08:58
  • So the question is really "flying from Cambridge to Chicago"? @Hiraphor –  Sep 17 '16 at 08:59
  • Essentially, John, but I don't think the location matters in this case, only the time difference – burn_burn_55 Sep 17 '16 at 09:00
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    Which Cambridge, though? The one in Massachusetts is quite a bit closer to Chigaco than the one in England (and neither has an airport of its own). – hmakholm left over Monica Sep 17 '16 at 09:01
  • No, the Latitude of the cities (and the speed) seems to be relevant here @Hiraphor –  Sep 17 '16 at 09:01
  • Henning - Cambridge, England – burn_burn_55 Sep 17 '16 at 09:02
  • John Ma, in an interview I don't think they're expected to know the latitudes of the cities. This can be worked out using the information given, apparently. – burn_burn_55 Sep 17 '16 at 09:03
  • Please check with your friend again, you might miss some information. –  Sep 17 '16 at 09:04
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    So the question assumes that the subject needs to be told that the earth rotates 360° in 24 hours, but just knows already what the latitudes of Cambridge and Chicago are and how fast the unspecified plane flies? – hmakholm left over Monica Sep 17 '16 at 09:05
  • John, this isn't a case of missed information, this question was written down in the interview and is pretty much word for word. – burn_burn_55 Sep 17 '16 at 09:06
  • Henning - either the Cambridge interviewers don't know what they're doing or there is a method of working this out without knowing already knowing the speeds, latitudes and longitudes. – burn_burn_55 Sep 17 '16 at 09:07
  • They might know what they are doing. They want the applicants to smell something wrong. –  Sep 17 '16 at 09:08
  • "Half its diameter", of course. – Glen O Sep 17 '16 at 14:51
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    Incidentally, the interviewer was probably testing you on your problem-solving skills, but not your mathematical problem-solving skills. You could have asked what the distance is between Cambridge and Chicago, as this is likely information the interviewer had, and problem-solving includes figuring out where to obtain information. – Glen O Sep 17 '16 at 14:55

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There is not enough information in the question to get anywhere near an estimate of the earth's radius.

Even if we assume that "setting your watch back 6 hours" means that the difference in longitude between the two points is exactly 90°, the question still doesn't specify the latitude of the two cities. And, of course, to get from a question where all the givens are time to an answer with the dimension of length, we would need to know the (ground) speed of the plane too.

To see concretely how there is too little information, Some other possible scenarios with a plane that flies with the same ground speed (and on the same earth) would be:

You fly from Frankfurt am Main to Santiago de Chile. It takes 15 hours, and when you arrive you set your watch back 6 hours.

or

You fly from Stockholm to Lagos. It takes 8 hours, and when you arrive you set your watch back 1 hour.

or even

You fly from Nuuk to Comodoro Rivadavia. It takes 15 hours, and when you arrive you set your watch back 1 hour.

If the problem was solvable with the given information, there would need to be a method that gave the same radius of the earth when given the inputs $(8,6)$ as for $(15,6)$ and $(8,1)$ and $(15,1)$.

Distance sources.

hmakholm left over Monica
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    I think this answers it, he got as far as setting it to 90 degrees but was lost after. Perhaps you had to determine the question wasn't possible and they just wanted to see the thinking process, but thanks for clarifying that. – burn_burn_55 Sep 17 '16 at 10:09
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To me the most straightforward way of seeing that this is unsolvable without further information is to analyse the units/dimensions (see dimensional analysis).

None of the given information contained a unit of distance. Hence, it must be impossible to calculate a distance from it.

Sometimes a distance is "hidden" in a compound unit (e.g. a speed in km/h or a fuel economy in miles per gallon) that can be "broken out" by using a second piece of information (in the previous examples, a time in hours or fuel consumption in gallons). But there's none of that here.

Since there's no information about scale here whatsoever, one can imagine in a galaxy far away, on a world the size of a pearl, the Microuniversity of Minicambridge asking the same question.

Silverfish
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  • Dimensional analysis is taught as a high school topic in the United Kingdom and has a reputation as a regular source of university interview questions in science, so my guess would be this being the angle the examiners were taking. I'd be surprised if the question was asked in this form verbatim - Oxford and Cambridge in recent decades have tried to make the interview process less intimidating to applicants from non-traditional or underprivileged backgrounds, and their publicity material frequently avows there will be "no trick questions". – Silverfish Sep 17 '16 at 12:32
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My guess is that the interviewer was not expecting to get an exact answer, but was looking to see how confident your friend was at making their own approximations and working using those. For example, here are two guestimates that, if correct, would provide the information that is needed as previous answers have pointed out:

  • Chicago and England have similar climates so there's probably not too much difference in latitude. They aren't close to the equator and aren't close to the pole, so let's guess about halfway between. i.e. latitude of 45 degrees.
  • I know that military jets go supersonic, but you're probably talking about a commercial jet here and they don't. Let's guess about half the speed of sound: i.e. about 150m/s

Note that these estimates are incredibly crude, but they're reasoned and would have given your friend something to do that the interviewer could have judged.

p.s. A good source to emulate for this sort of crude estimation is the xkcd What If? articles, which use guesses like these to insight into situations much further from normal human experience.

Chessanator
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    Commercial jets fly at about 80% of the speed of sound, but an 8-hour trip would be mostly at a high altitude where speed of sound is much less than at sea level. About 450 to 480 knots might be a reasonable guess. Knowing that 1 knot = 1 nautical mile/hour = 1852 m/hour (yes, I repeated all of this from memory) you could convert that to km/hour, which is useful if you want the radius in km rather than nautical miles. – David K Sep 17 '16 at 17:28
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Estimate plan flies at 500knts. Fly for 8 hours = 4000miles. Clock back 6 hours, so I'm 1/4 of the way round the earth. Therefore circumference of earth is 4x = 16,000miles at this lattitude. London is 53' so call it 45' for argument sake, using pythag/trig this means circumference is 0.7 (approx) of that at equator. Therefore 16000/0.7 = 23,000 (approx.) (actual is around 21,600 Nautical miles so not bad for an estimate)

Keith
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  • The plane will fly a great-circle route, not just follow a line of latitude (even if the latitudes were the same). But given the many other sources of error due to guesswork, this particular source of error isn't especially large. Good enough for an order-of-magnitude estimate, anyway. – David K Sep 17 '16 at 17:38
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The goal of this interview question wasn't a mathematics answer. It is really a question that is suppose to trigger you to ask more question.

The issues with which Cambridge was important, as with all sorts of pseudo distance units being thrown at you. The interviewee is suppose to be asking all sorts of questions to clarify and request more data. This is very typical in standard business interactions, where clients always think they want something, but actually rarely does, and really has no idea what they want.

Nelson
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