greg@harvard.ARPA (Greg) (11/04/85)
Johnny the adventurous flyer flies at constant altitude over the Pacific. He periodically turns left by one degree. After a while Johnny discovers that his plane is at the same position *and orientation* as when he started. Therefore he lands. He deduces that the path he took encloses 140 million/9*pi square kilometers. Now for some questions: 1) How many left turns did Johnny make? 2) What was the approximate air distance between two consecutive turns? You may assume that the Earth is a perfect sphere with a circumference of exactly 40,000 kilometers. If you know too much math, please don't post the spoiler right away. -- gregregreg
ab@unido.UUCP (Andreas Bormann) (11/09/85)
>/***** unido:net.puzzle / harvard!greg / 5:41 am Nov 4, 1985*/ >Subject: More interesting than the polar bear problem > >Johnny the adventurous flyer flies at constant altitude over the Pacific. >He periodically turns left by one degree. After a while Johnny discovers >that his plane is at the same position *and orientation* as when he started. >Therefore he lands. He deduces that the path he took encloses 140 million/9*pi >square kilometers. Now for some questions: > >1) How many left turns did Johnny make? >2) What was the approximate air distance between two consecutive turns? > >You may assume that the Earth is a perfect sphere with a circumference of >exactly 40,000 kilometers. After my calculations he made 352 left turns and the approximate air distance between two turning points was 22.237 kilometers. I assumed that Johnny flew an exact polygon like this: TAKEOFF 1. | +--==*====runway / \ 2.+ + 352. | | 3.+ + 351. \ / +--- 4. But when he reached his starting point he had to make one more 1deg-turn to get into the same orientation as the runway. So my solution might not be correct. Maybe there are an infinite set of solutions looking this way: TAKEOFF last turning point 1. | | +--==*=+==runway / \ 2.+ + n-1. | | 3.+ + n-2. \ / +--- 4. Possibly the number of left turns is 353 in any case... Andreas Bormann University of Dortmund [UniDo] West Germany Uucp: ab@unido.uucp Path: {USA}!ihnp4!seismo!mcvax!unido!ab {Europe}!{cernvax,diku,enea,ircam,mcvax,prlb2,tuvie,ukc}!unido!ab Bitnet: ab@ddoinf6.bitnet \ Missiles: -=>-->-*> N 51 29' 05" E 07 24' 42" /
greg@harvard.ARPA (Greg) (11/10/85)
ab@unido (Andreas Bormann) writes: > >Johnny the adventurous flyer flies at constant altitude over the Pacific. > >He periodically turns left by one degree. After a while Johnny discovers > >that his plane is at the same position *and orientation* as when he started. > >He deduces that the path he took encloses 140 million/9*pi square > >kilometers. ... > After my calculations he made 352 left turns and the approximate > air distance between two turning points was 22.237 kilometers. > I assumed that Johnny flew an exact polygon like this: > > TAKEOFF > 1. | > +--==*====runway > / \ > 2.+ + 352. > | | > 3.+ + 351. > \ / > +---... > 4. This is essentially the right answer (I counted 353, but that's just a matter of definition). Now for an even more interesting problem: Tom, who is Johnny's friend, is equally adventurous. He too goes flying over the Pacific and makes left turns by one degree. Indeed, he also makes 353 turns and then lands on the same runway he took off of. However, Tom does not make these turns periodically. He may go for hours in a straight line and then make several turns in rapid succession. Let X be the total area of the Pacific that Tom's plane encircles. What are the possible values of X? Again, the Earth is a perfect sphere with circumference 40,000 kilometers. Also, I should make clear that planes, when flying straight, travel in a great circle, and do not necessarily maintain a constant compass direction. -- gregregreg