judith@proper.UUCP (Judith Abrahms) (10/17/85)
The old Polar Bear Problem: You go somewhere, set up a tent, walk a mile
south, walk a mile due west, shoot a bear. Then you drag the bear a mile north
to your tent & have him for dinner. What color is the bear?
The sequel: (1) From how many points on Earth (assuming it's spherical, etc.)
can you make exactly these moves, i.e., walk 1 mile south, 1
mile west, 1 mile north, and be back where you started?
(2) Describe all of them.
Judith Abrahms
{ucbvax,ihnp4}!dual!proper!judithgwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (10/19/85)
> The old Polar Bear Problem: You go somewhere, set up a tent, walk a mile > south, walk a mile due west, shoot a bear. Then you drag the bear a mile north > to your tent & have him for dinner. What color is the bear? > > The sequel: (1) From how many points on Earth (assuming it's spherical, etc.) > can you make exactly these moves, i.e., walk 1 mile south, 1 > mile west, 1 mile north, and be back where you started? > > (2) Describe all of them. Are there any polar bears in Antarctica? Are there even any near the North Pole?
bs@faron.UUCP (Robert D. Silverman) (10/19/85)
> The old Polar Bear Problem: You go somewhere, set up a tent, walk a mile > south, walk a mile due west, shoot a bear. Then you drag the bear a mile north > to your tent & have him for dinner. What color is the bear? > > The sequel: (1) From how many points on Earth (assuming it's spherical, etc.) > can you make exactly these moves, i.e., walk 1 mile south, 1 > mile west, 1 mile north, and be back where you started? > > (2) Describe all of them. > > Judith Abrahms > {ucbvax,ihnp4}!dual!proper!judith The problem is trivial. There are an infinite number of such points lying on an infinite number of concentric circles centered on the south pole. The point is that you can walk N times around a circle whose radius is 1/(2 PI N) and still walk only 1 mile. Walking due west keeps you on the circle. Bob Silverman (they call me Mr. 9)
ask@cbdkc1.UUCP (A.S. Kamlet) (10/21/85)
> The old Polar Bear Problem: You go somewhere, set up a tent, walk a mile > south, walk a mile due west, shoot a bear. Then you drag the bear a mile north > to your tent & have him for dinner. What color is the bear? > > The sequel: (1) From how many points on Earth (assuming it's spherical, etc.) > can you make exactly these moves, i.e., walk 1 mile south, 1 > mile west, 1 mile north, and be back where you started? > (2) Describe all of them. > Judith Abrahms {ucbvax,ihnp4}!dual!proper!judith Well, there's the North Pole. And there's a bunch of points near the South Pole. [Rot 13] Pbafvqre n pvepyr bs yngvghqr jvgu pvephzsrerapr = 1/A zvyr [ A=1,2, ... ] N frg bs fhpu pvepyrf vf ybpngrq irel pybfr gb gur Fbhgu Cbyr. 1. Fgneg ng nal cbvag ybpngrq rknpgyl 1 zvyr abegu bs nal fhpu pvepyr. 2. Zbir fbhgu bar zvyr naq lbh ernpu gur pvepyr. 3. Zbir jrfg (be rnfg) rknpgyl A gvzrf, r.t., 1 zvyr, naq lbh erghea gb gur cbvag jurer lbh ragrerq gur pvepyr. 4. Gura zbir abegu 1 zvyr nybat gur cngu lbh bevtvanyyl gbbx, naq lbh ner ng gur fgnegvat cbvag. -- Art Kamlet AT&T Bell Laboratories Columbus {ihnp4 | cbosgd}!cbrma!ask
c160-3ay@ucbzooey.BERKELEY.EDU (Ranjit Bhatnagar) (10/21/85)
In article <367@faron.UUCP> bs@faron.UUCP (Robert D. Silverman) writes: >> The old Polar Bear Problem: ... >> The sequel: (1) From how many points on Earth (assuming it's spherical, etc.) >> can you make exactly these moves, i.e., walk 1 mile south, 1 >> mile west, 1 mile north, and be back where you started? >> (2) Describe all of them. >> Judith Abrahms > >The problem is trivial. There are an infinite number of such points lying >on an infinite number of concentric circles centered on the south pole. >The point is that you can walk N times around a circle whose radius is >1/(2 PI N) and still walk only 1 mile. Walking due west keeps you on the >circle. > >Bob Silverman (they call me Mr. 9) But what color is the bear? Will a penguin do? By the way: heat causes metal to expand. If you have a piece of metal with a spherical hole in it, does the hole expand, contract, or remain the same when the metal is heated? What about a square hole? I don't know the answer! ...ranjit bhatnagar (Disclaimer! If they knew what I was doing, they'd stop me.)
moews_b@h-sc1.UUCP (david moews) (10/21/85)
> By the way: heat causes metal to expand. If you have a piece of metal > with a spherical hole in it, does the hole expand, contract, or remain > the same when the metal is heated? What about a square hole? I don't > know the answer! > > ...ranjit bhatnagar Since the piece of metal expands uniformly when it is heated, the hole must expand. If you use a square hole, consider the portion of the metal that runs along an edge of the square hole. If this rod-shaped portion of the metal piece was heated by itself, it would expand in length; so it must also expand in length when it is part of the hole. Thus, the length of the side of the hole must get bigger and the hole expands. David Moews ...harvard!h-sc1!moews_b moews_b%h-sc1@harvard.arpa "The Cray-5 can execute an infinite loop in under a minute."
dim@whuxlm.UUCP (McCooey David I) (10/21/85)
> The old Polar Bear Problem: You go somewhere, set up a tent, walk a mile > south, walk a mile due west, shoot a bear. Then you drag the bear a mile north > to your tent & have him for dinner. What color is the bear? > > The sequel: (1) From how many points on Earth (assuming it's spherical, etc.) > can you make exactly these moves, i.e., walk 1 mile south, 1 > mile west, 1 mile north, and be back where you started? > > (2) Describe all of them. > > Judith Abrahms > {ucbvax,ihnp4}!dual!proper!judith How about a more difficult sequel like the following: Where on the earth can one walk 1 mile south, 1 mile west, 1 mile north, AND 1 mile east, and end up at the starting point? If you think you have a solution, there should be more... It would be nice if some mathematically inclined readers could contribute exact and complete solutions (to both sequels).
datanguay@watdaisy.UUCP (David Tanguay) (10/22/85)
> By the way: heat causes metal to expand. If you have a piece of metal > with a spherical hole in it, does the hole expand, contract, or remain > the same when the metal is heated? What about a square hole? I don't > know the answer! > > ...ranjit bhatnagar Physically speaking, it beats me, too, but coopers bind their barrels by heating a metal ring, placing it over the barrel pieces, and then cooling the ring. This would mean that the hole would expand when the metal is heated. This looks like a job for second year calculus students. David Tanguay
perkins@bnrmtv.UUCP (Henry Perkins) (10/22/85)
> The sequel: (1) From how many points on Earth (assuming it's spherical, etc.) > can you make exactly these moves, i.e., walk 1 mile south, 1 > mile west, 1 mile north, and be back where you started? > (2) Describe all of them. > Judith Abrahms {ucbvax,ihnp4}!dual!proper!judith There are infinitely many. The points are at (1) the north pole, and (2) a circle of points 1 mile north of the circle with circumference 1 mile centered around the south pole. {hplabs,amdahl,3comvax}!bnrmtv!perkins --Henry Perkins
halle@hou2b.UUCP (J.HALLE) (10/22/85)
>> By the way: heat causes metal to expand. If you have a piece of metal >> with a spherical hole in it, does the hole expand, contract, or remain >> the same when the metal is heated? What about a square hole? I don't >> know the answer! >> >> ...ranjit bhatnagar >Physically speaking, it beats me, too, but coopers bind their barrels by >heating a metal ring, placing it over the barrel pieces, and then cooling the >ring. This would mean that the hole would expand when the metal is heated. >This looks like a job for second year calculus students. > >David Tanguay This looks like a job for a first semester high school physics student. As you heat the metal, it expands in all dimensions. It stays exactly congruent (assuming uniform heating). Thus the hole expands, a spherical bubble inside expands, a square hole stays square, etc.
dave@inset.UUCP (Dave Lukes) (10/22/85)
In article <361@proper.UUCP> judith@proper.UUCP (judith) writes: >The old Polar Bear Problem: You go somewhere, set up a tent, walk a mile >south, walk a mile due west, shoot a bear. Then you drag the bear a mile north >to your tent & have him for dinner. What color is the bear? > >The sequel: (1) From how many points on Earth (assuming it's spherical, etc.) > can you make exactly these moves, i.e., walk 1 mile south, 1 > mile west, 1 mile north, and be back where you started? > > (2) Describe all of them. > I dunno about anyone else, but I though this was junior school (?? ``3rd. grade'' is what you colonials say, I think) stuff: 1) a) The North Pole b) anywhere 1 + 1/(2pi) miles from the south pole i.e. an arbitrarily large number of places. 2) What do you mean ``describe''?? Location?? Scenery?? I've never been to either place, but I suspect they're uniformly cold and icy. Dave Lukes. ...!ukc!inset!dave ``There are NO new ideas in the world, only new ways of misusing old ones.''
jsl@princeton.UUCP (Jong Lee) (10/22/85)
> In article <367@faron.UUCP> bs@faron.UUCP (Robert D. Silverman) writes: > >> The old Polar Bear Problem: ... > >> The sequel: (1) From how many points on Earth (assuming it's spherical, etc.) > >> can you make exactly these moves, i.e., walk 1 mile south, 1 > >> mile west, 1 mile north, and be back where you started? > >> (2) Describe all of them. > >> Judith Abrahms > > the bear could be any color right? I mean, you could start exactly on the north pole, and follow the prescribed route to get back to N; so he could be white. But as Mr. 9 said, there are infinite # of soln's around the south pole, where no polar bears exist, just penguin opuses. Sun_Man > By the way: heat causes metal to expand. If you have a piece of metal > with a spherical hole in it, does the hole expand, contract, or remain > the same when the metal is heated? What about a square hole? I don't > know the answer! > the hole expands as well....also true for square "holes". > ...ranjit bhatnagar > > (Disclaimer! If they knew what I was doing, they'd stop me.) *** REPLACE THIS LINE WITH YOUR MESSAGE ***
hes@ecsvax.UUCP (Henry Schaffer) (10/24/85)
> > The old Polar Bear Problem: You go somewhere, set up a tent, walk a mile > > How about a more difficult sequel like the following: > > Where on the earth can one walk 1 mile south, 1 mile west, 1 mile > north, AND 1 mile east, and end up at the starting point? > > If you think you have a solution, there should be more... It would be nice > if some mathematically inclined readers could contribute exact and complete > solutions (to both sequels). One set of solutions is a set of small circles close to the North Pole. They are of a size so that the 1 mile east takes one around the pole n times (1, 2, ...) and enough extra to end up at the starting point. (Sorry, I don't feel like doing the math tonight.) Another set of solutions is the circle 1/2 mile north of the equator. --henry schaffer
ken@turtlevax.UUCP (Ken Turkowski) (10/24/85)
In article <855@whuxlm.UUCP> dim@whuxlm.UUCP (McCooey David I) writes: > Where on the earth can one walk 1 mile south, 1 mile west, 1 mile > north, AND 1 mile east, and end up at the starting point? > >If you think you have a solution, there should be more... It would be nice >if some mathematically inclined readers could contribute exact and complete >solutions (to both sequels). Once you have an idea that the problem takes place on a sperical geometry, the answer is easy: 1/2 mile north of the south pole In a half mile you reach the pole; continue in the same direction and follow the rest of the steps, and you trace out a bow-tie path. However, several philosophical questions the occurs: After you reach the south pole in the first step, are you still going south? At the south pole, is there any east, west or south? All directions from there seem to be north. -- Ken Turkowski @ (CADLINC --> CIMLINC), Menlo Park, CA UUCP: {amd,decwrl,hplabs,seismo,spar}!turtlevax!ken ARPA: turtlevax!ken@DECWRL.ARPA
franka@mmintl.UUCP (Frank Adams) (10/24/85)
In article <749@inset.UUCP> dave@inset.UUCP (Dave Lukes) writes: >In article <361@proper.UUCP> judith@proper.UUCP (judith) writes: >>The old Polar Bear Problem: You go somewhere, set up a tent, walk a mile >>south, walk a mile due west, shoot a bear. Then you drag the bear a mile north >>to your tent & have him for dinner. What color is the bear? >> >>The sequel: (1) From how many points on Earth (assuming it's spherical, etc.) >> can you make exactly these moves, i.e., walk 1 mile south, 1 >> mile west, 1 mile north, and be back where you started? >> >> (2) Describe all of them. >> >I dunno about anyone else, but I though this was junior school >(?? ``3rd. grade'' is what you colonials say, I think) stuff: > >1) a) The North Pole > b) anywhere 1 + 1/(2pi) miles from the south pole >i.e. an arbitrarily large number of places. Actually, this isn't yet a complete list. Change (b) to "anywhere 1 + 1/(2n*pi) miles from the south pole, where n is a positive integer". This isn't exactly right, either; it assumes the Earth is flat in the neighborhood of the South Pole. The exact distances are a little larger. Frank Adams ihpn4!philabs!pwa-b!mmintl!franka Multimate International 52 Oakland Ave North E. Hartford, CT 06108
norman@lasspvax.UUCP (Norman Ramsey) (10/25/85)
In article <10755@ucbvax.ARPA> c160-3ay@ucbzooey.UUCP (Ranjit Bhatnagar) writes: >By the way: heat causes metal to expand. If you have a piece of metal >with a spherical hole in it, does the hole expand, contract, or remain >the same when the metal is heated? What about a square hole? I don't >know the answer! The hole expands. Thermal expansion dilates a whole object. Some people like to think about metal doughnuts; the doughnut does the same thing on heating whether the hole is present or not. I personally would rather think about dilates, or just renormalizing (there's that word again) the measure of distannce to obtain the original object. -- Norman Ramsey ARPA: norman@lasspvax -- or -- norman%lasspvax@cu-arpa.cs.cornell.edu UUCP: {ihnp4,allegra,...}!cornell!lasspvax!norman BITNET: (in desperation only) ZSYJ at CORNELLA US Mail: Dept Physics, Clark Hall, Cornell University, Ithaca, New York 14853 Telephone: (607)-256-3944 (work) (607)-272-7750 (home)
jacob@chalmers.UUCP (Jacob Hallen) (10/26/85)
The colour of the bear can be any one except white! Proof: - Polar bears live only near the north pole. - There is one possible point near the north pole. - There are an infinite number of points near the south pole. - The chance of picking the point near the north pole is zero, because one (number of points near north pole) divided by infinity plus one (total number of points) is zero. Jacob Soon to be elected twit of the year.
judith@proper.UUCP (Judith Abrahms) (10/26/85)
In article <> dim@whuxlm.UUCP (McCooey David I) writes: >How about a more difficult sequel like the following: > > Where on the earth can one walk 1 mile south, 1 mile west, 1 mile > north, AND 1 mile east, and end up at the starting point? What's wrong with the circle of latitude 1/2 mi. above the equator? Am I missing something? J.A. {ucbvax,ihnp4}!dual!proper!judith
rvdb@hou2c.UUCP (R.VANDERBEI) (10/26/85)
it's almost true everywhere - almost.
Wizard@codas.UUCP (Wistful Wizard) (10/27/85)
> Where on the earth can one walk 1 mile south, 1 mile west, 1 mile > north, AND 1 mile east, and end up at the starting point? Why, right here of course, but I'm not telling where that is!
hopp@nbs-amrf.UUCP (Ted Hopp) (10/29/85)
> By the way: heat causes metal to expand. If you have a piece of metal > with a spherical hole in it, does the hole expand, contract, or remain > the same when the metal is heated? What about a square hole? I don't > know the answer! > > ...ranjit bhatnagar If you heat the metal enough, it will glow. The hole will then be a black hole, and therefore contract forever. Oh, I'm sorry. I thought we were in net.physics.newman ;-) Actually, the hole (spherical, square, circular, or whatever) will expand. Imagine heating the metal that filled the hole when the metal was cool while you are heating the metal with the hole. The "filler" will fit back into the hole when both are hot. Since the filler chunk expanded (being metal), the hole must have expanded. As to why the filler should fit back in, imagine heating the metal plus filler without separating the filler - if it isn't going to fit in when heated separately, why does it exactly fill where the hole would have been? -- Ted Hopp {seismo,umcp-cs}!nbs-amrf!hopp
verma@ucla-cs.UUCP (10/29/85)
*******************__This_line_was_intentionally_left_blank__******************* In article <934@turtlevax.UUCP> ken@turtlevax.UUCP (Ken Turkowski) writes: { lots of stuff we've seen 100's of times in 100's of not so unique solutions } { to the polar bear problem and the new polar bear problem. } > >Once you have an idea that the problem takes place on a sperical geometry, >the answer is easy: > > 1/2 mile north of the south pole > >In a half mile you reach the pole; continue in the same direction and >follow the rest of the steps, and you trace out a bow-tie path. > >However, several philosophical questions the occurs: > >After you reach the south pole in the first step, are you still going south? > >At the south pole, is there any east, west or south? >All directions from there seem to be north. >-- I don't follow you, but first lets address your second point. I think that most people will agree with these definitions. (underlying these is the notion of surface distance) Def 1: Line of longitude: A shortest path on the surface of the earth which connects the north pole to the south pole. Def 2: Line of latitude: Any line on the surface of the earth that is perpendicular to every line of longitude. (if I have these backwards, I am deeply sorry.) Def 3: Northern movement: Movement along a line of longitude which decreases ones surface distance to the south pole. Def 4: Southern movement: Movement along a line of longitude which decreases ones surface distance to the south pole. These last two definitions lead to the following: Cor 1: Northern movement is just forward movement on a line of longitude when facing the north pole. Cor 2: Southern movement is just forward movement on a line of longitude when facing the south pole. <the proof is left as an exercise> These lead to the terminology in the following definitions: Def 5: Western movement is just forward movement on a line of latitude when the north pole is to your right. Def 6: Eastern movement is just forward movement on a line of latitude when the south pole is to your right. Lemma 1: All of the paths which radiate outward from the south pole and contain no east/west movement are lines of longitude. <again an exercise> Now we can address your question (second one) more easily. Thm 1: Any movement from the point of the south pole is northern movement. pf: (I should do this with epsilon neighborhoods, but...) While at the south pole (a point) the only way to leave is to take move in a radial direction for at least some small distance. Thus all paths originating at the pole will at least locally lie on a line of longitude. Also we are leaving the south pole, therefore must be facing the north pole. Hienceforth (how's that!!!) we will be moving north. That almost satisfies me, and I hope it is correct. But back to your first point; I do not understand your solution even if we consider moving past the pole as continued southern movement. First, note that since the distances we are talking about are small (<2000 mi) compaired to the radius of the earth, we can assume we are in a plane. But remember your directions though, North is any direction out from the south pole, west is counter-clockwise, and east clockwise around circles centered at the south pole. So start at point A on the circle of radius 1/2 mile. Follow its diameter, then move counter- clockwise for 1 mile. We are now ~1/3 the way back (actually 1/pi). Cross again? and we will not be home. I think you ment to go 1/2 circumference of the earth >> 1 mile. If not please re-explain. Thank you, TS Verma.
bs@faron.UUCP (Robert D. Silverman) (10/29/85)
> it's almost true everywhere - almost.
Do you really mean 'it's almost true everywhere' or do you mean
'it's true almost everywhere' ?
I hate to clue everyone in but:
IF your answer means that the Borel measure of the set of starting points
is 1 you're wrong. It is zero. The set of starting points is that set
such that the radius of a great-circle running E-W is the same as that
of another great-circle running E-W which is 1 mile south. The only place
this happens is the great-circle 1/2 mile north of the equator. Moving
1 mile south places you on the great-circle 1/2 mile south of the equator
and this obviously has the same radius as the original circle.
Thus, of the entire set of great circles (cardinality C) only 1 satisfies
the conditions (i.e. measure is zero)
Postings which claim the circles 1 mile north of the equator are solutions
are wrong. This is easy to see because lines of longitude are closer
together 1 mile north of the equator than they are at the equator. Thus,
if you travel 1 mile south to the equator, 1 mile west, and then 1 mile
north you will be closer than 1 mile to your starting point.
At latitude 90-theta, an East-West great circle has radius 2 PI r sin(theta)
where r is the Earth's radius. Why can't people do simple high school geometry?
Thus, it's FALSE almost everywhere.
Bob Silverman (they call me Mr. 9)bs@faron.UUCP (Robert D. Silverman) (10/29/85)
> > it's almost true everywhere - almost. > > Do you really mean 'it's almost true everywhere' or do you mean > 'it's true almost everywhere' ? > > I hate to clue everyone in but: > > IF your answer means that the Borel measure of the set of starting points > is 1 you're wrong. It is zero. The set of starting points is that set > such that the radius of a great-circle running E-W is the same as that > of another great-circle running E-W which is 1 mile south. The only place > this happens is the great-circle 1/2 mile north of the equator. Moving > 1 mile south places you on the great-circle 1/2 mile south of the equator > and this obviously has the same radius as the original circle. > Thus, of the entire set of great circles (cardinality C) only 1 satisfies > the conditions (i.e. measure is zero) > > Postings which claim the circles 1 mile north of the equator are solutions > are wrong. This is easy to see because lines of longitude are closer > together 1 mile north of the equator than they are at the equator. Thus, > if you travel 1 mile south to the equator, 1 mile west, and then 1 mile > north you will be closer than 1 mile to your starting point. > > At latitude 90-theta, an East-West great circle has radius 2 PI r sin(theta) > where r is the Earth's radius. Why can't people do simple high school geometry? > > Thus, it's FALSE almost everywhere. > > Bob Silverman (they call me Mr. 9) There are also some 'peculiar shaped' solutions near the south pole. One walks a mile south. Then walking a mile west wraps one some multiple N times around a great circle. This does not bring one back to the point where one started the East-West walk but rather leaves one at some other point on that circle. Then walking 1 mile north places one back on the original great circle, only one mile east. Note that N is not an integer. ------- <-- 1st great circle distance = 1 mile | | 1 mile north --> | | <-- 1 mile south | | / \ <-- imagine folded 90 degress toward you / \ | | \ / <-- 2nd great circle -------- The exact set of starting points would be a tedious, but fairly simple problem in trigonometry to work out. An equation for the simple case (where one does not wrap around each circle more than one full turn) is: 1 2 PI r sin( 90 - t1) - 1 ------------------ = ------------------------------- 2 PI r sin(90 - t2) 2 PI r sin(90 -t1) One simply finds angles t1 and t2 that satisfy this equation. Bob Silverman (they call mr Mr. 9)
dim@whuxlm.UUCP (McCooey David I) (10/30/85)
> In article <> dim@whuxlm.UUCP (McCooey David I) writes: > >How about a more difficult sequel like the following: > > > > Where on the earth can one walk 1 mile south, 1 mile west, 1 mile > > north, AND 1 mile east, and end up at the starting point? > > What's wrong with the circle of latitude 1/2 mi. above the equator? Am I > missing something? Yes, you are missing some of the other solutions. Dave McCooey AT&T Bell Labs, Whippany, NJ
bundy@oasys.UUCP (11/01/85)
Judith Abrahms: > > How about a more difficult sequel like the following: > > Where on the earth can one walk 1 mile south, 1 mile west, 1 mile > north, AND 1 mile east, and end up at the starting point? > > If you think you have a solution, there should be more... It would be nice > if some mathematically inclined readers could contribute exact and complete > solutions (to both sequels). This sounds pretty sensitive to local geography, especially over a one mile distance. Anyone care to post a general set of solutions for the Himalaya alone? (No flames, you did say the Earth, not some abstract (and incorrect) mathematical representation) Bruce Bundy {ucbvax,allegra,hao}!nbires!oasys!bundy
hal@ecsvax.UUCP (Hal Hunnicutt) (11/01/85)
How about net.puzzle.polarbears? -- "You see, Elvis can't read contracts. All Elvis knows is, no Ferrari, no more rides with the top down." --Sonny Crockett
levy@ttrdc.UUCP (Daniel R. Levy) (11/01/85)
In article <42@nbs-amrf.UUCP>, hopp@nbs-amrf.UUCP (Ted Hopp) writes: >> If you have a piece of metal >> with a spherical hole in it, does the hole expand, contract, or remain >> the same when the metal is heated? What about a square hole? I don't >> know the answer! >> ...ranjit bhatnagar > >Actually, the hole (spherical, square, circular, or whatever) will >expand. Imagine heating the metal that filled the hole when the metal >was cool while you are heating the metal with the hole. The "filler" >will fit back into the hole when both are hot. Since the filler >chunk expanded (being metal), the hole must have expanded. As to >why the filler should fit back in, imagine heating the metal plus >filler without separating the filler - if it isn't going to fit in >when heated separately, why does it exactly fill where the hole would >have been? Isn't this begging the question? Suppose that internal forces/stresses developed as a result of the expansion process. (This happens, for example, with ordinary glass when it is heated, often resulting in shattering [BTW, can anyone tell why it is that ordinary glass will break when heated, but the same glass was successfully cooled into that shape from a molten blob or sheet?].) Anyhow, some glasses, like Pyrex, are much less subject to this since they expand much less when heated than ordinary glass. The internal forces would constrain the "filler" piece of material to be a slightly different shape when an integral part of the whole than if heated by itself. Just ramblin' on... -- ------------------------------- Disclaimer: The views contained herein are | dan levy | yvel nad | my own and are not at all those of my em- | an engihacker @ | ployer or the administrator of any computer | at&t computer systems division | upon which I may hack. | skokie, illinois | -------------------------------- Path: ..!ihnp4!ttrdc!levy
rvdb@hou2c.UUCP (R.VANDERBEI) (11/01/85)
>> it's almost true everywhere - almost. >Do you really mean 'it's almost true everywhere' or do you mean >'it's true almost everywhere' ? >I hate to clue everyone in but: >IF your answer means that the Borel measure of the set of starting points >is 1 you're wrong. It is zero. The set of starting points is that set >such that the radius of a great-circle running E-W is the same as that >of another great-circle running E-W which is 1 mile south. The only place >this happens is the great-circle 1/2 mile north of the equator. Moving >1 mile south places you on the great-circle 1/2 mile south of the equator >and this obviously has the same radius as the original circle. >Thus, of the entire set of great circles (cardinality C) only 1 satisfies >the conditions (i.e. measure is zero) > >Postings which claim the circles 1 mile north of the equator are solutions >are wrong. This is easy to see because lines of longitude are closer >together 1 mile north of the equator than they are at the equator. Thus, >if you travel 1 mile south to the equator, 1 mile west, and then 1 mile >north you will be closer than 1 mile to your starting point. ... but real close! >At latitude 90-theta, an East-West great circle has radius 2 PI r sin(theta) >where r is the Earth's radius. Why can't people do simple high school geometry? > >Thus, it's FALSE almost everywhere. > >Bob Silverman (they call me Mr. 9) You changed my answer as well as the definition of great-circle! (The only great-circle running E-W is the equator.)
judith@proper.UUCP (Judith Abrahms) (11/02/85)
>The colour of the bear can be any one except white! > >Proof: > >- Polar bears live only near the north pole. >- There is one possible point near the north pole. >- There are an infinite number of points near the south pole. >- The chance of picking the point near the north pole is zero, > because one (number of points near north pole) > divided by infinity plus one (total number of points) is zero. ^^^^^^^^ ^^^^ ^^^ >Jacob >Soon to be elected twit of the year. And inarguably the man for the job! (:-)
jbs@mit-eddie.UUCP (Jeff Siegal) (11/03/85)
In article <541@ttrdc.UUCP> levy@ttrdc.UUCP (Daniel R. Levy) writes: >...with ordinary glass when it is heated, often resulting in >shattering [BTW, can anyone tell why it is that ordinary glass will >break when heated, but the same glass was successfully cooled into that >shape from a molten blob or sheet?].) Anyhow, some glasses, like Pyrex, >are much less subject to this since they expand much less when heated than >ordinary glass. The internal forces would constrain the "filler" piece >of material to be a slightly different shape when an integral part of the >whole than if heated by itself. Just ramblin' on... The glass doesn't shatter becuase from expansion, but rather, from _uneven_ expansion. If the glass is heated sufficiently evenly or sufficiently slowly (so the heat is conducted throughout the glass), it will not shatter. Thermal expansion really is like a photo-reducing and photo-increasing copy machine, only one that works in three dimensions. Jeff Siegal - MIT EECS
mouse@mcgill-vision.UUCP (der Mouse) (11/03/85)
[ selected lines ] > such that the radius of a great-circle running E-W is the same as that > of another great-circle running E-W which is 1 mile south. The only place > this happens is the great-circle 1/2 mile north of the equator. Moving > 1 mile south places you on the great-circle 1/2 mile south of the equator > Thus, of the entire set of great circles (cardinality C) only 1 satisfies > At latitude 90-theta, an East-West great circle has radius 2 PI r sin(theta) a great circle. This does not bring one back to the point where one started Then walking 1 mile north places one back on the original great circle, only Correct me if I'm wrong, but isn't a great circle a circle with its center at the center of the earth (yes, I know the earth isn't a sphere, but this discussion is pretending it is)? Everyone here seems to be using it to mean a circle of constant latitude. -- der Mouse {ihnp4,decvax,akgua,etc}!utcsri!mcgill-vision!mouse philabs!micomvax!musocs!mcgill-vision!mouse Hacker: One responsible for destroying / Wizard: One responsible for recovering it afterward
bs@faron.UUCP (Robert D. Silverman) (11/04/85)
> [ selected lines ] > > > such that the radius of a great-circle running E-W is the same as that > > of another great-circle running E-W which is 1 mile south. The only place > > this happens is the great-circle 1/2 mile north of the equator. Moving > > 1 mile south places you on the great-circle 1/2 mile south of the equator > > Thus, of the entire set of great circles (cardinality C) only 1 satisfies > > At latitude 90-theta, an East-West great circle has radius 2 PI r sin(theta) > a great circle. This does not bring one back to the point where one started > Then walking 1 mile north places one back on the original great circle, only > > Correct me if I'm wrong, but isn't a great circle a circle with its > center at the center of the earth (yes, I know the earth isn't a sphere, > but this discussion is pretending it is)? Everyone here seems to be > using it to mean a circle of constant latitude. > -- > der Mouse > > {ihnp4,decvax,akgua,etc}!utcsri!mcgill-vision!mouse > philabs!micomvax!musocs!mcgill-vision!mouse > > Hacker: One responsible for destroying / > Wizard: One responsible for recovering it afterward Oops!!! Bad terminology. You are indeed correct. A great circle does in fact have the center of the sphere as it's center. I should have said 'circle of constant lattitude.' The math is right but the names were wrong. I typed my response to the problem too quickly :-) Bob Silverman (they call me Mr. 9)
judith@proper.UUCP (Judith Abrahms) (11/05/85)
>I hate to clue everyone in but: ... >such that the radius of a great-circle running E-W is the same as that >of another great-circle running E-W which is 1 mile south. The only place >this happens is the great-circle 1/2 mile north of the equator. Moving >1 mile south places you on the great-circle 1/2 mile south of the equator ... > ... Why can't people do simple high school geometry? A great circle is one whose radius is the same as that of the sphere it's drawn on. The only East-West great circle on the planet is the equator. Do you mean circles of latitude? J.A.
judith@proper.UUCP (Judith Abrahms) (11/05/85)
In article <> bundy@oasys.UUCP writes: > > Judith Abrahms: a) I didn't post this stuff that follows! >> How about a more difficult sequel like the following: >> >> Where on the earth can one walk 1 mile south, 1 mile west, 1 mile >> north, AND 1 mile east, and end up at the starting point? >This sounds pretty sensitive to local geography, especially over a one >mile distance. Anyone care to post a general set of solutions for the >Himalaya alone? b) Well, my original posting did include an assumption that the planet was spherical and whatnot... I'd expect that to carry over to related puzzles, no? > ... (No flames, you did say the Earth, not some abstract >(and incorrect) mathematical representation) c) *FLAME ON* YAAAAARRRRRRGGGHHHHHHHHHHHHHHHHHHHHHH!!!!!!!!!!!!!!!! *FLAME OFF!* J.A.
hopp@nbs-amrf.UUCP (Ted Hopp) (11/07/85)
Regarding my answer to the question about holes in metal being heated, Dan Levy (article <541@ttrdc.UUCP>) wonders about my gedanken experiment of watching the "hole" when it is filled (i.e., when the hole isn't there): > Isn't this begging the question? Suppose that internal forces/stresses > developed as a result of the expansion process. (This happens, for > example, with ordinary glass when it is heated, often resulting in > shattering [BTW, can anyone tell why it is that ordinary glass will > break when heated, but the same glass was successfully cooled into that > shape from a molten blob or sheet?].) Anyhow, some glasses, like Pyrex, > are much less subject to this since they expand much less when heated than > ordinary glass. The internal forces would constrain the "filler" piece > of material to be a slightly different shape when an integral part of the > whole than if heated by itself. This is beginning to sound like it should be net.physics. In the real world, you are right. There are combinations of materials, geometries, and heating processes for which a hole will end up smaller after heating even when the total volume of the material is larger. (I.e., I'm not even allowing those wierd materials [no one is allowed to mention water at freezing] that get smaller as they heat.) The effects I am thinking about are indeed caused by internal stresses, primarily a result of the material not being in thermal equilibrium as it is heated. In the puzzle world, though, one must assume that objects and processes are in some sense "ideal". To me, this meant that heating is uniform throughout the material at all times and that the material is homogeneous and isotropic (there is no preferred direction of expansion). The condition on the material is easy to accept; the condition on the heating process is sort of implied when the material was stated to be metal (known for its high thermal conductivity). The main reason glass shatters when heated is because thermal gradients cause uneven expansion. Forces are highest where the temperature gradient is highest. Rapid cooling can be just as bad as rapid heating - try touching a hot lightbulb with a cold wet cloth. (But be wearing gloves and safety glasses! Better yet, don't do it, just take my word for it. I did it once by accident when cleaning an oven.) A dramatic example of how important the heating/cooling process can be is in how large telescope mirrors are made. After the glass is poured, it is cooled extremely slowly. It can require many months for the glass to cool to room temperature. This is done so slowly just to prevent the glass from cracking due to stresses due to thermal gradients. Anyway, a perfectly homogeneous, isotropic, infinite thermal conductivity, positive thermal coefficient material (hope I have them all) with a hole in it will have a larger hole at higher temperatures. Almost all run-of- the-mill metals (cast iron, aluminum, etc.) will show the same behavior in the real world. -- Ted Hopp {seismo,umcp-cs}!nbs-amrf!hopp
dim@whuxlm.UUCP (McCooey David I) (11/07/85)
> The colour of the bear can be any one except white! > > Proof: > > - Polar bears live only near the north pole. > > - There is one possible point near the north pole. > > - There are an infinite number of points near the south pole. > > - The chance of picking the point near the north pole is zero, > because one (number of points near north pole) > divided by infinity plus one (total number of points) is zero. > > > Jacob > > Soon to be elected twit of the year. You are forgetting one important point: The fact is, the bear IS there, where ever you are, so we must ask the following question: What kind of bear stands the best chance of surviving near the south pole? Answer: A polar bear. So the color should be white. Back to you Jacob...
shaver@isucs1.UUCP (11/11/85)
> Correct me if I'm wrong, but isn't a great circle a circle with its > center at the center of the earth [...]? [...] Everyone here seems to be > using it to mean a circle of constant latitude. {That's correct, but here is an 'offical' definition} As defined by George Abell in the fourth edition of his "Exploration of the Universe": Great Circle: "A circle on the surface of a sphere that is the curve of intersection of the sphere with a plane passing through its center" Further defined on page 103: "A great circle is any circle on the surface of a sphere whose center is at the center of the sphere. The earth's equator is a great circle on the earth's surface halfway between the North and South poles. We can also imagine a series of great circles that pass through the North and South Poles. These circles are called meridians; they intersect the equator at right angles." /\ Dave Shaver -=*=- Located at Iowa State University -- Ames, IA \/ UUCP: {okstate||umn-cs||csu-cs}!isucs1!shaver CSNET: shaver@iowa-state