Problem: Antipodal points -------- John McCarthy posed what was supposedly Martin Gardner's favorite: As of ~ 30 years ago, what single geographical name applied to two antipodal localities? Solutions: Antipodal points ---------- - "The Pole"? Surely it's got to be more sophisticated than that. - (I'd have thought georgia would work. (CAL)) - Formosa, northeasternmost Argentina and Formosa, now Taiwan. >Two place names, consecutive entries in the 1946 (e.g.) Britannica, >each nonemptily intersecting the antipodal image of the other. The >population of the smaller was about 300,000 in 1981. > >This would make a neat data-retrieval exercise, except that one of >the placenames has changed. >(Bill Gosper) - Wales! And New South Wales. Nice try, but since Wales is near the Prime Meridian (whoa, geographers think zero is prime!), NSW would have to be within 4 degrees of the International Date Line. (Latitudes differ, too.) And would the Britannica ever stick NSW under Wales, New South? You get partial credit. The answer is indeed a country and a province. (And a provincial capital.) However, if you ever heard the name of the province, you'd say, "Wow, what a coincidence." So, I'll bet you never heard of it. Problem: USA N-E-S-W-most points -------- As geographico-mathematical problems go, what are the northernmost, the southernmmost, the easternmost, the westernmost states in the US? (Trick question). (Rollo Silver) Solution: USA N-E-S-W-most points --------- Northernmost: Alaska (Pt. Barrow) Southernmost: Hawaii (South Cape) The trick is that "easternmost" and "westernmost" have no natural global definition; but if measured from the prime meridian: Westernmost: Alaska (Amatignak Island, in the Aleutians, 179 degrees W) Easternmost: Alaska (Semisopochnoi Island, ditto, 179.5 E) Problem: USA-Canada -------- Find an point on land in the USA such that if you go due North, South, East, OR West from it, the first non-USA land you get to is in Canada. (John Conway) Solution: USA-Canada --------- There are several places in northern Maine that will work. Vanceboro, for example, lies in a pocket that protrudes eastward into New Brunswick, which takes care of N,S,E. Heading west, across Maine, you reach Quebec. There is a tiny place in Minnesota, detached from the rest of it, sticking into Canada, adjacent to a lake on its immediate east. Problem: USA-path -------- If you follow the shortest path joining two points in one state, what's the maximum number of other states that you can pass through? (Dean Hickerson) Solution: USA-path --------- Go from a point in Maryland just south of where the Delaware-Maryland border reaches the Atlantic, to the middle of the western border of Maryland. You'll pass through Delaware, Virginia and West Virginia, and the District of Columbia as well. Is that the best? Dean Hickerson's answer follows, rot13'd for those who haven't given up: Vs lbh geniry va n fgenvtug yvar sebz gur abegurnfg pbeare bs Arj Lbex fgngr gb gur rnfgrea raq bs Svfuref Vfynaq (abegurnfg bs Ybat Vfynaq), lbh'yy cnff guebhtu Irezbag, Arj Unzcfuver (whfg oneryl), Irezbag ntnva, Znffnpuhfrggf, naq Pbaarpgvphg, n gbgny bs sbhe fgngrf. Vs lbh qba'g zvaq lbhe qrfgvangvba orvat va gur bprna, lbh pna vapernfr gung gb svir fgngrf: Geniry sebz gur abegurnfg pbeare bs Arj Lbex gb n cbvag ba gur fgngr yvar orgjrra Arj Lbex naq Eubqr Vfynaq, juvpu rkgraqf fbhgu naq rnfg sebz n cbvag va gur bprna orgjrra Svfuref Vfynaq naq gur fbhgujrfg pbeare bs gur ynaq cneg bs Eubqr Vfynaq. Gura, va nqqvgvba gb gur fgngrf yvfgrq nobir, lbh'yy nyfb cnff guebhtu n cneg bs gur bprna gung'f jvguva Eubqr Vfynaq. (Ng yrnfg, guvf nccrnef gb jbex ba zl znc bs gur nern. V unira'g purpxrq gur yrtny qrsvavgvbaf bs gur fgngr obhaqnevrf.) Problem: Grounded tetrahedron -------- Subject: Re: geo-metrical question From: fredh@ix.netcom.com (Fred W. Helenius) Date: Thu, 3 Oct 1996 11:27:34 -0700 Andy Latto wrote: >The novel _Earth_, by David Brin, makes the claim that it in fact is >very difficult to find a land tetrahedron, and that it can only be >done using Easter Island as one of the vertices. I can't remember what >the other three vertices are, but they are all described in the book. >When I read the book, I looked at the four sites on a globe, and the >claim that they formed a tetrahedron and that no other land >tetrahedrons existed looked at least plausible. Nope, no islands needed: 33.87 S, 151.20 E Sydney, Australia 24.80 S, 70.00 W northern Chile, S of Antofagasta 66.69 N, 151.70 W central Alaska, near Bettles Field 3.35 N, 39.92 E northeast Kenya There's enough room to play with in Alaska and East Africa that I suspect this can be adjusted to work with the real, non-spherical Earth. -- Fred W. Helenius - - - - - - - - - - - - > I was moved to wonder: What is the largest physically realized tetrahedron? At a party last year, I learned the undispued answer. Auckland, NZ; Cape Town, South Africa; Tegucigalpa, Honduras; and Irkutsk, Russia are the vertices of a regular tetrahedron inscribed in the globe. (Well, to within a degree or two.) Michael Kleber Subject: Re: grounded tetrahedron (geo-metry) From: Dan Hoey Date: Thu, 3 Oct 96 13:39:18 EDT I do see this procedure: 1. Inflate all land masses by epsilon. 2. Walk around all boundaries in steps of epsilon. 3. At each point P on the boundary walk, draw the circle C(P) that contains all base vertices of regular tetrahedrons with apex P. 4. Intersect C(P) with itself mod 2Pi/3. If nonempty, you have a tetrahedron on the inflated land. 5. Nudge the tetrahedra, if any, so they fit on the real land. The reason this works is that any tetrahedron can be extremalized into one that has a vertex on a boundary. I don't see how to make any hay out of extremalizing tetrahedra into ones with vertices on two boundaries, but it might make step 4 nicer. Dan Hoey Curiosities: ------------ The (ex-"Free") City of Konstanz is reachable by foot/car from Switzerland, but not from Germany, of which it is a part. You have to swim or take a ferry across Lake Constance (lovely, by the way!) to get there from the rest of Germany. The City of Los Angeles is multiply-connected. I don't recall how many holes it has, but it's quite a few. Beverly Hills is completely surrounded by LA, for example. (During the LA riots, a TV newsperson asked a BH cop whether they had had any looting; the cop replied "No, we don't allow that sort of thing here", without even the slightest trace of irony.) -- Henry Baker