tag:blogger.com,1999:blog-264226589944705290.post8050975787360092323..comments2021-12-14T05:53:12.175-08:00Comments on God Plays Dice: The napkin ring problemMichael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-264226589944705290.post-90493903759884487182010-04-10T18:24:08.300-07:002010-04-10T18:24:08.300-07:00The effects are gradual and the result will vary f...The effects are gradual and the result will vary for each user.(<a href="http://www.male-sexual.com/vimax-pills.html" rel="nofollow">vimax pills</a>) However, further review show that most users experience benefits within the first week. In the first month you can expect erections that stay longer than usual and your penile may even start to become wider. <a href="http://www.cucumaerot.com" rel="nofollow">penis enlargement pills</a> - http://www.cucumaerot.comAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-87579339217992054462008-08-18T04:25:00.000-07:002008-08-18T04:25:00.000-07:00This problem is somewhat simpler if you just compu...This problem is somewhat simpler if you just compute the volume of a h-cyclinder in a r-sphere. The answer is a function of h, but not of r.<BR/>Ref:<BR/>-- L. A. Graham, Ingenious Mathematical problems and methods, 34 p.23<BR/>-- T. M. Apostol, A Century of Calculus II, p.321<BR/>-- C. W. Trigg, Mathematical Quickies, 217 p.59<BR/>-- David Wells, The Penguin Book of Curious and Interesting Puzzles, 323 p.109<BR/>-- Martin Gardner, Hexaflexagons and other..., 7 p.113<BR/>-- Howard Eves, Great Moments in Mathematics Before 1650, p.210<BR/>-- G. Polya, Mathematics and Plausible Reasoning (Vol.I), p.191, 11.5 p.200alphachaphttps://www.blogger.com/profile/12188341952298447851noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-56851115724264426992008-04-24T08:37:00.000-07:002008-04-24T08:37:00.000-07:00This was a question in my calculus exam in the las...This was a question in my calculus exam in the last year of high-school. Nobody had been taught the necessary pre-requisites, and I don't think anyone got it right!Unknownhttps://www.blogger.com/profile/03657901505827400722noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-62140336696263966392008-04-20T06:23:00.000-07:002008-04-20T06:23:00.000-07:00My students also struggled with this problem earli...My students also struggled with this problem earlier in the semester. I haven't watched it recently (so I'm not sure how coherent it is), but I recorded <A HREF="http://www.screencast.com/t/G71EDVDHhPh" REL="nofollow">this video live</A> in class to help the online students with the problem.Maria H. Andersenhttps://www.blogger.com/profile/04686325011770339309noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-4999205445465268702008-04-16T20:54:00.000-07:002008-04-16T20:54:00.000-07:00i *just* finished teaching volumes in my calculus ...i *just* finished teaching volumes in my calculus class. guess what problem we're going to be working on tomorrow?<BR/><BR/>thanks!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-64251527968025215682008-04-16T11:33:00.000-07:002008-04-16T11:33:00.000-07:00Oh, I learned a similar thing from Martin Gardner'...Oh, I learned a similar thing from Martin Gardner's "aha!" books. The area between two concentric circles depends only on the length of the chord in the big circle that's just tangent to the small circle.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-47381681254860910922008-04-16T01:29:00.000-07:002008-04-16T01:29:00.000-07:00I believe I was surprised when I first encountered...I believe I was surprised when I first encountered this.<BR/><BR/>In an attempt to develop an intuition about the problem, I think it's interesting to imagine a sphere of clay with a tiny hollow pin through a diameter (the radius of the pin is almost zero). Next suppose that the pin has some clever special construction so that it can expand from within to a cylinder of arbitrary radius. Imagine further that the clay adheres to the pin/cylinder at both points where the pin penetrates it.<BR/><BR/>It's not too hard to imagine that as the clay "napkin ring" stretched and deformed into a "napkin ring" around the expanding cylinder, it would resemble the residue of a sphere that had stared with whatever radius the napkin ring ends up having. And of course the clay would keep its volume, due to conservation of ass and a reasonable supposition that the density shouldn't change.<BR/><BR/>Of course these kinds of appeals to imagination can be used nearly equally easily to convince your of something that's either false or true, but at least I hope it shows that it really doesn't totally defy intuition how the napkin ring problem works out, after a little meditation on it.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-37326241343725564582008-04-15T20:05:00.000-07:002008-04-15T20:05:00.000-07:00Argh, I don't have time to be playing with calculu...Argh, I don't have time to be playing with calculus right now! Shame on you and your interesting math concepts!Anonymousnoreply@blogger.com