tag:blogger.com,1999:blog-2954359812249072053.post1679756513568010152..comments2023-02-28T14:16:14.782-08:00Comments on Cary Millsap: The String PuzzleCary Millsaphttp://www.blogger.com/profile/16697498718050285274noreply@blogger.comBlogger22125tag:blogger.com,1999:blog-2954359812249072053.post-34181341594360117292016-02-26T11:25:08.782-08:002016-02-26T11:25:08.782-08:00I love this problem. Never heard of it before.
He...I love this problem. Never heard of it before.<br /><br />Here is the short answer:<br /><br /><b>π*2*r</b> = <i>C</i> (the number doesn't matter 25000)<br />-- r - radius<br />-- <i>C</i> - Circumference<br /><br />the circumference after adding 4" to the radius is<br /><br />π*2*(r + 4") =<b>π*2*r</b> + π*2*4" = <i>C</i> + π*2*4"<br /><br />so the original Circumference <i>C</i> will increase with π*2*4" which is approximately <b>25.12 inches</b>Anonymoushttps://www.blogger.com/profile/02976437275215545272noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-89307715137024356582014-08-07T07:52:44.436-07:002014-08-07T07:52:44.436-07:00about collective human intelligence and legacy... ...about collective human intelligence and legacy... this problem and solution approach reminded me of line/method of eduction followed in vedic mathematics... though I never got a chance to learn that myself even though I like in India... but this approach is very akin to what is being discussed here (the heart and soul of that)...Garyhttps://www.blogger.com/profile/13531035767823877715noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-4148834414099871392014-06-04T09:50:39.318-07:002014-06-04T09:50:39.318-07:00Gabe, I'm sorry it's taken me so long to r...Gabe, I'm sorry it's taken me so long to respond to your comment. I love it.<br /><br />—CaryCary Millsaphttps://www.blogger.com/profile/16697498718050285274noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-74802153393020520742014-04-01T12:23:58.620-07:002014-04-01T12:23:58.620-07:00To give your boys a visual solution to this proble...To give your boys a visual solution to this problem consider the following.<br /><br />Start with a string with the ends tied up and shaped as a circle. The length of the string is 2 * π * Radius.<br /><br />Twist the string to create a perfect 8 shape. You get two smaller, equal, tangential circles and it is easy to figure out that the radius for each of these two circles is half the size of the radius for the original circle.<br /><br />Twist the string to create an imperfect 8 shape. You get two smaller, tangential circles and it is easy to figure out that the sum of radiuses for these two circles is equal to the radius of the original circle.<br /><br />Now that one has the visual image of an imperfect 8 shape (that is, two tangential circles ... say smaller one at the top and big one at the bottom) consider the formula for calculating the "perimeter" of the 8 shape:<br /><br />2 * π * R_top + 2 * π * R_bottom = 2 * π * (R_top + R_bottom)<br /><br /><br />But 2 * π * (R_top + R_bottom) is also the formula for calculating the circumference of a circle having the centre in the centre of the bottom circle in the 8 shape and a radius going through the tangential point to the centre of the top circle in the 8 shape.<br /><br />Expressing this differently ... for any two concentric circles, the difference between the circumference of the outer one and the circumference of the inner one is equal to the circumference of a circle having the centre on any point on the outer circle and tangential to the inner circle.<br /><br />In the image of the globe you have in your article consider having a small circle, right at the top, tangential to the earth circle and centered on the "suspended" circle. The sum of the circumferences for the two smaller circles (making up the 8 shape) is equal to the circumference of the "suspended" circle.<br /><br />Doing the math in reverse … for any two concentric circles, the difference between the circumference of the outer one and the circumference of the inner one is equal to:<br /><br />2 * π * (R_outer - R_inner)<br /><br />For your particular case: 2 * π * 4 inches.<br /><br />Gabe?https://www.blogger.com/profile/16363120205602353079noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-32080492813271927072013-02-25T12:26:49.205-08:002013-02-25T12:26:49.205-08:00Rahul,
Neither boy has taken a lot of interest in...Rahul,<br /><br />Neither boy has taken a lot of interest in the puzzle since I posted this. I think they have too much money already to be interested for bribery's sake. :-)<br /><br />Now that you've asked, my older boy is trying to save money for something he wants. I'll remind him that this offer still stands.<br /><br />CaryCary Millsaphttps://www.blogger.com/profile/16697498718050285274noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-44263167776912973382013-02-20T15:56:02.858-08:002013-02-20T15:56:02.858-08:00Cary,
Did your second one solve the puzzle before...Cary,<br /><br />Did your second one solve the puzzle before the first?Anonymoushttps://www.blogger.com/profile/06918188219934919942noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-38605535613899781802012-07-04T18:42:50.577-07:002012-07-04T18:42:50.577-07:00Thanks Cary for posting this problem. It was a lot...Thanks Cary for posting this problem. It was a lot of fun, and I had NO idea that the answer would end up looking like that! I posted it to my Facebook page (referencing your blog of course) to ask my friends to solve it too. :) <br /><br />Cheers,<br /><br />JeremyJeremyhttps://www.blogger.com/profile/01707263788648681366noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-54945120063150406542012-07-04T18:41:59.132-07:002012-07-04T18:41:59.132-07:00This comment has been removed by the author.Jeremyhttps://www.blogger.com/profile/01707263788648681366noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-54606833343079200422012-05-17T08:03:56.554-07:002012-05-17T08:03:56.554-07:00Andrew,
Certainly that would be ok.
—CaryAndrew,<br /><br /><i>Certainly</i> that would be ok.<br /><br />—CaryCary Millsaphttps://www.blogger.com/profile/16697498718050285274noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-79720716568562761922012-05-15T15:36:40.109-07:002012-05-15T15:36:40.109-07:00Cary,
My Dad is a Calculus/Math professor ... He l...Cary,<br />My Dad is a Calculus/Math professor ... He loved it! <br />Would it be ok... if he uses this example in his class... <br />He proposed a similiar algebriac challenge to his students... <br /><br />His response : <br /><br />Assumptions and Variables<br /> <br />Perfect Circle with an exact perimeter = 25,000 miles<br /> <br />Then a second circle with a radius 4 inches longer than the first<br /> <br />Radius original = 25,000 divided by 2*pi gives the original radius in miles. This result hast o be multiplied by 5280 * 12 to convert to inches<br /> <br />25,000 * 5289 *12 / (2^pi). Lets call this number N. The length of the second (slightly longer circumfrance is given b y<br /> <br />(Old radius +4)* 2*pi is the total length <br /> <br />The incremental length would be the New minus Old<br /> <br />Since OLD is just a big number compared to the contribution of the additional radius, some simplications can be applied to prevent<br />errors related to subtracting two large and close to one another errors from creeping in<br /> <br /><br />Dadasliwxhttps://www.blogger.com/profile/09822655014150263006noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-7255965407919795172012-04-23T10:47:59.440-07:002012-04-23T10:47:59.440-07:00Cary
Here is another interesting problem. With the...Cary<br />Here is another interesting problem. With the numbers 1,2 and 3 used only once, get to 19 using the arithetic operators, sqrt and factorial.<br />MaheshMaheshhttps://www.blogger.com/profile/03984773849590215351noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-38033551699112860792012-04-16T13:00:00.309-07:002012-04-16T13:00:00.309-07:00*sigh* Well, I just discovered Vi Hart is a female...*sigh* Well, I just discovered Vi Hart is a female and not just a site LOL<br /><br />The 5th link down on Google's search results...Chrishttps://www.blogger.com/profile/13394883098550592871noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-16803080937106808542012-04-16T12:53:49.728-07:002012-04-16T12:53:49.728-07:00Cary - I had to go lookup Vi Hart. I had never he...Cary - I had to go lookup Vi Hart. I had never heard of it :)<br /><br />Interesting site. <br /><br />I was just remembering about the "Tau" day versus "Pi" day and the relationship of Tau to Pi and the media spin on it a while back...Chrishttps://www.blogger.com/profile/13394883098550592871noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-78271007691439603702012-04-16T09:37:30.666-07:002012-04-16T09:37:30.666-07:00Chris, I'm with you. I'm guessing that vir...Chris, I'm with you. I'm guessing that virtually anyone who reads my blog probably has a crush on Vi Hart. :-)Cary Millsaphttps://www.blogger.com/profile/16697498718050285274noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-88069829430886700282012-04-16T08:49:18.491-07:002012-04-16T08:49:18.491-07:00C'mon Cary, Everyone knows 2π is really τ [tau...C'mon Cary, Everyone knows 2π is really τ [tau] so C=τr (C=tau*r)...<br /><br />Get with the times ;)Chrishttps://www.blogger.com/profile/13394883098550592871noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-23248722705662922072012-04-04T08:44:16.140-07:002012-04-04T08:44:16.140-07:00Nice to see that you won't stoop to product pl...Nice to see that you won't stoop to product placement Mr Millsap.Adrianhttps://www.blogger.com/profile/00943497054353493755noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-80179230120777258982012-04-01T13:02:58.881-07:002012-04-01T13:02:58.881-07:00Excellent!
Wonder if it holds for other kinds of ...Excellent!<br /><br />Wonder if it holds for other kinds of shapes?<br /><br />http://www.oraclemusings.com/?p=168Dominic Delmolinohttps://www.blogger.com/profile/04315832976299102261noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-86571268355993304542012-03-31T08:55:35.497-07:002012-03-31T08:55:35.497-07:00Hello Cary,
I think that there a big change how m...Hello Cary,<br /><br />I think that there a big change how math is taught at school. I remember that during my primary school years in Poland I have learned that math is not only numbers but it include formulas with all transformations plus basic algebra. Now when I work with my sons (12 and 14 years as well) in Ireland I can see that math is now about numbers only and I think that this is wrong approach as they can't solve problems described in abstract way like this "One brick is the one kilogram plus half a brick heavy What is the weight of one brick?"<br /><br />Anyway I like your ending conclusion. I think that prove in IT is so important like prove in math.<br /><br />regards,<br />MarcinMarcin Przepiorowskihttps://www.blogger.com/profile/15133397892511680504noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-28854219212045484112012-03-31T07:38:49.690-07:002012-03-31T07:38:49.690-07:00Hello, Cary!
It seems to me that your personal co...Hello, Cary!<br /><br />It seems to me that your personal conclusions are more profound than the task by itself. This fact is very interesting that the difference between two lenghts of the strings disposed at the constant distance is the same. Let me note that it is true not only for spherical bodies but for every bodies with the similar type of its shape...<br /><br />The main conclusion to my mind is the advantage of taking the final formula than numerical calculations.<br /><br />So, if most programmers were powerful mathematicians, our (in the whole world) programms (including Oracle) would be much better. Do you agree with me?Vladimir Whttps://www.blogger.com/profile/08156880103830307212noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-16763371517019391792012-03-31T01:10:09.666-07:002012-03-31T01:10:09.666-07:00Cary, I think you should pay up! If you hadn't...Cary, I think you should pay up! If you hadn't confused him by giving him an actual value of 25000 for the diameter he would have got the answerFergal Tahenyhttps://www.blogger.com/profile/17178161424593516169noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-10175248496293308662012-03-30T12:26:14.516-07:002012-03-30T12:26:14.516-07:00My kids (ages 12 and 14) are both doing interestin...My kids (ages 12 and 14) are both doing interesting problems in school right now. "A rectangular prism has volume 42 cubic feet. What will be the volume of the prism if you multiply its side lengths by a factor of 1/2?" ...Stuff like that.<br /><br />They're doing perimeters (with linear relationships, exactly as in your problem) areas (quadratic), and volumes (cubic). The problem is, though, that they're just following a mechanical process that their teacher has shown them. (If perimeter, do thing A; else, if area do thing B; else, if volume do thing C; otherwise, <i>run</i>!)<br /><br />I like for them to see <i>why</i> there's a cubic relationship when there's a volume involved: because when you scale by some factor <i>f</i>, your V = L × H × W becomes V′ = fL × fH × fW, which is V′ = (f^3)(L × H × W) = (f^3)V.<br /><br />I feel like the only way they'll actually be able to remember this stuff and apply it to their lives is to <i>see why</i> it works the way it does.Cary Millsaphttps://www.blogger.com/profile/16697498718050285274noreply@blogger.comtag:blogger.com,1999:blog-2954359812249072053.post-36798259960840855122012-03-30T11:58:48.736-07:002012-03-30T11:58:48.736-07:00I love this problem!
I was asked the inverse ques...I love this problem!<br /><br />I was asked the inverse question (if you add x to the length how much will the string raise off the earth) years ago by physics professor. The method to arrive at the answer is deceptively simple and the actual value for the answer defies intuition. I have asked at least 100 people (mostly engineering and comp sci majors) this question over the last 25 or so years and only 2 or 3 arrived at the correct answer…most gave up before arriving at an answer. Some spent hours and filled up pages of paper before giving up.<br /><br />With a basic understanding of differentials, the solution is even easier. The relationship between circumference and radius is linear (no squared or higher order terms). As a result, the change in circumference is directly related to the change in radius (in this case 4 inches) multiplied by 2pi.<br /><br />This is shown below. In mathematical notation, a delta symbol (triangle) would represent the change.<br /><br />C=2*pi*r<br />(change in)C=2*pi*(change in)r<br />(change in)C=6.28*4=25.12 <br /><br />Most people take the same approach that your son did…they make an assumption about the size of the earth and start from there. As you noted, the original circumference doesn’t matter. I always give a hint that the answer is the same whether the earth, a basketball or a golf ball is used for the problem. Even with that hint, most people do not arrive at the correct answer.Unknownhttps://www.blogger.com/profile/18196966317127611134noreply@blogger.com