==> pickover/pickover.06.s <== ------------------------- Cliff Pickover, So here I am, waiting to see if one of my long Grobner basis calculations is going to finish before the machine goes down. This is a good time to read news, and I came across this trivial problem in rec.games.puzzles. I'm not sure why I'm responding, perhaps the hour, or perhaps curiousity to see what will come of this, but I could have done this the day in high school when I learned how to compute cos(pi/5). The ratio between side lengths of successive pentagrams is r = (3+sqrt(5))/2 = 1 + golden ratio = 2.618... . The smallest N for which r^N > 5.48e10 (slightly more accurate value for sun's diameter in inches) is 26, with r^26 = 7.37e10. The smallest N for which 5[r^(N+1)-1]/(r-1) > 5.48e10 is 24, with 5(r^25 - 1)/(r-1) = 8.70e10. This seems too trivial to post, but do with this response as you like. Bob Holt ------------------------- I just started reading 'rec.puzzles', so have just seen this one and the one before (#5)... and to be honest I'm not sure why you put this one out, since it is pretty straightforward. >Start with a five sided star formed with 5 line segments, each 1 inch >long. Continually nest stars so that the assembly of stars gets bigger >and bigger. The analytical (and general) answer to this problem comes from the basic relationship of a "chord" of a regular pentagon, which is defined as follows: _=*=_ _=/ / \=_ _=/ | \=_ _=/ | \=_ * / * | | <-- "chord" | \ | / | / | \ | / | / | *-------------* compared to the length of one of the sides is the golden ratio, i.e. _ 1 + \/5 --------- . 2 It can then be derived that the length of the "chord" (i.e. segment length) of the next bigger Star compared to the length of the "chord" of its incribed Star is the square of the golden ratio, or the golden ratio plus one, same thing. >Questions: >1. How many nestings N are required to make star N >have an edge-length equal to the diameter of the sun (4.5E9 feet)? Back-of-envelope calculations as follows: 4.5E9 * 12 = total of 5.22E10 inches. ratio of Star sizes approx. 2.618. The best answer is 27 nested Stars, although it produces a Star with a "chord" length of 7.366E10 inches, i.e. a bit bigger. The first, and smallest Star, is assumed to be the one with "chord" length of 1 inch. >2. How many nestings N are required to make the cumulative length >of lines of all the nested stars equal to the diameter of the sun? This is just the sum of a geometric sequence with the ratio being the golden ratio squared (or the golden ratio plus one). _ / 1 + \/5 \ 2 So, S = 1 inch, and S = S | --------- | 0 n n-1 \ 2 / The sum is just the standard geometric summation, which I can't remember offhand, but the contributing terms in the sum (other than the last term), are less than one 1.6th of the total (by conincidence the inverse of the golden ratio ;-). This means that the 25th Star (term) is the determining factor, and in this case is the answer with a total length of 8.694E10 inches amoung all of them, and 5.373E10 inches for just the sum of the segments of the 25th Star (again, counting the first one as side length of 1 inch, or sum of 5 inches). Well, that's it, I guess. Sorry to be so exhaustive, but I like to use analytical methods to be sure I have the right answer. My .signature explains most of what you need to know. What I mean by "Honorary Grad Student" is that I have been taking Grad math classes since my sophomore year, and for all intensive purposes might as well be one. My Snail-mail address is 1521 S.W. 66th Ave., Portland, OR 97225. As to info about myself... I love learning about things, and mathematics and consequently computers tend to be a great focus. Anyway, if you have any more puzzles, pass them along... I am still pondering on that sequence (puzzle #5) that you posted. Thanks for your time. Erich -- "I haven't lost my mind; I know exactly where it is." / -- Erich Stefan Boleyn -- \ --=> *Mad Genius wanna-be* <=-- { Honorary Grad. Student (Math) } Internet E-mail: \ Portland State University / WARNING: INTERESTED AND EXCITABLE