October 2010 - mathfun@mailman.xmission.com

Re: [math-fun] Elementary geometry puzzle
by Dan Asimov 17 Oct '10

17 Oct '10

OK, so "the latter" means the arcs as it should, but then "the line segment" means its endpoints, and "it" suddenly refers back to the circle. O...kay. Then indeed it is a very elementary geometry puzzle. --Dan ----------- Mike wrote: << I think they're meant to be arcs centered on the endpoints, i.e. the point where they intersect at the top together with the endpoints of the segment form an equilateral triangle. On Sun, Oct 17, 2010 at 11:41 AM, Dan Asimov <dasimov(a)earthlink.net> wrote: > The wording seems to have gone awry: > > << > a small circle surrounded by a pair of larger arcs tangent > to it, the latter centred on a line segment also tangent to it >>> > > since "the latter" must refer to the circle, which is the > former, not the latter. Assuming this is what's intended: > > Seems to me the inner circle C can have any radius r, > and there will always be a corresponding picture > satisfying the stated conditions. > > Then as long as the two arcs (convex outward) ending > at the segment's endpoints have a (common) radius > > R > max{1,r}, > > they can be swung around from outside C until > becoming tangent to it -- and then extended > to meet at the perpendicular bisector of the > segment. > > So there appear to be two degrees of freedom here. > > --Dan > ----------- > > Fred wrote: > > << > Chris Maslanka (Guardian newspaper, Saturday October 16th 2010) > diagrams the following bilaterally symmetric architectural decoration, > comprising a small circle surrounded by a pair of larger arcs tangent > to it, the latter centred on a line segment also tangent to it; and asks, > given the segment has unit length, what radius has the circle? > > > @ > @ @ > @ * * @ > @ * * @ > @ * * @ > @ * * @ > @ * * @ > @ * * @ > @ * * @ > @----------*-*----------@ >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Elementary geometry puzzle
by Dan Asimov 17 Oct '10

17 Oct '10

The wording seems to have gone awry: << a small circle surrounded by a pair of larger arcs tangent to it, the latter centred on a line segment also tangent to it >> since "the latter" must refer to the circle, which is the former, not the latter. Assuming this is what's intended: Seems to me the inner circle C can have any radius r, and there will always be a corresponding picture satisfying the stated conditions. Then as long as the two arcs (convex outward) ending at the segment's endpoints have a (common) radius R > max{1,r}, they can be swung around from outside C until becoming tangent to it -- and then extended to meet at the perpendicular bisector of the segment. So there appear to be two degrees of freedom here. --Dan ----------- Fred wrote: << Chris Maslanka (Guardian newspaper, Saturday October 16th 2010) diagrams the following bilaterally symmetric architectural decoration, comprising a small circle surrounded by a pair of larger arcs tangent to it, the latter centred on a line segment also tangent to it; and asks, given the segment has unit length, what radius has the circle? @ @ @ @ * * @ @ * * @ @ * * @ @ * * @ @ * * @ @ * * @ @ * * @ @----------*-*----------@ >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Elementary geometry puzzle
by Fred lunnon 17 Oct '10

17 Oct '10

Chris Maslanka (Guardian newspaper, Saturday October 16th 2010) diagrams the following bilaterally symmetric architectural decoration, comprising a small circle surrounded by a pair of larger arcs tangent to it, the latter centred on a line segment also tangent to it; and asks, given the segment has unit length, what radius has the circle? Fred Lunnon @ @ @ @ * * @ @ * * @ @ * * @ @ * * @ @ * * @ @ * * @ @ * * @ @----------*-*----------@

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Benoît Mandelbrot, Novel Mathematician, Dies at 85
by Henry Baker 17 Oct '10

17 Oct '10

FYI -- I really enjoyed Mandelbrot's books & learned a lot from them. http://www.nytimes.com/2010/10/17/us/17mandelbrot.html October 16, 2010 BenoÃ®t Mandelbrot, Novel Mathematician, Dies at 85 By JASCHA HOFFMAN BenoÃ®t B. Mandelbrot, a maverick mathematician who developed an innovative theory of roughness and applied it to physics, biology, finance and many other fields, died on Thursday in Cambridge, Mass. He was 85. His death was caused by pancreatic cancer, his wife, Aliette, said. He had lived in Cambridge. Dr. Mandelbrot coined the term ÂfractalÂ to refer to a new class of mathematical shapes whose uneven contours could mimic the irregularities found in nature. ÂApplied mathematics had been concentrating for a century on phenomena which were smooth, but many things were not like that: the more you blew them up with a microscope the more complexity you found,Â said David Mumford, a professor of mathematics at Brown University. ÂHe was one of the primary people who realized these were legitimate objects of study.Â In a seminal book, ÂThe Fractal Geometry of Nature,Â published in 1982, Dr. Mandelbrot defended mathematical objects that he said others had dismissed as ÂmonstrousÂ and Âpathological.Â Using fractal geometry, he argued, the complex outlines of clouds and coastlines, once considered unmeasurable, could now Âbe approached in rigorous and vigorous quantitative fashion.Â For most of his career, Dr. Mandelbrot had a reputation as an outsider to the mathematical establishment. From his perch as a researcher for I.B.M. in New York, where he worked for decades before accepting a position at Yale University, he noticed patterns that other researchers may have overlooked in their own data, then often swooped in to collaborate. ÂHe knew everybody, with interests going off in every possible direction,Â Professor Mumford said. ÂEvery time he gave a talk, it was about something different.Â Dr. Mandelbrot traced his work on fractals to a question he first encountered as a young researcher: how long is the coast of Britain? The answer, he was surprised to discover, depends on how closely one looks. On a map an island may appear smooth, but zooming in will reveal jagged edges that add up to a longer coast. Zooming in further will reveal even more coastline. ÂHere is a question, a staple of grade-school geometry that, if you think about it, is impossible,Â Dr. Mandelbrot told The New York Times earlier this year in an interview. ÂThe length of the coastline, in a sense, is infinite.Â In the 1950s, Dr. Mandelbrot proposed a simple but radical way to quantify the crookedness of such an object by assigning it a Âfractal dimension,Â an insight that has proved useful well beyond the field of cartography. Over nearly seven decades, working with dozens of scientists, Dr. Mandelbrot contributed to the fields of geology, medicine, cosmology and engineering. He used the geometry of fractals to explain how galaxies cluster, how wheat prices change over time and how mammalian brains fold as they grow, among other phenomena. His influence has also been felt within the field of geometry, where he was one of the first to use computer graphics to study mathematical objects like the Mandelbrot set, which was named in his honor. ÂI decided to go into fields where mathematicians would never go because the problems were badly stated,Â Dr. Mandelbrot said. ÂI have played a strange role that none of my students dare to take.Â BenoÃ®t B. Mandelbrot (he added the middle initial himself, though it does not stand for a middle name) was born on Nov. 20, 1924, to a Lithuanian Jewish family in Warsaw. In 1936 his family fled the Nazis, first to Paris and then to the south of France, where he tended horses and fixed tools. After the war he enrolled in the Ãcole Polytechnique in Paris, where his sharp eye compensated for a lack of conventional education. His career soon spanned the Atlantic. He earned a masterÂs degree in aeronautics at the California Institute of Technology, returned to Paris for his doctorate in mathematics in 1952, then went on to the Institute for Advanced Study in Princeton, N.J., for a postdoctoral degree under the mathematician John von Neumann. After several years spent largely at the Centre National de la Recherche Scientifique in Paris, Dr. Mandelbrot was hired by I.B.M. in 1958 to work at the Thomas J. Watson Research Center in Yorktown Heights, N.Y. Although he worked frequently with academic researchers and served as a visiting professor at Harvard and the Massachusetts Institute of Technology, it was not until 1987 that he began to teach at Yale, where he earned tenure in 1999. Dr. Mandelbrot received more than 15 honorary doctorates and served on the board of many scientific journals, as well as the Mandelbrot Foundation for Fractals. Instead of rigorously proving his insights in each field, he said he preferred to Âstimulate the field by making bold and crazy conjecturesÂ Â and then move on before his claims had been verified. This habit earned him some skepticism in mathematical circles. ÂHe doesnÂt spend months or years proving what he has observed,Â said Heinz-Otto Peitgen, a professor of mathematics and biomedical sciences at the University of Bremen. And for that, he said, Dr. Mandelbrot Âhas received quite a bit of criticism.Â ÂBut if we talk about impact inside mathematics, and applications in the sciences,Â Professor Peitgen said, Âhe is one of the most important figures of the last 50 years.Â Besides his wife, Dr. Mandelbrot is survived by two sons, Laurent, of Paris, and Didier, of Newton, Mass., and three grandchildren. When asked to look back on his career, Dr. Mandelbrot compared his own trajectory to the rough outlines of clouds and coastlines that drew him into the study of fractals in the 1950s. ÂIf you take the beginning and the end, I have had a conventional career,Â he said, referring to his prestigious appointments in Paris and at Yale. ÂBut it was not a straight line between the beginning and the end. It was a very crooked line.Â

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Re: [math-fun] [seqfan] Swapping knights
by Richard Guy 17 Oct '10

17 Oct '10

I'm sorry that I didn't make it clear that the problem was to keep the order of the knights, i.e., White knight Si is to swap places with Black knight si for i = 1, 2, ..., n. R. On Sun, 17 Oct 2010, Lior Manor wrote: > Hi, > Maybe I got it wrong, but it seems that the following is an 8 steps solution > for n=3: > > s s s s s S s s S s S s S s S S S S S S S S S s s > s S s s s s S S S S S S S S S s S S s S s S s s > s s s s s > > Am I missing something? > > Lior > > On Fri, Oct 15, 2010 at 21:53, Richard Guy <rkg(a)cpsc.ucalgary.ca> wrote: > I don't believe that the following sequence is in OEIS. > > The (least) number of moves needed to swap > n pairs of knights on a 3 by n chessboard. > > I.e., the number of chess knight moves needed > to get from > > S S S ... S s s s ... s > - - - - to - - - - > s s s ... s S S S S > > where s, S are chess knights of opposite > colors. Not possible for n = 1, 2, 3. > > n = 4: 32 moves are necessary and sufficient. > > n = 5, 6, 7, ... are Rikki-Tikki-Tavi questions > on p.250 (R187) of The Inquisitive Problem Solver. > > R. > > > _______________________________________________ > > Seqfan Mailing list - http://list.seqfan.eu/ > > > > > -- > Lior > --------------------------------------------------------------------------- > ----------------------- > “Crime doesn’t pay directly; it goes through escrow”, Dogbert > >

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[math-fun] Flipping 4 Penney's game sequences
by rcs＠xmission.com 16 Oct '10

16 Oct '10

This is from Ed Pegg. --Rich -------------- I managed to calculate sequences for all of the 4 flip games, in Penney's game. http://en.wikipedia.org/wiki/Penney's_game The count of ways HHHH beats HHHT on move 4, 5, 6, 7 ... leads to the sequence 1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136,5768,10609,19513 This is A000073 , the Tribonacci numbers. In the pasted list below, all the various types of 4-flip games are listed along with the odds of winning, a generating function or recurrence equation, and 30 terms of the sequence. If anyone would like it as a nicely formatted text file, write to me. Outlier: HHTT beats HTHT seems excessively complicated. HHHH beats TTTH has a chance of winning of 7/22, or about 1/Pi. It's A123908. --Ed Pegg Jr { A000012 {HHHT beats THHH,1/8, (1-n)^(-1), {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}}, A000071 {HHHT beats TTTH,1/2, n/(1-2n+n^3), {1,2,4,7,12,20,33,54,88,143,232,376,609,986,1596,2583,4180,6764,10945,17710,28656,46367,75024,121392,196417,317810,514228,832039,1346268,2178308,3524577,5702886,9227464,14930351}}, A000073 {HTTT beats TTTH,7/8, (1-n(1+n+n^2))^(-1), {1,2,4,7,13,24,44,81,149,274,504,927,1705,3136,5768,10609,19513,35890,66012,121415,223317,410744,755476,1389537,2555757,4700770,8646064,15902591,29249425,53798080,98950096}}, A000073 {HHHH beats HHHT,1/2, -(n/(-1+n+n^2+n^3)), {1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136,5768,10609,19513,35890,66012,121415,223317,410744,755476,1389537,2555757,4700770,8646064,15902591,29249425,53798080,98950096}}, A003476 {HTHT beats THHH,7/16, (1+n+2n^2)/(2-2n-4n^3), {1,2,3,5,9,15,25,43,73,123,209,355,601,1019,1729,2931,4969,8427,14289,24227,41081,69659,118113,200275,339593,575819,976369,1655555,2807193,4759931,8071041,13685427,23205289}}, A003479 {HTHT beats HTTT,1/2, n/((-1+n)(-1+n+2n^3)), {1,2,3,6,11,18,31,54,91,154,263,446,755,1282,2175,3686,6251,10602,17975,30478,51683,87634,148591,251958,427227,724410,1228327,2082782,3531603,5988258,10153823,17217030,29193547}}, A005251 {HHTH beats TTTT,5/8, (1+(-1+n)n)/(1+n(-2+n-n^2)), {1,2,4,7,12,21,37,65,114,200,351,616,1081,1897,3329,5842,10252,17991,31572,55405,97229,170625,299426,525456,922111,1618192,2839729,4983377,8745217,15346786,26931732,47261895}}, A005314 {HHHT beats HTHH,1/2, n/(1+n(-2+n-n^2)), {1,2,3,5,9,16,28,49,86,151,265,465,816,1432,2513,4410,7739,13581,23833,41824,73396,128801,226030,396655,696081,1221537,2143648,3761840,6601569,11584946,20330163,35676949}}, A078044 {HTHT beats HHHT,3/8, -((n(1+n))/(-1+n+2n^3)), {1,2,2,4,8,12,20,36,60,100,172,292,492,836,1420,2404,4076,6916,11724,19876,33708,57156,96908,164324,278636,472452,801100,1358372,2303276,3905476,6622220,11228772,19039724}}, A005251 {HHHH beats THHT,1/4, {-a(n)+a(n-1)+a(n-2)+a(n-4)=0, a(1)=1,a(2)=1,a(3)= 2,a(4)=4}, {1,1,2,4,7,12,21,37,65,114,200,351,616,1081,1897,3329,5842,10252,17991,31572,55405,97229,170625,299426,525456,922111,1618192,2839729,4983377,8745217,15346786,26931732}}, A014743 {HHTH beats HHHH,3/5, {-a(n)-a(1+n)-a(3+n)-a(4+n)+a(5+n)=0, a(1)=1,a(2)=2,a(3)=3,a(4)=5,a(5)=9}, {1,2,3,5,9,17,31,56,101,183,332,602,1091,1977,3583,6494,11770,21332,38662,70071,126997,230170,417161,756064,1370293,2483524,4501148,8157897,14785402,26797116,48567190}}, A077930 {HHTH beats HHHT,1/3, {-a(n)-a(2+n)-a(3+n)+a(4+n)=0, a(1)=1,a(2)=1,a(3)=2,a(4)=3}, {1,1,2,3,6,10,18,31,55,96,169,296,520,912,1601,2809,4930,8651,15182,26642,46754,82047,143983,252672,443409,778128,1365520,2396320,4205249,7379697,12950466,22726483,39882198}}, A079959 {HHHH beats HTTH,4/11, {-a(n)-a(2+n)-a(4+n)-a(5+n)+a(6+n)=0, a(1)=1,a(2)=1,a(3)=2,a(4)=3,a(5)=6,a(6)=10}, {1,1,2,3,6,10,19,33,60,106,191,340,610,1089,1950,3485,6236,11150,19946,35670,63802,114107,204091,365018,652857,1167652,2088402,3735179,6680529,11948378,21370166,38221375}}, A079976 {HHHH beats HHTH,2/5, {-a(n)-a(1+n)-a(3+n)-a(4+n)+a(5+n)=0, a(1)=1,a(2)=1,a(3)=2,a(4)=3,a(5)=6}, {1,1,2,3,6,11,20,36,65,118,214,388,703,1274,2309,4185,7585,13747,24915,45156,81841,148329,268832,487232,883061,1600463,2900685,5257212,9528190,17268926,31298264,56725087}}, A116732 {HHTT beats HHHT,1/3, { a(n)-a(1+n)-a(2+n)-a(3+n)+a(4+n)=0, a(1)=1,a(2)=1,a(3)=2,a(4)=4}, {1,1,2,4,6,11,19,32,56,96,165,285,490,844,1454,2503,4311,7424,12784,22016,37913,65289,112434,193620,333430,574195,988811,1702816,2932392,5049824,8696221,14975621,25789274}}, A123908 {HHHH beats TTTH,7/22, {-a(n)-2a(1+n)-2a(2+n)-a(3+n)+a(5+n)=0, a(1)=1,a(2)=1,a(3)=2,a(4)=3,a(5)=6}, {1,1,2,3,6,10,17,30,52,90,156,271,470,815,1414,2453,4255,7381,12804,22211,38529,66836,115940,201120,348881,605201,1049837,1821143,3159121,5480100,9506282,16490465,28605867}}, A128588 {HHTH beats TTTH,7/16, {-a(n)-a(1+n)+a(2+n)=0, a(1)=1,a(2)=2,a(3)=4}, {1,2,4,6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194,5168,8362,13530,21892,35422,57314,92736,150050,242786,392836,635622,1028458,1664080,2692538,4356618,7049156,11405774}}, {HHTT beats TTTH,7/12, -((n(1+n+n^2))/(-1+n+n^2+n^3-n^4)), {1,2,4,7,12,21,36,62,107,184,317,546,940,1619,2788,4801,8268,14238,24519,42224,72713,125218,215636,371343,639484,1101245,1896436,3265822,5624019,9685032,16678437,28721666}}, {HTHH beats THHH,7/12, -((n(1+n+n^2))/(-1+n+n^2+n^4)), {1,2,4,6,11,19,34,59,104,182,320,561,985,1728,3033,5322,9340,16390,28763,50475,88578,155443,272784,478702,840064,1474209,2587057,4539968,7967089,13981266,24535412,43056646}}, {HHHT beats THTH,9/16, (1+n^2)/(2(-1+n)(-1+n+2n^3)), {1,2,4,7,12,21,36,61,104,177,300,509,864,1465,2484,4213,7144,12113,20540,34829,59056,100137,169796,287909,488184,827777,1403596,2379965,4035520,6842713,11602644,19673685,33359112}}, {HTTT beats HHHT,3/7, (n(1+n))/((-1+n)(-1+n+n^2)(1+n+n^2)), {1,2,3,6,10,16,27,44,71,116,188,304,493,798,1291,2090,3382,5472,8855,14328,23183,37512,60696,98208,158905,257114,416019,673134,1089154,1762288,2851443,4613732,7465175}}, {HHHT beats HTHT,5/8, (n+n^3)/((-1+n)(-1+n+2n^3)), {1,2,4,8,14,24,42,72,122,208,354,600,1018,1728,2930,4968,8426,14288,24226,41080,69658,118112,200274,339592,575818,976368,1655554,2807192,4759930,8071040,13685426,23205288}}, {HTTT beats TTHT,7/12, (n-n^3+n^4)/(1-2n+n^3-n^4+n^5), {1,2,3,6,11,20,35,62,109,192,337,592,1039,1824,3201,5618,9859,17302,30363,53284,93507,164094,287965,505344,886817,1556256,2731039,4792640,8410497,14759394,25900931,45452966}}, {HHHT beats HHTT,2/3, n/(1-2n+2n^4-n^5), {1,2,4,8,14,25,44,76,132,228,393,678,1168,2012,3466,5969,10280,17704,30488,52504,90417,155706,268140,461760,795190,1369385,2358196,4061012,6993404,12043228,20739449,35715070}}, {HHHT beats HHTH,2/3, n/(1-2n+n^3-n^4+n^5), {1,2,4,7,13,23,41,72,127,223,392,688,1208,2120,3721,6530,11460,20111,35293,61935,108689,190736,334719,587391,1030800,1808928,3174448,5570768,9776017,17155714,30106180,52832663}}, {HHTH beats THHH,5/12, -((n(-1+n^2+n^3))/(1-2n+n^3-n^4+n^5)), {1,2,3,4,7,12,21,36,63,110,193,338,593,1040,1825,3202,5619,9860,17303,30364,53285,93508,164095,287966,505345,886818,1556257,2731040,4792641,8410498,14759395,25900932,45452967}}, {HHHH beats TTHH,1/4, { a(n)-a(2+n)-a(3+n)-a(4+n)+a(5+n)=0, a(1)=1,a(2)=1,a(3)=1,a(4)=2,a(5)=4}, {1,1,1,2,4,6,11,20,35,62,111,197,350,623,1108,1970,3504,6232,11083,19711,35056,62346,110881,197200,350716,623741,1109311,1972887,3508739,6240221,11098106,19737755,35103195}}, {HHTT beats HHHH,3/5, { a(n)-a(2+n)-a(3+n)-a(4+n)+a(5+n)=0, a(1)=1,a(2)=2,a(3)=3,a(4)=6,a(5)=11}, {1,2,3,6,11,19,34,61,108,192,342,608,1081,1923,3420,6082,10817,19238,34214,60849,108219,192465,342295,608765,1082676,1925517,3424493,6090391,10831636,19263844,34260354,60931341}}, {HHHH beats HHTT,2/5, { a(n)-a(2+n)-a(3+n)-a(4+n)+a(5+n)=0, a(1)=1,a(2)=1,a(3)=2,a(4)=4,a(5)=7}, {1,1,2,4,7,12,22,39,69,123,219,389,692,1231,2189,3893,6924,12314,21900,38949,69270,123195,219100,389665,693011,1232506,2191987,3898404,6933232,12330612,21929742,39001599,69363549}}, {HHTT beats TTTT,3/4, { a(n)-a(2+n)-a(3+n)-a(4+n)+a(5+n)=0, a(1)=1,a(2)=2,a(3)=4,a(4)=8,a(5)=14}, {1,2,4,8,14,25,45,80,142,253,450,800,1423,2531,4501,8005,14237,25320,45031,80087,142433,253314,450514,801230,1424971,2534282,4507169,8015908,14256129,25354235,45091990,80195185}}, {HTHT beats HHTT,4/9, {(-5-n)a(n)+(-3-n)a(2+n)+(2+n)a(3+n)=0, a(1)=1,a(2)=2,a(3)=3}, {1,2,3,6,11,18,30,50,81,130,208,330,520,816,1275,1984,3077,4758,7337,11286,17322,26532,40563,61908,94336,143540,218112,331008,501749,759726,1149159,1736534,2621751,3954826,5960902}}, {HHHT beats TTHH,5/12, {1+a(n)-a(1+n)-a(2+n)-a(3+n)+a(4+n)=0, a(1)=1,a(2)=2,a(3)=3,a(4)=5}, {1,2,3,5,8,13,22,37,63,108,185,318,547,941,1620,2789,4802,8269,14239,24520,42225,72714,125219,215637,371344,639485,1101246,1896437,3265823,5624020,9685033,16678438,28721667}}, {HHHH beats HTTT,4/11, {-1-a(n)-2a(1+n)-2a(2+n)-a(3+n)+a(5+n)=0,a(1)=1,a(2)=1,a(3)=2,a(4)=4,a(5)=7}, {1,1,2,4,7,12,21,37,64,111,193,335,581,1008,1749,3034,5263,9130,15838,27474,47659,82674,143414,248779,431555,748615,1298616,2252698,3907736,6778716,11758980,20398201,35384583}}, {HHTT beats THHT,5/12, {-1-a(n)-a(2+n)-a(3+n)+a(4+n)=0, a(1)=1,a(2)=1,a(3)=2,a(4)=4}, {1,1,2,4,8,14,25,44,78,137,241,423,743,1304,2289,4017,7050,12372,21712,38102,66865,117340,205918,361361,634145,1112847,1952911,3427120,6014177,10554145,18521234,32502500,57037912}}, {HHHT beats HTTH,4/7, {-1-a(n)-a(3+n)-a(4+n)+a(5+n)=0, a(1)=1,a(2)=2,a(3)=4,a(4)=7,a(5)=12}, {1,2,4,7,12,21,36,62,106,181,309,527,899,1533,2614,4457,7599,12956,22089,37660,64207,109467,186631,318188,542480,924876,1576824,2688332,4583345,7814158,13322380,22713363,38724076}}, {HTTT beats THHT,1/2, {-1-a(n)-a(3+n)-a(4+n)+a(5+n)=0, a(1)=1,a(2)=2,a(3)=3,a(4)=6,a(5)=10}, {1,2,3,6,10,18,31,53,91,155,265,452,771,1315,2242,3823,6518,11113,18947,32303,55074,93896,160084,272928,465316,793319,1352532,2305936,3931397,6702650,11427367,19482550,33215854}}, {HHTT beats HTHT,5/9, {-2a(n)+(-4-n)a(1+n)-2a(2+n)-na(3+n)+(1+n)a(4+n)=0, a(1)=1,a(2)=2,a(3)=4,a(4)=8}, {1,2,4,8,14,24,41,68,111,180,289,460,728,1146,1795,2800,4352,6742,10414,16044,24659,37818,57885,88440,134899,205448,312448,474548,719861,1090734,1650908,2496260,3770910,5691360}}, {HHTH beats HTTH,1/2, {-2a(n)+2a(1+n)-2a(3+n)+a(4+n)=0, a(1)=1,a(2)=2,a(3)=4,a(4)=6}, {1,2,4,6,10,16,28,48,84,144,248,424,728,1248,2144,3680,6320,10848,18624,31968,54880,94208,161728,277632,476608,818176,1404544,2411136,4139136,7105536,12197888,20939776,35946752}}, {HTHH beats TTHT,7/16, {-2a(n)+2a(1+n)-2a(3+n)+a(4+n)=0, a(1)=1,a(2)=2,a(3)=3,a(4)=5}, {1,2,3,5,8,14,24,42,72,124,212,364,624,1072,1840,3160,5424,9312,15984,27440,47104,80864,138816,238304,409088,702272,1205568,2069568,3552768,6098944,10469888,17973376,30854400}}, {HHTH beats THTT,9/16, {-2a(n)+2a(1+n)-2a(3+n)+a(4+n)=0, a(1)=1,a(2)=2,a(3)=4,a(4)=7}, {1,2,4,7,12,20,34,58,100,172,296,508,872,1496,2568,4408,7568,12992,22304,38288,65728,112832,193696,332512,570816,979904,1682176,2887744,4957312,8510080,14609024,25078912,43052288}}, {HHHT beats THHT,9/16, {-2-a(n)-a(1+n)+a(2+n)=0, a(1)=1,a(2)=2,a(3)=4}, {1,2,4,8,14,24,40,66,108,176,286,464,752,1218,1972,3192,5166,8360,13528,21890,35420,57312,92734,150048,242784,392834,635620,1028456,1664078,2692536,4356616,7049154,11405772,18454928}}, {HTTT beats HHHH,7/11, {-3-a(n)-2a(1+n)-2a(2+n)-a(3+n)+a(5+n)=0,a(1)=1,a(2)=2,a(3)=4,a(4)=7,a(5)=13}, {1,2,4,7,13,23,40,70,122,212,368,639,1109,1924,3338,5791,10046,17427,30231,52442,90971,157807,273747,474867,823748,1428949,2478786,4299929,7459050,12939150,22445432,38935897,67541764}}, {HHHT beats TTTT,15/22,{-4-a(n)-2a(1+n)-2a(2+n)-a(3+n)+a(5+n)=0,a(1)=1,a(2)=2,a(3)=4,a(4)=8,a(5)=14}, {1,2,4,8,14,25,44,77,134,233,405,703,1220,2117,3673,6372,11054,19176,33265,57705,100101,173645,301221,522526,906422,1572363,2727565,4731484,8207665,14237766,24698130,42843633}}, {HHHT beats HTTT,4/7, {-4-n-a(n)-2a(1+n)-a(2+n)+a(4+n)=0, a(1)=1,a(2)=2,a(3)=4,a(4)=8}, {1,2,4,8,14,24,41,68,112,184,300,488,793,1286,2084,3376,5466,8848,14321,23176,37504,60688,98200,158896,257105,416010,673124,1089144,1762278,2851432,4613721,7465164,12078896,19544072}}, {HHTH beats THTH,9/16, {-a(n)+a(1+n)-2a(2+n)+a(3+n)=0, a(1)=1,a(2)=2,a(3)=4,a(4)=6}, {1,2,4,6,10,18,32,56,98,172,302,530,930,1632,2864,5026,8820,15478,27162,47666,83648,146792,257602,452060,793310,1392162,2443074,4287296,7523680,13203138,23169892,40660326,71353898}}, {HTHT beats THHT,7/16, {-a(n)+a(1+n)-2a(2+n)+a(3+n)=0, a(1)=1,a(2)=2,a(3)=2,a(4)=4}, {1,2,2,4,8,14,24,42,74,130,228,400,702,1232,2162,3794,6658,11684,20504,35982,63144,110810,194458,341250,598852,1050912,1844222,3236384,5679458,9966754,17490434,30693572,53863464}}, {HHTH beats THHT,5/12, {-a(n)+a(2+n)-2a(4+n)+a(5+n)=0, a(1)=1,a(2)=1,a(3)=2,a(4)=4,a(5)=7}, {1,1,2,4,7,13,23,41,73,130,232,414,739,1319,2354,4201,7497,13379,23876,42609,76040,135701,242172,432180,771268,1376404,2456329,4383562,7822900,13960739,24914320,44462069,79346961}}, {HTHH beats THHT,7/12, {-a(n)+a(2+n)-2a(4+n)+a(5+n)=0, a(1)=1,a(2)=2,a(3)=3,a(4)=6,a(5)=10}, {1,2,3,6,10,18,32,57,102,182,325,580,1035,1847,3296,5882,10497,18733,33431,59661,106471,190008,339088,605136,1079925,1927233,3439338,6137839,10953581,19547749,34884892,62255541}}, {HHTT beats HTHH,3/7, {-a(n)+a(3+n)-2a(5+n)+a(6+n)=0, a(1)=1,a(2)=2,a(3)=3,a(4)=5,a(5)=8,a(6)=13}, {1,2,3,5,8,13,22,38,66,115,200,347,601,1040,1799,3112,5384,9316,16121,27898,48279,83549,144584,250205,432982,749278,1296630,2243827,3882960,6719495,11628145,20122608,34822351}}, {HHTT beats TTHT,9/14, {-a(n)+a(3+n)-2a(5+n)+a(6+n)=0, a(1)=1,a(2)=2,a(3)=4,a(4)=8,a(5)=14,a(6)=24}, {1,2,4,8,14,24,41,70,120,207,358,620,1074,1860,3220,5573,9644,16688,28877,49970,86472,149640,258954,448124,775485,1341986,2322320,4018795,6954558,12034920,20826530,36040488}}, {HHTH beats TTHH,5/14, {-a(n)+a(3+n)-2a(5+n)+a(6+n)=0, a(1)=1,a(2)=2,a(3)=3,a(4)=4,a(5)=6,a(6)=9}, {1,2,3,4,6,9,15,26,46,81,142,247,428,740,1279,2211,3824,6616,11449,19814,34291,59344,102698,177721,307547,532210,920990,1593777,2758042,4772815,8259400,14292968,24734111,42802599}}, {HTHH beats HHTT,4/7, {-a(n)+a(3+n)-2a(5+n)+a(6+n)=0, a(1)=1,a(2)=2,a(3)=4,a(4)=7,a(5)=12,a(6)=20}, {1,2,4,7,12,20,34,58,100,173,300,520,901,1560,2700,4672,8084,13988,24205,41886,72484,125435,217068,375640,650050,1124918,1946680,3368745,5829640,10088240,17457785,30210848}}, {HHHH beats HTHT,5/12, {-a(n)-2a(1+n)-a(3+n)+a(4+n)=0, a(1)=1,a(2)=1,a(3)=2,a(4)=4}, {1,1,2,4,7,12,22,40,71,127,229,411,736,1321,2372,4255,7633,13698,24580,44101,79130,141988,254770,457131,820237,1471765,2640797,4738402,8502169,15255528,27373129,49115869,88129094}}, {HTHT beats HHHH,7/12, {-a(n)-2a(1+n)-a(3+n)+a(4+n)=0, a(1)=1,a(2)=2,a(3)=3,a(4)=5}, {1,2,3,5,10,18,31,56,102,182,325,585,1051,1883,3378,6065,10882,19521,35029,62858,112782,202361,363106,651528,1169032,2097605,3763767,6753359,12117601,21742740,39013225,70001786}}, {HTHT beats TTTT,5/8, {-a(n)-2a(1+n)-a(3+n)+a(4+n)=0, a(1)=1,a(2)=2,a(3)=3,a(4)=6}, {1,2,3,6,11,19,34,62,111,198,356,640,1147,2057,3693,6627,11888,21331,38278,68681,123231,221118,396758,711901,1277368,2292002,4112562,7379199,13240571,23757697,42628657,76488998}}, {HHHH beats THTH,3/8, {-a(n)-2a(1+n)-a(3+n)+a(4+n)=0, a(1)=1,a(2)=1,a(3)=2,a(4)=3}, {1,1,2,3,6,11,19,34,62,111,198,356,640,1147,2057,3693,6627,11888,21331,38278,68681,123231,221118,396758,711901,1277368,2292002,4112562,7379199,13240571,23757697,42628657,76488998}}, {HTTT beats TTTT,15/16,{-a(n)-a(1+n)-a(2+n)+a(3+n)=0, a(1)=1,a(2)=2,a(3)=4,a(4)=8}, {1,2,4,8,14,26,48,88,162,298,548,1008,1854,3410,6272,11536,21218,39026,71780,132024,242830,446634,821488,1510952,2779074,5111514,9401540,17292128,31805182,58498850,107596160}}, {HHHH beats HTHH,3/10, {-a(n)-a(1+n)-a(3+n)-a(4+n)+a(5+n)=0, a(1)=1,a(2)=1,a(3)=1,a(4)=2,a(5)=4}, {1,1,1,2,4,8,14,25,45,82,149,270,489,886,1606,2911,5276,9562,17330,31409,56926,103173,186991,338903,614229,1113231,2017624,3656749,6627505,12011714,21770074,39456161,71510489}}, {HTHH beats HHHH,7/10, {-a(n)-a(1+n)-a(3+n)-a(4+n)+a(5+n)=0, a(1)=1,a(2)=2,a(3)=4,a(4)=6,a(5)=11}, {1,2,4,6,11,20,37,67,121,219,397,720,1305,2365,4286,7768,14079,25517,46247,83818,151912,275326,499002,904393,1639125,2970756,5384209,9758360,17686087,32054328,58095380,105292277}}, {HTTH beats HHHH,7/11, {-a(n)-a(2+n)-a(4+n)-a(5+n)+a(6+n)=0, a(1)=1,a(2)=2,a(3)=4,a(4)=6,a(5)=11,a(6)=19}, {1,2,4,6,11,19,35,62,112,199,357,637,1141,2039,3649,6524,11671,20871,37332,66766,119418,213579,382000,683216,1221966,2185527,3908911,6991233,12504110,22364086,39999073,71539919}}, {HHTH beats HTHT,5/7, {-a(n)-a(3+n)+a(4+n)-2a(5+n)+a(6+n)=0, a(1)=1,a(2)=2,a(3)=4,a(4)=7,a(5)=12,a(6)=21}, {1,2,4,7,12,21,38,69,125,226,408,736,1328,2397,4327,7811,14100,25452,45943,82931,149698,270219,487771,880473,1589337,2868903,5178640,9347933,16873900,30458980,54981330,99246483}}, {HTHT beats HHTH,2/7, {-a(n)-a(3+n)+a(4+n)-2a(5+n)+a(6+n)=0, a(1)=1,a(2)=1,a(3)=1,a(4)=2,a(5)=4,a(6)=7}, {1,1,1,2,4,7,13,24,43,77,139,251,453,818,1477,2666,4812,8686,15679,28302,51088,92219,166464,300483,542400,979083,1767337,3190210,5758630,10394870,18763720,33870283,61139053}}, {HTTH beats HHHT,3/7, {-a(n)-a(3+n)-a(4+n)+a(5+n)=0, a(1)=1,a(2)=2,a(3)=3,a(4)=5,a(5)=8}, {1,2,3,5,8,14,24,41,70,119,203,346,590,1006,1715,2924,4985,8499,14490,24704,42118,71807,122424,208721,355849,606688,1034344,1763456,3006521,5125826,8739035,14899205,25401696}}, {HTHH beats THTH,5/14, {-a(n)-a(3+n)+a(4+n)-2a(5+n)+a(6+n)=0, a(1)=1,a(2)=1,a(3)=2,a(4)=3,a(5)=5,a(6)=9}, {1,1,2,3,5,9,17,31,56,101,182,328,592,1069,1930,3484,6289,11352,20491,36988,66767,120521,217552,392702,708864,1279566,2309737,4169293,7525967,13585080,24522350,44265153,79902773}}, {HTTH beats THHH,1/2, {-a(n)-a(3+n)-a(4+n)+a(5+n)=0, a(1)=1,a(2)=2,a(3)=4,a(4)=6,a(5)=10}, {1,2,4,6,10,17,29,50,85,145,247,421,718,1224,2087,3558,6066,10342,17632,30061,51251,87378,148971,253981,433013,738245,1258636,2145852,3658469,6237334,10634048,18130018,30909918}}, {HTHT beats THTT,9/14, {-a(n)-a(3+n)+a(4+n)-2a(5+n)+a(6+n)=0, a(1)=1,a(2)=2,a(3)=3,a(4)=6,a(5)=11,a(6)=19}, {1,2,3,6,11,19,34,62,112,202,365,659,1189,2146,3874,6993,12623,22786,41131,74245,134019,241917,436683,788254,1422873,2568420,4636240,8368850,15106563,27268770,49222700,88851613}}}

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Re: [math-fun] [seqfan] Swapping knights
by Richard Guy 16 Oct '10

16 Oct '10

I sh'd've numbered the knights. The problem was (sh'd've been) How many moves needed to swap Si with si for i = 1,2,...,n. The 8 half moves you mention swap the knights, but leave them in a different order. R. On Sat, 16 Oct 2010, Maximilian Hasler wrote: > I'm sorry to bother you, but I don't see why the 3x3 case is not possible. > > The following 8 half-moves seem to do it: > 1. a1-c2 b3-a1 > 2. c1-b3 c3-a2 > 3. b1-c3 a3-b1 > 4. c2-a3 a2-c1 > > see > https://docs.google.com/present/view?id=d34bxwj_519c2s3jcr9 > or enclosed PDF for illustration. > > Unless I'm wrong, at least n=3k seems as simple as n=3 (same moves k times). > (Maybe there's a shorter solution, I did not search for.) > > Maximilian > > > On Fri, Oct 15, 2010 at 3:53 PM, Richard Guy <rkg(a)cpsc.ucalgary.ca> wrote: >> n pairs of knights on a 3 by n chessboard. >> I.e., to get from >> >> S S S ... S s s s ... s >> - - - - to - - - - >> s s s ... s S S S S >> >> where s, S are chess knights of opposite >> colors. Not possible for n = 1, 2, 3. >> >> n = 4: 32 moves are necessary and sufficient. >> n = 5, 6, 7, ... are Rikki-Tikki-Tavi questions >

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[math-fun] Vertex-regular tessellations of 3-space by regular and Archimedean polyhedra
by Dan Asimov 16 Oct '10

16 Oct '10

There are of course 3 regular tilings of the plane by copies of one regular polygon. There are 8 more if several regular polygons are used, with the regularity condition being that the group of tile-preserving isometries of the plane is transitive on the set of all vertices: the tiling is "vertex-regular". (9 more if you distinguish between mirror images, but it's more natural not to.) * * * The natural generalization to 3-space would ask for all vertex-regular tilings of space by: a) any collection of regular polyhedra OR, what is perhaps more interesting, b) any collection of regular and/or Archimedean polyhedra. (There are 13 Archimedean polyhedra if we don't distinguish between mirror images.) Has anyone seen a classification of either case a), or better, case b) ??? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Is 312 a black hole for N?
by Eric Angelini 16 Oct '10

16 Oct '10

Hello Math-fun, a little puzzle slightly adapted from here (in French), thanks to my friend Alexandre Wajnberg: http://dev.ulb.ac.be/urem/IMG/pdf/PRO12102010.pdf For any natural number "n" (base-10-written) we build "next-n" like this: - write the quantity of digits of "n" - concatenate the quantity of even digits in "n" - concatenate to the former concatenation the quantity of odd digits in "n" Example for "n" = 80322057626942 - there are 14 digits in "n": --> 14 - there are 10 even digits in "n": --> 1410 - there are 4 odd digits in "n": --> 14104 If we iterate the construction, we get the sequence: 80322057626942, 14104, 532, 312, 312, 312, ... Another example: 5771, 404, 330, 312, 312, 312, ... Is 312 the end of all such sequences? Best, É.

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[math-fun] Swapping knights
by Richard Guy 15 Oct '10

15 Oct '10

I don't believe that the following sequence is in OEIS. The (least) number of moves needed to swap n pairs of knights on a 3 by n chessboard. I.e., the number of chess knight moves needed to get from S S S ... S s s s ... s - - - - to - - - - s s s ... s S S S S where s, S are chess knights of opposite colors. Not possible for n = 1, 2, 3. n = 4: 32 moves are necessary and sufficient. n = 5, 6, 7, ... are Rikki-Tikki-Tavi questions on p.250 (R187) of The Inquisitive Problem Solver. R.

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