February 2003 - mathfun@mailman.xmission.com

Re: [math-fun] Bomb iraq
by Mike Stay 25 Feb '03

25 Feb '03

I apologize; I regretted sending the note as soon as I had done so. I hereby promise to leave all future moderation to the moderator (as is "proper netiquette"... d'oh!) -- Mike Stay staym(a)clear.net.nz

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Re: [math-fun] Bomb iraq
by Mike Stay 25 Feb '03

25 Feb '03

Richar Guy wrote: >Apologies for not sending this out sooner. I wish you hadn't. As has been stated before, this isn't a political mailing list. I feel the war with Iraq is justified and necessary, but this isn't the place to discuss it. It's completely improper netiquette. So even if it seems funny, please abstain from posting topics that aren't Fun Math to this list. Thanks, -- Mike Stay staym(a)clear.net.nz

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[math-fun] Fractions test
by Chris Landauer 25 Feb '03

25 Feb '03

hihi, all - i used to give as a puzzle the problem of finding all 2-digit pairs that used that kind oc cancellation - it was a relatively hard exercise in the careful use of fairly easy reasoning steps i have not seen it published, except maybe in a number puzzle book more later, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal(a)aero.org

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[math-fun] Re: Unsolved Problems D22: On a paper by O.Schulz
by Richard Guy 24 Feb '03

24 Feb '03

Rainer, Many thanks for your interest. Please feel free to distribute this reply to other newsgroups. I'm just doing the third edition of UPINT, so your message is of double interest. Anyone who has wind of this or other items which they would be willing to sell or donate to the Strens Collection, please let me know (Leon Bankoff's widow recently donated a couple of thousand items, some from a couple or more centuries ago, but you don't all have to be quite so generous). Addenda & corrigenda for UPINT also most welcome. R. On Mon, 24 Feb 2003, Rainer Rosenthal wrote: > Dear Richard Guy, > > in your book "Unsolved Problems in Number Theory" > you ask (in chapter D22 (Heron Tetrahedra)): > > O. Schultz, Ueber Tetraeder mit rationalen Masszahlen > der Kantenlaengen und des Volumens > > Dickson appealed for a copy of this last. Did he ever > get one? Would they be willing to donate it, or offer > it for sale, to the Strens Collection? > > Some days ago we discussed rational tetrahedra in the german > newsgroup de.sci.mathematik and Hermann Kremer, who is well > known for his searching-the-net did succeed in fining even > an online reference. Please note that "Schultz" should be > correctly spelled "Schulz" (without any t in the name). > > I post a copy of this mail to sci.math, where there may be > some additional echo ... > > For those who don't know your book, I give its title here: > > Richard K.Guy > Unsolved Problems in Number Theory Volume 1 > Springer Verlag 1994 > ISBN 0-387-94289-0 > > With best regards (and thanks for your wonderful book) > Rainer Rosenthal > r.rosenthal(a)web.de > > > ----- Original Message ----- > From: Hermann Kremer <hermann.kremer(a)online.de> > Newsgroups: de.rec.denksport,de.sci.mathematik > Sent: Friday, February 21, 2003 2:18 AM > Subject: Re: Ganzzahliges Tetraeder > > > > Rainer Rosenthal schrieb in Nachricht ... > > >Hermann Kremer wrote > > > > [ .. ] > > > > > > > >O. Schultz, Ueber Tetraeder mit rationalen Masszahlen der Kantenlängen > > > und des Volumen, Halle, 1914, 292 ff. > > > > > >Dickson hat dringend um eine Kopie der letztgenannten Abhandlung gebeten. > > >Ob er wohl je eine erhalten hat? Kennt jemand solch eine Kopie (z.B. > > >Herr Kremer aus der deutschen Newsgroup de.sci.mathematik?) Würde dieser > > >jemand sie spendieren oder sie gegen Geld für die Strens Sammlung > hergeben? > > > > Ja, er kennt, und sogar ohne Geld im WWW ... Allerdings für einen Herrn > Otto > > Schulz ohne t ... ist aber der gemeinte ;-)) > > > > Die Arbeit ist die Inaugural-Dissertation des Otto Schulz 1912 in > Leipzig. > > Zuerst mal die Rezension aus dem JfM: > > ---------------------------------------------------- > > Electronic Research Archive for Mathematics Jahrbuch Database > > > > JFM 45.0295.01 > > Schulz, O. > > Über Tetraeder mit rationalen Masszahlen der Kantenlängen und des > Volumens. [D] > > Leipzig, 1912, 292 S. Published: 1914 > > > > Die Arbeit ist aus einer Anregung hervorgegangen, die der Verf. im > > mathematischen Seminar der Universität Berlin durch Herrn Geheimrat > > Schwarz empfangen hat. Einleitend werden die Hülfsmittel aus der Theorie > der > > diophantischen Gleichungen besprochen, auch wird eine kurze Geschichte des > > Tetraederproblems gegeben. Nachdem die grundlegenden Untersuchungen wie > die > > Formulierung des Problems geleistet sind, werden zunächst rationale > Dreiecke und > > Vierecke, d. h. Tetraeder vom Volumen Null, behandelt. Es folgt die > Theorie von > > speziellen rationalen Tetraedern, hierauf die der allgemeinen rationalen > > Tetraeder, wobei nun die Darlegung der Ankum'schen Beobachtung einen > > breiten Raum einnimmt, dass nämlich das Tetraederproblem durch geeignete > > Transformation der Volumenformel in eine diophantische Aufgabe zweiten > Grades > > übergeführt werden kann, von der eine partikuläre Lösung ein Tetraeder vom > > Volumen Null ist. Ein Anhang gibt ein Literaturverzeichnis und zwei > Tabellen, > > deren eine Tetrader mit kleinzahligen Kanten, deren andere Ecken mit einem > > rationalen v. Staudt'schen Eckensinus gibt. > > > > [ Haentzschel, Prof. (Berlin) ] > > > > Subject heading: Dritter Abschnitt. Niedere und höhere Arithmetik. Kapitel > 2. > > Zahlentheorie. A. Allgemeines. > > > > Jahrbuch Project: Copyright (c) 2003 European Mathematical Society > > > > ---------------------------------------------------- > > > > Die Buchform der Dissertation ist nachstehend rezensiert: > > ---------------------------------------------------- > > Electronic Research Archive for Mathematics Jahrbuch Database > > > > JFM 45.0755.03 > > Schulz, O[tto]. > > Über Tetraeder mit rationalen Masszahlen .... [B] > > Halle, Gebauer-Scheretschke. XII + 292 S. $8^\circ$. > > Published: 1914 > > > > Die Arbeit behandelt eine von Herrn H[ermann] A[mandus] Schwarz oft > > gestellte Aufgabe: Tetraeder mit rationalen Masszahlen der Kantenlängen > > und des Volumens zu finden. > > > > Eine allgemeine Lösung dieses Problems ist bis jetzt noch nicht gefunden > > worden. > > > > Der Verf. gibt eine eingehende Darstellung der Geschichte dieses Problems > und > > ein vollständiges Verzeichnis der betreffenden Literatur. > > > > Im zweiten Abschnitt werden solche Tetraeder behandelt, deren > Bestimmungsstücke > > einer oder mehreren Bedingungsgleichungen unterworfen sind. Darunter > finden > > sich auch rationale Tetraeder mit verschwindendem Volumen, d. h. Vierecke > mit > > rationalen Seiten und Diagonalen. > > > > Von einigen Typen, z. B. Trapezen, Tetraedern mit einer Symmetrieebene > > oder -achse u. a. m., ist die allgemeine Lösung gegeben. > > Von andern Typen sind nur partikuläre Lösungen angeführt, von denen die > > rationalen Höhentetraeder zu erwähnen sind. > > > > Im dritten Abschnitt ist das allgemeine rationale Tetraeder behandelt. > Nach einer > > von Herrn Ankum gemachten Entdeckung kann man aus einem gegebenen > > rationalen Tetraeder durch Einführung eines Parameters unendlich viele > neue > > rationale Lösungen erhalten. Denn behält man ein Dreieck bei und > vergrössert > > oder verkleinert man die drei übrigen Kanten um eine Grösse $s$, so > erscheint > > das Quadrat des Volumens $V$ als Funktion zweiten Grades von > > $s$ $$V^2 = As^2 + Bs+V_0^2$$, worin $V_0$ das Volumen, $A$ und $B$ > > Funktionen der Kanten des Ausgangstetraeders sind. Hieraus kann man > leicht > > $s$ und $V$ rational darstellen. Wird eine andere Ecke bevorzugt, so > erhält man > > eine zweite Schar rationaler Tetraeder, und so können beliebig viele > Parameter > > sukzessive eingeführt werden. Der Verf. stellt einige interessante > Untersuchungen > > über diese Methode an und zeigt z. B., dass zwei an verschiedenen Ecken > > nacheinander vorgenommene Transformationen nicht vertauschbar sind. Es > > werden auch noch andere Methoden zur sukzessiven Einführung von > Parametern > > angegeben, darunter auch eine, bei der der Radius der Umkugel dem > > ursprünglichen Rationalitätsbereich erhalten bleibt. > > > > Zum Schluss wird eine vollständige Lösung des Problems im Bereich der > > Grösse $\sqrt {-1}$ gegeben. > > Vgl. auch d. Bd. S. 295. > > > > [ Neiss, ] > > Subject heading: Achter Abschnitt. Reine, elementare und synthetische > Geometrie. > > Kapitel 3. Elementare Geometrie (Planimetrie, Trigonometrie, > Stereometrie). > > > > Jahrbuch Project: Copyright (c) 2003 European Mathematical Society > > ---------------------------------------------------- > > > > und die gibt es auch online: > > http://134.76.163.65/simple_search.html ... dann etwas Handarbeit: > > --> [Click]: Author Schulz --> [Click]: Submit --> [Click]: | 21-40 > ---> 29 > > > > > > Das ganze Opus hat rund 300 Seiten und ist daher leider zu lang, um > > komplett als PDF-Datei heruntergeladen zu werden ... > > > >

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[math-fun] identity from a stranger
by N. J. A. Sloane 24 Feb '03

24 Feb '03

This just came in from a total stranger. Anyone care to answer it? NJAS >From efknightjr(a)hotmail.com Fri Feb 21 07:02:09 2003 >Delivered-To: njas(a)research.att.com >X-Authentication-Warning: mail-red.research.att.com: postfixfilter set sender to efknightjr(a)hotmail.com using -f >X-Originating-IP: [64.40.53.47] >From: "EFKnightJr" <EFKnightJr(a)hotmail.com> >To: <njas(a)research.att.com> >Subject: Arctan(Phi) >Date: Fri, 21 Feb 2003 04:08:03 -0800 >X-Priority: 3 >X-MSMail-Priority: Normal >X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2600.0000 >X-OriginalArrivalTime: 21 Feb 2003 12:02:02.0197 (UTC) FILETIME=[0B472850:01C2D9A1] >X-Spam-Status: No, hits=-95.8 required=5.0 > tests=DEAR_SOMEBODY,DEAR_SOMETHING,HTML_WITH_BGCOLOR,MAILTO_LINK, > SPAM_PHRASE_05_08,USER_AGENT_OE,USER_IN_ALL_SPAM_TO > version=2.43-cvs > > >This is a multi-part message in MIME format. > >------=_NextPart_000_000D_01C2D95E.D462E780 >Content-Type: text/plain; > charset="iso-8859-1" >Content-Transfer-Encoding: quoted-printable > > >Dear Sir, > >Can you tell me if the following series expression >is a new result? > >arctan(Phi) =3D (1/2) * Sigma(k=3D0 to Infinity) >( ((-1/4)^k) * >( 1/(4k+1) + 2/(4k+2) + (1/2)/(4k+3) ) ) > >where Phi is the Golden Ratio and Sigma is summation notation. > >I developed the expression after investigating a cone >associated with the dodecahedron. The cone's base is the >circle inscribed in a face of the dodecahedron, and the >cone's vertex is the center of the dodecahedron. The angle >between the cone's axis and side is (1/2)*arctan(2), and >the tangent of that angle is the reciprocal of Phi. > >That leads to an expression relating Phi, Pi, and arctan(2), >namely arctan(Phi) =3D (1/2) * ( Pi - arctan(2) ) > >To obtain the expression in question at the start of >this email, I substituted in expressions for Pi and for >arctan(2) published in an article by Victor Adamchik. > >The same type of reasoning also leads to a similar >expression useful for base 16 computations, namely > >arctan(Phi) =3D (1/2) * Sigma(k=3D0 to Infinity) >( ((1/16^k)) * >( 3/(8k+1) - (1/2)/(8k+3) - 2/(8k+4) - >(3/4)/(8k+5) - 1/(8k+6) + (1/8)/(8k+7) ) ) > >Again, I would like to know if this is a new result. > >Thank your for your time. > >Sincerely, > >Jack R. Bench, Jr. >Email alias: EFKnightJr(a)Hotmail.Com > > >------=_NextPart_000_000D_01C2D95E.D462E780 >Content-Type: text/html; > charset="iso-8859-1" >Content-Transfer-Encoding: quoted-printable > ><!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"> ><HTML><HEAD> ><META http-equiv=3DContent-Type content=3D"text/html; = >charset=3Diso-8859-1"> ><META content=3D"MSHTML 6.00.2600.0" name=3DGENERATOR> ><STYLE></STYLE> ></HEAD> ><BODY bgColor=3D#c0c0c0><FONT face=3DArial size=3D2> ><DIV><BR>Dear Sir,</DIV> ><DIV> </DIV> ><DIV>Can you tell me if the following series expression<BR>is a new=20 >result?</DIV> ><DIV> </DIV> ><DIV>arctan(Phi) =3D (1/2) * Sigma(k=3D0 to Infinity)<BR>( ((-1/4)^k) = >*<BR>(=20 >1/(4k+1) + 2/(4k+2) + (1/2)/(4k+3) ) )</DIV> ><DIV> </DIV> ><DIV>where Phi is the Golden Ratio and Sigma is summation = >notation.</DIV> ><DIV> </DIV> ><DIV>I developed the expression after investigating a cone<BR>associated = >with=20 >the dodecahedron. The cone's base is the<BR>circle inscribed in a face = >of the=20 >dodecahedron, and the<BR>cone's vertex is the center of the = >dodecahedron. The=20 >angle<BR>between the cone's axis and side is (1/2)*arctan(2), and<BR>the = >tangent=20 >of that angle is the reciprocal of Phi.</DIV> ><DIV> </DIV> ><DIV>That leads to an expression relating Phi, Pi, and = >arctan(2),<BR>namely=20 >arctan(Phi) =3D (1/2) * ( Pi - arctan(2) )</DIV> ><DIV> </DIV> ><DIV>To obtain the expression in question at the start of<BR>this email, = >I=20 >substituted in expressions for Pi and for<BR>arctan(2) published in an = >article=20 >by Victor Adamchik.</DIV> ><DIV> </DIV> ><DIV>The same type of reasoning also leads to a similar<BR>expression = >useful for=20 >base 16 computations, namely</DIV> ><DIV> </DIV> ><DIV>arctan(Phi) =3D (1/2) * Sigma(k=3D0 to Infinity)<BR>( ((1/16^k)) = >*<BR>(=20 >3/(8k+1) - (1/2)/(8k+3) - 2/(8k+4) -<BR>(3/4)/(8k+5) - 1/(8k+6) + = >(1/8)/(8k+7) )=20 >)</DIV> ><DIV> </DIV> ><DIV>Again, I would like to know if this is a new result.</DIV> ><DIV> </DIV> ><DIV>Thank your for your time.</DIV> ><DIV> </DIV> ><DIV>Sincerely,</DIV> ><DIV> </DIV> ><DIV>Jack R. Bench, Jr.<BR>Email alias: <A=20 >href=3D"mailto:EFKnightJr@Hotmail.Com">EFKnightJr(a)Hotmail.Com</A><BR></FO= >NT></DIV></BODY></HTML> > >------=_NextPart_000_000D_01C2D95E.D462E780-- >

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[math-fun] Rubik cube sequence?
by N. J. A. Sloane 21 Feb '03

21 Feb '03

Inspired by Jaap Scherphuis's excellent web site on generalizations of Rubik's cube, i've added several sequences to the OEIS giving the number of positions that are n moves away from the start in these puzzles. For example: %I A079761 %S A079761 1,9,54,321,1847,9992,50136,227536,870072,1887748,623800,2644 %N A079761 Number of positions that are n moves from the starting position in the 2 X 2 X 2 Rubik cube puzzle. %C A079761 A puzzle in the Rubik cube family. The total number of distinct positions is 3674160. A half-turn is considered to be one move. %D A079761 D. R. Hofstadter, Metamagical Themas, Basic Books, NY, 1985, p. 359. %H A079761 Jaap Scherphuis, <a href="http://www.geocities.com/jaapsch/puzzles/">Puzzle Pages</a> %K A079761 nonn,fini,full,new %O A079761 0,2 %A A079761 njas, Feb 20 2003 Does anyone know the analogous sequence for the 3 X 3 X 3 cube, or even the beginning of that sequence? NJAS

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[math-fun] Ulam data files
by Jud McCranie 18 Feb '03

18 Feb '03

I now have data files of 2,959,301 Ulam numbers <= 40,000,000, if anyone wants a copy them. One file is a text file with one number per line, 28MB (7MB zipped). The other has the number and the indices of the two Ulam numbers that make up the sum in a binary data file, 35MB (14MB). I can zip these if needed (recommended) or possibly get them into a different format (e.g. gaps, binary file of numbers only). Still no gaps larger than 587 and the density is holding up around 7.4%.

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[math-fun] Ulam gaps and term frequency
by Jud McCranie 16 Feb '03

16 Feb '03

Based on the 1,513,118 Ulam numbers <= 20,447,213, the frequency of gaps is: The density is holding up at about 7.4%, contrary to Ulam's expectation that the density goes to zero. gap freq percent 1 4 0.00 2 558959 36.94 3 207138 13.69 4 2 0.00 5 16811 1.11 7 3245 0.21 8 18431 1.22 9 3 0.00 10 1426 0.09 12 75087 4.96 13 2 0.00 15 51331 3.39 17 129419 8.55 19 3130 0.21 20 116950 7.73 22 59663 3.94 24 3066 0.20 25 82142 5.43 27 5511 0.36 29 1764 0.12 30 9868 0.65 32 265 0.02 34 18297 1.21 37 4988 0.33 39 33475 2.21 41 1898 0.13 42 49025 3.24 (omitted terms <= 42 did not occur) The largest gap found is 587 after 933709. Record-setting gaps are in A080287 and A80288. No larger gaps were found, even though the search went more than 20 times farther. --------------------------- I looked at which terms in the sequence are later used in the sum of other terms. The percentage here is the percentage of the terms generated that have this Ulam number as a term in the sum. Indices <= 59 that are omitted did not occur. Note that 2 or 3 are used in 50.631% of the terms. It could be conjectured that 50% of the terms use 2 or 3 in the sum. Index Ulam Freq Percent 1 1 3 0.000 2 2 558960 36.941 3 3 207139 13.690 4 4 3 0.000 5 6 2 0.000 6 8 24947 1.649 7 11 2 0.000 8 13 659 0.044 9 16 2 0.000 10 18 3 0.000 11 26 4 0.000 12 28 2 0.000 13 36 73144 4.834 15 47 177377 11.723 16 48 1 0.000 17 53 3342 0.221 18 57 19 0.001 19 62 1 0.000 20 69 153174 10.123 24 87 1 0.000 25 97 7767 0.513 26 99 1 0.000 27 102 127232 8.409 28 106 2 0.000 33 145 1 0.000 36 175 1 0.000 37 177 1 0.000 38 180 5 0.000 40 189 3 0.000 42 206 3411 0.225 44 219 2 0.000 46 236 1 0.000 47 238 1 0.000 48 241 1 0.000 51 258 1 0.000 52 260 1 0.000 53 273 44036 2.910 54 282 1 0.000 55 309 1 0.000 56 316 72 0.005 59 339 90452 5.978

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[math-fun] Ulam(1,2)
by David Wilson 15 Feb '03

15 Feb '03

Based on the definition alone, I would have guessed that the Ulams would thin out to density zero (apparently Ulam thought so too); I would never have never guessed a fairly uniform density with wave-like density. Klaus's pictures really show the phenomenon.

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Re: [math-fun] Generalizing strong induction
by asimovd＠aol.com 14 Feb '03

14 Feb '03

Bob Mayans wrote: << ... This is the so-called epsilon induction, and it requires not the Axiom of Choice but the Axiom of Restriction. Suppose that for any set A, if P(a) holds for each a in A, then P(A) holds. Then P(A) holds for all sets... >> I wrote: << I think we'd also have to suppose P(Ø) to conclude P(A) holds for all sets. Michael Kleber wrote: << Nope! It's the base-case-free induction. ... I remember learning that trick from Gerald Sacks as an undergraduate, and walking around in a delighted haze at the cleverness for the rest of the day... >> Thanks, Michael (and Bob M. via private e-mail) for setting me straight. Unfortunately, I managed to be in a haze *before* I grasped this incredibly clever kind of induction. --Dan

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