81. BBC - Radio 4 - Another 5 Numbers - The Number Four There could be just one map out there that wasn t. A proof was needed that couldbe applied to all maps. This was seemingly supplied by alfred kempe in 1879. http://www.bbc.co.uk/radio4/science/another51.shtml

@import url('/includes/tbenh.css') ; Home TV Radio Talk ... A-Z Index 1 June 2004 Text only BBC Homepage BBC Radio Radio 4 PROGRAMME FINDER: A-Z Directory Listen Again What's On Listings Presenters PROGRAMME GENRES: Arts and Drama Science History Factual TOP PROGRAMMES THIS WEEK: Reith Lectures The Archers Today Programme Woman's Hour ... Help Like this page? Send it to a friend! ANOTHER 5 NUMBERS: The Number Four MISSED A PROGRAMME? Go to the Listen Again page Simon Singh investigates another five very important numbers. Monday 27 October 2003 3.45-4.00pm Simon Singh's journey begins with the number 4, which for over a century has fuelled one of the most elusive problems in mathematics: is it true that any map can be coloured with just 4 colours so that no two neighbouring countries have the same colour? This question has tested some of the most imaginative minds - including Lewis Carroll's - and the eventual solution has aided the design of some of the world's most complex air and road networks. Listen again to Programme 1: The Number Four Most people's memories of geography at school will include two things. Firstly, there was learning by rote the groundnut crop yield figures for Senegal in the mid '70s, and secondly, there was colouring-in maps. Many hours were spent shading-in maps of Birmingham, scrounging around for different colour pens to distinguish Smethick from Oldbury. Now, a puzzle which owes more to maths than geography posits the question, how many different colour pens did you really need to pilfer from your mate's pencil-case in order to do Brum, so that no two adjacent suburbs had the same colour. Memories of youthful exuberance would suggest every pen in your pal's possession, but mathematically, the answer is 4.

82. The Four Colour Theorem On 17 July 1879 alfred Bray kempe announced in Nature that he had aproof of the Four Colour Conjecture. kempe was a London barrister http://physics.rug.ac.be/Fysica/Geschiedenis/HistTopics/The_four_colour_theorem.

The four colour theorem Previous topic Next topic History Topics Index The Four Colour Conjecture first seems to have been made by Francis Guthrie . He was a student at University College London where he studied under De Morgan . After graduating from London he studied law but by this time his brother Frederick Guthrie had become a student of De Morgan . Francis Guthrie showed his brother some results he had been trying to prove about the colouring of maps and asked Frederick to ask De Morgan about them. De Morgan was unable to give an answer but, on 23 October 1852, the same day he was asked the question, he wrote to Hamilton in Dublin. De Morgan wrote:- A student of mine asked me today to give him a reason for a fact which I did not know was a fact - and do not yet. He says that if a figure be anyhow divided and the compartments differently coloured so that figures with any portion of common boundary line are differently coloured - four colours may be wanted, but not more - the following is the case in which four colours are wanted. Query cannot a necessity for five or more be invented. ...... If you retort with some very simple case which makes me out a stupid animal, I think I must do as the Sphynx did.... Hamilton replied on 26 October 1852 (showing the efficiency of both himself and the postal service):- I am not likely to attempt your quaternion of colour very soon.

83. Publikationen Translate this page H3 (gemeinsam mit Gerhard Ammerer und alfred Stefan Weiß für Heinz Dopsch (Wien/München2001) 512 S. H4 (gemeinsam mit Michael kempe) Coping with the http://www.sbg.ac.at/ges/people/rohr/publ2000.htm

Publikationen (Stand: 4. Mai 2004) Universitäre Prüfungsarbeiten, Monographien Herausgeberschaft Einzelbeiträge in Zeitschriften und Sammelbänden Lernbehelfe ... Rezensionen Universitäre Prüfungsarbeiten [U1] Die Vita Columbani des Ionas von Bobbio (phil. Diplomarbeit Wien 1989) 126 Bl . (vgl. den überarbeiteten Teilabdruck unter [A3]) [U2] Vorstudien zum Theoderich-Panegyricus des Ennodius (Hausarbeit am Institut für Österreichische Geschichtsforschung Wien 1992) 98 Bl [U3] Der Theoderich-Panegyricus des Ennodius. Überlieferungs- und Wirkungsgeschichte - Allgemeine Einführung - Edition - Übersetzung - Erläuterungen ( phil . Diss. Wien 1994) 402 Bl . (vgl. die überarbeitete Druckfassung unter [M1]) Monographien [M1] Der Theoderich-Panegyricus des Ennodius (MGH Studien und Texte 12, Hannover 1995) XXXVII, 309 S. [M2] Festkultur des Mittelalters (Lebensbilder des Mittelalters 2, Graz 2002) 166 S. Einzelbeiträge in Zeitschriften und Sammelbänden (Monographien und Einzelbeiträge) [A1] Überlegungen zu Datierung und Anlaß des Theoderich-Panegyricus . In: Ethnogenese und Überlieferung. Angewandte Methoden der Frühmittelalterforschung (Hg. Karl Brunner/Brigitte Merta, Veröffentlichungen des Instituts für Österreichische Geschichtsforschung 31, Wien/München 1994) S. 95-106.

84. Z Historie Matematiky A Fyziky (3) 17. cervence 1879 alfred Bray kempe oznámil v casopise Nature, ženalezl dukaz problému ctyr barev. kempe byl právní obhájce http://www.gymtc.cz/natura/2001/8/20010805.html

Z historie matematiky a fyziky (3) zpracovali: Jiøí Svršek, Roman Bartoš Typografické poznámky k matematickým vztahùm jsou uvedeny na konci tohoto textu. 9. Problém ètyø barev Problém ètyø barev pochází od Francise Guthrieho , který byl studentem na University College v Londýnì, kde studoval u De Morgana . Po dokonèení studia zaèal studovat práva, ale jeho bratr Frederick Guthrie se stal také studentem u De Morgana. Frederick Guthrie nejprve vykonával praxi obhájce a v roce 1861 odejel do Jižní Afriky jako profesor matematiky. Zde publikoval nìkolik matematických èlánkù a zaèal se zajímat o botaniku. Jeden druh vøesu [Erica Guthriei] byl pojmenován po nìm. V dobì, kdy Frederick ještì studoval u De Morgana, jeho bratr mu ukázal nìkteré výsledky své práce a také ho požádal, aby ze De Morgana zeptal na možnost dùkazu problému ètyø barev. De Morgan na domnìnku o ètyøech barvách neznal odpovìï, ale 23. øíjna 1852 zaslal dopis Hamiltonovi do Dublinu. Domnìnka o ètyøech barvách spoèívá v následující úloze. Libovolný obrazec je rozdìlen na èásti a každá èást má být obarvena jednou ze ètyø barev tak, aby se na spoleèné hranici èástí nevyskytovaly dvì stejné barvy. De Morgan zaslal nìkolika matematikùm dotaz, zda by nebyli ochotni zabývat se øešením problému ètyø barev.

85. Telekom - Presse Translate this page Michael Krammer. Gernot Hochfellner. alfred Veider. Jürgen Horvath. Hubertus Hofkirchner. AlexanderArtner. Jürgen kempe. Mark Sorenson. Rona Newmarkt. Irina Simmons. http://www.telekom-presse.at/services/personalia_13887.html

Alfred Zmek Canon Österreich besetzt Bereich Marketing Professional Solutions mit Alfred Zmek. (Wien, 15.04.2004) Alfred Zmek (40) und sein Team sind ab sofort für die Vermarktung von High-End Drucklösungen für den B2B Markt bei Canon Österreich verantwortlich. Sowohl internationale Marketingstrategien als auch regionale Kampagnen sollen umgesetzt werden. Außerdem verantwortet Zmek die preisliche Gestaltung und die Positionierung der angebotenen Lösungen am heimischen Markt. Neuorganisation Der Geschäftsbereich Professional Solutions wurde im Rahmen einer Neuorganisation zu Jahresbeginn 2004 geschaffen. Alfred Zmek und seine vier Mitarbeiter erarbeiten Vermarktungskonzepte und Gesamtlösungen für Druckereien, Dienstleistungsunternehmen aus der Druck- und Kopiererbranche und Agenturen. Zmek ist bereits seit 1986 für Canon tätig. Bis 1993 war er als technischer Trainier für digitale Farbsysteme zuständig. Danach trainierte Zmek zwei Jahre lang die Vertriebs-Mannschaft. Bis 2003 verantwortete er das sechsköpfige Pre-Sales Consultant-Team mit dem Spezialgebiet digitale Drucksysteme. ed © Telekom-Presse Reaktionen auf diesen Artikel Es wurden noch keine Reaktionen in diesem Forum eingegeben.

86. KCL: ISS Archives Summary Guide: FRANKLIN, Alfred White (1905-1984) 3.2 CONTEXT AREA 3.2.1 Name of creator(s) Franklin alfred White 19051984 onchild abuse and neglect, London, 1978 with C Henry kempe and Christine Cooper http://www.kcl.ac.uk/depsta/iss/archives/collect/10fr30-1.html

King's College London Archives Services - Summary Guide FRANKLIN, Alfred White (1905-1984) 3.1 IDENTITY STATEMENT AREA 3.1.1 Reference code(s) : GB 0100 KCLCA KH/PP6 3.1.2 Title: FRANKLIN, Alfred White (1905-1984) 3.1.3 Date(s) 3.1.4 Level of description : Collection (fonds) 3.1.5 Extent and medium of the unit of description (quantity, bulk or size) : 1 envelope 3.2 CONTEXT AREA 3.2.1 Name of creator(s) 3.2.2 Administrative/Biographical history : Born, 1905; educated, Clare College, Cambridge and St Bartholomew's Hospital, London; Lawrence Scholarship and Gold Medal, St Bartholomew's Hospital, 1933 and 1934; Temple Cross Research Fellow, Johns Hopkins Hospital, 1934-1935; Honorary Consulting Physician, Department of Child Health, St Bartholomew's Hospital; Honorary Consulting Paediatrician, Queen Charlotte's Maternity Hospital, London; co-founder, The Osler Club, London; President, British Society for Medical History, 1974-1976; President, International Society for Prevention of Child Abuse and Neglect, 1981-1982; died, 20 September 1984. Publications: Editor of The care of invalid and crippled children (Oxford University Press, London, 1960); editor of

87. RUDOLF KEMPE Translate this page En 1960, Bicknell déclara à alfred Diez, agent de kempe en Autriche, que des investissementsconsidérables avaient été consentis pour les enregistrements http://www.abeillemusique.com/produit.php?cle=7554

88. Stothard S Monumental Effigies Of Great Britain Stothard devoted much time and money to seeing that his work reached publicationwhich it finally did in 1832 with text by alfred John kempe, Anna s brother. http://www.devon.gov.uk/library/locstudy/stothard.html

Devon Library and Information Services Devon's heritage: D731 Local Studies Service Stothard's Monumental effigies of Great Britain A Bassett and lady at Atherington Devon Geoffrey Plantagenet, Earl of Maine and Anjou, Died 1149. from an enamelled tablet formerly in the church of St. Julian at Mans. Devon Libraries recently received a donation from Mr and Mrs Kempe of The monumental effigies of Great Britain by Charles Stothard. It consists of over 140 beautifully etched plates of medieval sculpted effigies in churches and cathedrals across England. Charles Stothard (1786-1821) was the son of the artist Thomas Stothard. He became a student at the Royal Academy and exhibited there from 1811. In that year he began his series of monumental effigies, which was dedicated to the Prince Regent in 1817. Nine parts, each with twelve engravings appeared up to 1820. In the meantime he was employed by Daniel Lysons for the Magna Britannia series. In 1815 he was appointed historical draughtsman to the Society of Antiquaries and the following year he went to Normandy to make drawings of the Bayeux Tapestry. Through his detailed knowledge on medieval costume he was able to confirm that it was indeed a near contemporary account of the conquest and not a century or more later as some had thought. On 28 May 1821 he was on a ladder in Bere Ferrers church in Devon, drawing a stained glass window, when he made the fatal mistake of stepping back to admire his work. He is buried in the churchyard there. His widow Anna Eliza Stothard devoted much time and money to seeing that his work reached publication which it finally did in 1832 with text by Alfred John Kempe, Anna's brother. Anna had in the meantime married Edward Atkyns Bray, vicar of Tavistock, and became a well-known antiquary and historical novelist, her medieval settings being probably inspired in part by the work of her first husband.

89. Kaczynski, Theodore The Unabomber Manifesto Industrial Society Kemp, William Kemps Nine Daies Wonder (text in the UK); kempe, alfred Bray Howto Draw a Straight Line A Lecture on Linkages (frame and JavaScript-dependent http://www.cartage.org.lb/en/themes/BookLibrary/books/internet_biblio/k.htm

Kaczynski, Theodore: The Unabomber Manifesto: Industrial Society and Its Future (HTML at emf.net) Kadrey, Richard, contrib.: Dream Protocols , by Lee Ballentine (PDF at Probook) Kadrey, Richard: Metrophage (text files in the UK) Kaganoff, Tessa: Collaboration, Technology, and Outsourcing Initiatives in Higher Education: A Literature Review (PDF at rand.org) Kaganoff, Tessa: Combining Service and Learning in Higher Education: Evaluation of the Learn and Serve America Higher Education Program , also by Maryann J. Gray, Elizabeth Ondaatje, Ronald Fricker, Sandra A. Geschwind, Charles A. Goldman, Abby E. Robyn, Melora Sundt, Lori Vogelgesang, and Stephen P. Klein (PDF files at rand.org) Kaganoff, Tessa: Workplace Skills in Practice: Case Studies of Technical Work , also by Cathleen Stasz, K. Ramsey, R. Eden, and E. Melamid (PDF files at rand.org) Kahane, J., trans.: Socialism: An Economic and Sociological Analysis

90. AIM25: King's College London College Archives: FRANKLIN, Alfred White (1905-1984 Name of creator(s) Franklin alfred White 19051984 Honorary Consulting Paediatrician. abuseand neglect, London, 1978 with C Henry kempe and Christine http://www.aim25.ac.uk/cgi-bin/search2?coll_id=988&inst_id=6

91. Object-Oriented Programming With Ada 95 (Tutorial) Programming with Ada 95 on Friday 6 October 1995, from 930 to 1730, in FrankfurtPRESENTERS Stephane Barbey, Magnus kempe, and alfred Strohmeier, of the http://www.cs.kuleuven.ac.be/~dirk/ada-belgium/events/95/951006-oopa.html

Conference announcements Call for Participation - Object-Oriented Programming with Ada 95 (Tutorial) To the Ada-Belgium home page. Last update: 95/08/24. Dirk Craeynest

92. Kempe Universality kempe Linkages for his Universality Theorem. Page constructed by Dessislava Michaylova and Joseph O'Rourke, Smith College This material is from Folding and Unfolding in Computational Geometry, a book being written by Erik D. kempe's Universality Theorem says that a mechanical linkage can be Below we show kempe's construction for multiplying angles by an integer http://www.cs.smith.edu/~orourke/FUCG/Kempe.html

Kempe Linkages for his Universality Theorem Page constructed by: Dessislava Michaylova and Joseph O'Rourke , Smith College This material is from Folding and Unfolding in Computational Geometry , a book being written by Erik D. Demaine and Joseph O'Rourke. Thanks to Amanda Toop for help with Java, and Marty Demaine for help help locating the references. This page is by the authors. Kempe's Universality Theorem Kempe's Universality Theorem says that a mechanical linkage can be constructed so that one joint follows any given plane algebraic curve. His clever construction proceeds roughly as follows. Let phi(x,y) = be the equation of the curve to be followed. The variables x and y are represented by cosines of angles between links in the linkage. Powers of x and y then convert to cosines of sums of angles, e.g., cos (a) cos (b) = (1/2) cos (a+b) + cos (a-b) so that finally phi(x,y) becomes a sum of cosines of angles, where each angle is of the form r a +- s b +- c where a, b, c are angles between links, and r and s are integers that result from summing the same angles. The implication of this is that if angles can be multiplied by an integer, and summed, then the phi-equation can be followed. Below we show Kempe's construction for multiplying angles by an integer.

93. Hifi-Forum.de: Musik: Klassik: Rudolf Kempe - - - Ein Mann Ohne Eigenschaften ?? http://www.hifi-forum.de/index.php?action=browseT&back=5&sort=lpost&forum_id=68&

94. MathComp Database - Browse - List 1, Kelmer, Boaz. 2, Kelso, JA Scott. 1, Kelton, W. David. 8, Kemeny, John G. 1,Kemer, Aleksandr Robertovich. 1, Kemp, Murray C. 1, Kemp, Rainer. 1, kempe, AlfredBray. http://ram0.huji.ac.il/ALEPH/ENG/JSL/JMC/JMC/SCAN-R/0419923

MathComp database - Browse - AUTHOR list - ALL DOCUMENTS The numbers in the list below indicate the number of documents listed under a term. To display the documents, click on an eye . To move up or down the list, click on the arrow. Kelly, John J. (John Joseph), 1940 Nov. 23-1995 Kelly, L. M. (Leroy Milton), 1914- Kelly, Paul (Paul H. J.) Kelly, Paul Joseph Kelmer, Boaz Kelso, J. A. Scott Kelton, W. David Kemeny, John G. Kemer, Aleksandr Robertovich Kemp, Murray C. Kemp, Rainer Kempe, Alfred Bray

95. The Pimp's Bible: The Sweet Science Of Sin Alfred Bilbo Gholson The Pimp s Bible The Sweet Science of Sin alfred Bilbo Gholson. Authoror Artist alfred Bilbo Gholson. Title The Pimp s Bible The http://www.housebritegardens.co.uk/Alfred-Bilbo-Gholson-The-Pimps-Bible-The-Swe-

The Pimp's Bible: The Sweet Science of Sin Alfred Bilbo Gholson Author or Artist : Alfred Bilbo Gholson Title: The Pimp's Bible: The Sweet Science of Sin Gholson Alfred Bilbo Alfred Bilbo Gholson Subject: Biography Autobiography General Category: Biography General Format: Paperback Roy Winstanley-James Woodforde - Parson and Diarist... Bryant's Map of Norfolk in 1826... Richard Huntsman-Elizabeth Fry - Quaker and Prison Reformer... Christopher Bird-Silent Sentinels... ... Robert Moss-Dream Gates: A Journey into Active Dreaming...

96. Historia Matematica Mailing List Archive: Re: [HM] Kempe's Proof Next message Franz Lemmermeyer Re HM kempe s proof ; Previous messageAlfred Ross HM The Mathematics of Plato s Academy . I wrote http://sunsite.utk.edu/math_archives/.http/hypermail/historia/jun99/0144.html

Re: [HM] Kempe's proof Jim Propp propp@math.mit.edu Wed, 23 Jun 1999 02:34:01 -0400 (EDT) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: Franz Lemmermeyer: "Re: [HM] Kempe's proof" Previous message: Alfred Ross: "[HM] The Mathematics of Plato's Academy" I wrote: John Conway replied: To me, it makes a great deal of difference! I would like to subscribe to the view that says that mathematicians nowadays possess an acquired critical faculty, called "mathematical rigor", that prevents them from making mistakes like this, at least on a communal level. Incidents in which fallacious arguments survive widespread scrutiny challenge this rosy picture. One can abandon the picture (as I suspect John will do) and cheerfully admit that the modern mathematics community may be making all kinds of mistakes that 21st century graduate students (and 22nd century undergrads) will snicker about. Or one can try to defend the idealistic view (as I tend to do on most days), by saying "Well, of course 19th century mathematicians had lower standards of rigor than we do, so they were fooled by false

97. Alfred Molina more Written by Larry McMurtry , alfred Molina Published by Simon Schuster Audio (November 2003) ISBN 0743527887 Price $39.95. http://facultyofcinema.com/search_Alfred_Molina/searchBy_Author.html

STUDENT BOOKS ONLINE Faculty Of Cinema Student Books Online Back To Cinema Search Featured Universities ... Golden Globe Awards Looking for more books? Computers Solutions Cinema Ellington Duke Ellington History Consensus ... Herbert Zettl Links For Students Lesson Tutor Shopping Links Shopping Directory Digital Cameras Big Shoes Consumer Electronics ... The Intelligencer: A Novel When I heard the author on the radio talking about her real-life experiences as a PI, her ex-CIA boss, and U.S. attorney father, I decided to give this book a try. I like authenticity, and too many thrillers are painfully over the top...you know, so implausible you roll your eyes. But this one, wow! I loved it. It's got a very authentic, informative feel, but is also inventive, fresh and exciting. Kept me up till dawn!The Marlowe chapters come to life so vividly. Really enjoyed Marlowe's ban... more... Written by Leslie Silbert Jan Maxwell Alfred Molina Published by Audioworks (February 2004) ISBN 0743536487 Price $26.00

98. Telekom - Presse Translate this page alfred Veider. alfred Veider ist neuer Leiter der Transport AutomationSolutions Division der Alcatel Austria AG. (Wien, 19.11.2003 http://www.telekom-presse.at/services/personalia_11543.html

Alfred Veider Alfred Veider ist neuer Leiter der Transport Automation Solutions Division der Alcatel Austria AG. (Wien, 19.11.2003) Mit Wirkung vom 1. November 2003 wurde Alfred Veider zum neuen Leiter des Geschäftsbereichs Transport Automation Solutions der Alcatel Austria AG bestellt. Veider (43) Absolvent der Technischen Universitaet Wien (Technische Physik) und der Höheren Technischen Lehranstalt Saalfelden (Maschinenbau), trat 1989 in das Forschungszentrum des Unternehmens ein. Seit Mitte 2001 war er als Direktor Produkt Management und Produkt Strategie national und konzernweit verantwortlich für die Marktausprägung der Produkte und Dienstleistungen der Alcatel Austria AG / Transport Automation Solutions. Veider folgt Thomas Necker nach, der ab Februar 2004 die Funktion des Vorstandes Betrieb bei den Berliner Verkehrsbetrieben übernehmen wird. Veider blickt auf eine Karriere im Bereich Transport Automation Solutions der Alcatel Austria AG zurück. Er hatte maßgeblich zur Produkteinführung des elektronischen Stellwerks der neuen Generation ELEKTRA 2 in Österreich und in der Schweiz sowie zur erfolgreichen Einführung der Zugsicherung nach ETCS Standard in Österreich und zahlreichen Exportländern beigetragen. Vor 2001 war Veider zwei Jahre lang für den internationalen Vertrieb der Eisenbahnsicherungstechnik der Alcatel Austria AG verantwortlich, davor Abteilungsleiter in der technischen Entwicklung, zuständig für hochzuverlässige und sichere Echtzeitcomputersysteme. Veider hat ein Doktorat der Technischen Wissenschaften sowie ein Diplom im Fach Technische Physik von der Technischen Universität Wien. Sein Weg zu Alcatel führte über berufliche Stationen an der TU Wien, der Uni München und am IBM Forschungslabor in Zürich/Rüschlikon.

99. The Cornell Library Historical Mathematics Monographs The Cornell Library Historical Mathematics Monographs. http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Kemp009&seq

100. Browse The Cornell Library Historical Math Monographs Wilhelm Joseph Einführung in die grundlagen der geometrie (Volume 1). http://historical.library.cornell.edu/math/math_K.html

Sorted by Title A B C ... J K L M N O ... XYZ Authors "K" Kantor, Seligmann Theorie der endlichen gruppen von eindeutigen transformationen in der ebene Kempe, Alfred Bray How to draw a straight line; a lecture on linkages Killing, Wilhelm Joseph Einführung in die grundlagen der geometrie (Volume 1) Einführung in die grundlagen der geometrie (Volume 2) Lehrbuch der analytischen geometrie in homogenen koordinaten (Volume 1) Lehrbuch der analytischen geometrie in homogenen koordinaten (Volume 2) Die nicht-euklidischen Raumformen in analytischer Behandlung Klein, Felix Anwendung der Differential- und Integralrechnung auf Geometrie: eine Revision der Principien. Vorlesung gehalten während des Sommersemesters 1901 Ausgewählte kapitel aus der theorie der linearen differentialgleichungen zweiter ordnung (Volume 1) Ausgewählte kapitel aus der theorie der linearen differentialgleichungen zweiter ordnung (Volume 2) Ausgewählte kapitel der zahlentheorie ... Nicht-Euklidische geometrie (Volume 1) Nicht-Euklidische geometrie (Volume 2) Über die elliptischen normalcurven der N ordnung und zugehörige modulfunctionen der N stufe Ueber lineare differentialgleichungen der zweiten ordnung. Vorlesung, gehalten im sommersemester 1894

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z

Page 5     81-100 of 101    Back | 1  | 2  | 3  | 4  | 5  | 6  | Next 20