Extremal values: ---------------- - longest month of the year (September, one extra hour) - longest week of the century (8 days long, ISO week) - Friday the 13th. - January 20 is the inauguration day in the U.S. Although 20 = 13 (mod 7), it is more likely to fall on Sunday than any other day. [Skolnik] - perverse months, double dating: References: ???, Feast or Famine of Friday the 13th, AMM 98 (1991) 646-649 Heinz Bachmann; Kalenderarithmetik, 2nd Ed., Juris Verlag, Z\"urich, 1984, ISBN 3-260-05035-0, - Kap. 2: Tagesz\"ahlung und Wochentag, p26-42 Sect. 4.c: Die Verteilung der Wochentage auf die Monatstage - the 13th is most likely to be a Friday (equidistribution in the Julianian calendar) S. R. Baxter; - Friday the 13th -, Math. Gazette 53 (1969) 127-129 - the 13th is most likely to be a Friday B. H. Brown; (solver Raphael Robinson) E 36 (Friday the 13th), AMM 40 (1933) 607 - the 13th is most likely to be a Friday G. C. Bush; (solver C. V. Heuer) Friday the 13th's, AMM 70 (1963) 759 - The number of Friday 13th per year is at least one and at most three. Heinrich Hemme; Mathematisches Kabinett, Ein Aberglaube erweist sich als Wahrheit, Bild der Wissenschaft 1984 Heft 2, 142-143 - the 13th is most likely to be a Friday - the weekday distribution of the nth day, - the weekday distribution of the nth day of december. Heinrich Hemme; Mathematisches Kabinett, Kalenderr\"atsel, Bild der Wissenschaft 1987 Heft 5, 55 - 9 problems J. O. Irwin; Friday 13th, Math. Gazette, 55 (1971) 412-415, - The number of Friday 13th per year is at least one and at most three. - the 13th is most likely to be a Friday Maurice Kraitchik; Mathematical Recreations, George Allen and Unwin, 1943 p109-116 calendar Sidney Kravitz; More room for the calendar girl, Journal of Recreational Mathematics 6 (1973) 238-239 - double dating Martin Gardner; Puzzles from other worlds, 1981 (german: Denkspiele von anderen Planeten, Hugendubel, 1986) Problem 18: thirty days in september Martin Gardner; MG4SA: The Numerology of Dr. Matrix, Chap. 1-7 MG4SA: Simon & Schuster (1967) MG4SA: The Incredible Dr. Matrix, Chap. 1-18 MG4SA: Scribner (1976) MG4SA: The Magic Numbers of Dr. Matrix, Chap. 1-22 MG4SA: Prometheus Books (1985) MG4SA.13. each year has 2..4 perverse (needs six calendar lines) months MG4SA.20. calendar cubes problem: three cubes, three letters for the 12 months MG9SA.15.5 Two-Cube Calendar V. Frederick Rickey; Mathematics of the Gregorian Calendar, The Math. Intelligenzer, 7:1 (1985) 53-56, - Friday the 13th - leap year rules, why 97/400 - perverse months, double dating - history Wolfgang Alexander Schocken; The Calculated Confusion of Calendars, Vantage Press, New York, 1976 David Skolnik; A perpetual calendar formular, The Mathematics Teacher, 40 (1947) 36-37 Harold Watkins; Time counts; the Story of the Calendar, Neville Spearman, London, 1954 --------------------------------------------- CAL: Calender, Easter, Day of Week, Friday 13th CAL- Calendrical Calculations, (N. Dershowitz, E. M. Reingold), CAL Cambridge Univ. Press, 1997 CAL URL http://emr.cs.uiuc.edu/home/reingold/calendar-book/index.html CAL URL http://emr.cs.uiuc.edu/~reingold/calendar.C (C++ source) CAL- calendrical calculations, (N. Dershowitz, E. M. Reingold), CAL Software Practice and Exp. 20:9 (Sep 1990) 899-929 CAL Julian, Gregorian, Islam, Hebrew, Conversions, Gnu Lisp program CAL- calendrical calculations II, (E. Reingold, N. M. Dershowitz, S. Clamen), CAL Software Practice and Exp. 23:4 (Apr 1993) 383-404, CAL three historical calendars CAL- Chr. Zeller, Kalender-Formeln, Acta Mathematica, 9 (1887) 131-136 CAL- Hatcher, D. A., Simple Formulae for Julian Day Numbers and Calendar Dates, CAL Quarterly Journal of the Royal Astronomical Society, 25 (1984) 53-55. CAL- Hatcher, D. A., Generalized Equations for Julian Day Numbers and Calendar CAL Dates, Quarterly Journal of the Royal Astronomical Society, 26 (1985) CAL 151-155. Includes coefficients for various calendar systems, CAL including the Egyptian, Alexandrian, Roman, Gregorian, and Islamic. CAL- Keith & Craver; The ultimate perpetual calendar?, JoRM 22:4 (1990) 280-282 CAL day of the week as a 44 character expression in C. (illegal use of --) CAL The following 45 character C expression by Keith is correct. CAL dow(y,m,d) { return (d+=m<3?y--:y-2,23*m/9+d+4+y/4-y/100+y/400)%7; } CAL- Kalender-Formeln (C. Zeller), Acta Mathematica, 9 (1887) 131-136 CAL- The Day of the Week for Gregorian Calendars (A. D. Bradley) ?? 82-87 CAL- W. S. B. Wodhouse, Calendar, Encyclopaedia Britannica, CAL 11th and 13th ed., 4, 987-1004 CAL- A. W. Butkewitsch, M. S. Selikson - Ewige Kalender, Teubner (Leipzig) 1974 CAL Kleine Naturwissenschaftliche Bibliothek, Bd. 23 CAL- Ilan Vardi, Computational Recrations in Mathematica, (1991) CAL Chap 4: The Calender, p35-55 88h:01013 01A40 Swerdlow, Noel M.(1-CHI) The length of the year in the original proposal for the Gregorian calendar. J. Hist. Astronom. 17 (1986), no. 2, 109--118. From the text: "In an earlier note [same journal 5 (1974), part 1, 48--49; MR 58 #21098] we considered the origin of the length of the year in the Gregorian calendar, concentrating in particular on the earliest proposal by Petrus Pitatus in 1560 for the year eventually adopted in the reformed calendar, the general principles of which are usually attributed to Aloysius Lilius. At the time we had not seen the proposal of 1577, the Compendium novae rationis restituendi calendarium, that preceded the adoption of the calendar in 1582, and as we mistakenly believed that no copies of the proposal survived we made no to attempt to consult it. In fact, as we have since learned, some few copies of the original 1577 printing do survive, but more to the point, the text has always been readily available in countless libraries since it is printed in Christopher Clavius's comprehensive treatise on the calendar, originally published in 1603 and reprinted in Vol. V of the 1612 edition of his Opera mathematica. We have recently examined the 1577 proposal as reprinted "eisdem prorsus verbis", as Clavius says, in the 1612 edition, and found that it contains an interesting alternative plan for intercalation, taking account, after a fashion, of the variable tropical year of Copernicus. The plan was not adopted, for very sensible reasons, but does show that the possibility of incorporating the most up-to-date, although questionable, astronomy was at least considered." 88m:01024 01A40 (01A05 01A35) Dutka, Jacques On the Gregorian revision of the Julian calendar. Math. Intelligencer 10 (1988), no. 1, 56--64. In 1582, the Julian calendar (in use for over fifteen centuries) was transformed into the present day Gregorian one. The major change was that years not divisible by 400 were no longer treated as leap years, with the consequence that the mean Gregorian year was equal to $((365.25\times 400)-3)/400=365\frac{97}{400}$ days. The author explores the basis for this change and claims that it had its roots in the Roman Catholic Church's attempt to satisfy two demands: a more convenient method for determination of the date of Easter, and arresting the gradual drift of the vernal equinox (already mispredicted by some seven or eight days by the thirteenth century). The author claims that work basing itself on the Alfonsine tables (c. 1272) and its estimates of the length of the tropical year, especially the work of Aloysius Lilius of Calabria (1510?--1576), won over the Calendrical Commission appointed by Pope Gregory XIII. This article also discusses the role of continued fractions in this history and the obscurity sometimes surrounding explanations of the mean length of the Gregorian year. The article provides thirty-one bibliographical references, spanning several centuries and languages. 87i:00001 00A05 (01-01 11-01 11-04) Bachmann, Heinz Kalenderarithmetik. (German) [Calendar arithmetics] Second edition. Juris Druck und Verlag AG, Zurich, 1984. 111 pp. ISBN 3-260-05035-3 First published in 1984, this discussion of calendar arithmetics (now slightly improved and extended) presents a wealth of elementary formulas. Probably best known of all is Gauss's formula for computing the variable date of Easter. That is but one among many others (they could all be classified as elementary number theory) that are explained and derived in this volume. In Section 1 the author deals with the calendar year (Julian, Gregorian year, astronomical seasons), in Section 2 with counting the sequence of days in a longer period and computing the days of tHe weeks. In Section 3 the difficult problem of the lunar phases is discussed in detail (astronomical background, the mean phases, the clerical new moon calendar, etc.). Section 4 is devoted to the central problem of computing Easter dates in the Julian and the Gregorian calendars, including exceptionary rules, statistics about the distribution of Easter dates, and a derivation of the formula given by Gauss. Throughout numerous formulas are presented, while tables merely have an auxiliary function in this book (though Easter dates are tabulated for the years 1051--2702). As an appendix, a program in BASIC for computing the calendar of any year (including festival days dependent on the date of Easter) is reprinted. There is no bibliography, but several references in the footnotes refer to older and recent studies on chronology and calendar problems. Since the aim of this book is mathematical, the vast field of the history of the calendar is only mentioned where necessary for practical calendar arithmetics. Reviewed by C. J. Scriba 86d:01001 01-01 (00A69) Rickey, V. Frederick(1-BLGS) Mathematics of the Gregorian calendar. Math. Intelligencer 7 (1985), no. 1, 53--56. A brief account of the history of the Julian and Gregorian calendars, with some mathematical aspects of the Gregorian calendar. \{Reviewer's remarks: (1) It is asserted that Julius Caesar, in 45 B.C., "declared that every year divisible by 4 was to be a leap year with 366 days"---but the Romans did not then number their years. Rather, they named years after the consuls in office, and only later did the Romans start to number years from the presumed date of the foundation of Rome. (2) John Wallis is said (in this article) to have been appointed "as the first Savilian Professor with no more mathematical reputation than breaking a few coded messages for the king". But the Savilian Chairs in Geometry and in Astronomy were founded at Oxford in 1619, and Wallis was appointed as Savilian Professor of Geometry in 1649. Moreover, he had decrypted some cryptograms for the Parliamentarians, and not for King Charles I.\} Reviewed by Garry J. Tee Nixdorf-Museum zeigt «Software-Eisenbahn» Paderborn (dpa/gms) Das Heinz Nixdorf MuseumsForum in Paderborn hat sich etwas Besonderes einfallen lassen, um die Arbeitsweise von Computersoftware zu zeigen. Mit Hilfe einer Modelleisenbahn können sich Besucher den Wochentag ihrer Geburt «errechnen» lassen. Der dafür notwendige Programmablauf werde durch die Steuerung und Fahrtroute des Zuges erkennbar, teilte das Museum mit. Über eine Tastatur gibt der Besucher zunächst seine Geburtsdaten ein. Die Lok fährt mit drei Waggons, die das Datum auf den Ladeflächen anzeigen, los und passiert drei Kontrollstellen, an denen die Eingaben auf Wahrscheinlichkeit überprüft werden. Dann wird der Wochentag der Geburt ermittelt. Die «Software-Eisenbahn» stoppt an einem von sieben Bahnhöfen, die je einem Wochentag zugeordnet sind. Die verwendete Formel stammt aus dem Jahr 1885 und wurde von dem Geistlichen Christoph Zeller entwickelt. Die Besucher sollen anhand der Fahrtstrecke des Zuges und über digitale Anzeigen den Rechenvorgang der Software plastisch nachvollziehen können. Die Eisenbahn ist im Foyer des Heinz Nixdorf MuseumsForum dienstags bis freitags zu besichtigen. Der Eintritt ist frei. Schwäbische Zeitung Online http://www.szon.de/news/multimedia/aktuell/200503180852.html Software-Eisenbahn im Heinz Nixdorf MuseumsForum Kein Computer funktioniert ohne Software. Doch wie kann man dieses ebenso grundlegende wie unsichtbare Thema anschaulich in einem Museum präsentieren? Eine Antwort ist die neue Modelleisenbahn im Paderborner Heinz Nixdorf MuseumsForum (HNF). Besucher können mit Hilfe des Zuges den Wochentag ihrer Geburt ermitteln. Der dahinter stehende Programmablauf wird durch die Steuerung und Fahrtroute des Zuges ersichtlich. Die Besucher des HNF können über eine Zifferntastatur ihren Geburtstag eingeben. Daraufhin fährt eine Lok mit drei Waggons los, die das Datum auf den Ladeflächen anzeigt. Der Zug passiert drei Kontrollstellen, an denen die Angaben von Jahr, Monat und Tag überprüft werden. Bei scheinbaren Besucher-Methusalems mit Geburtsjahr 1815 oder anderen unzulässigen Werten fährt die Lok zum Startbahnhof zurück und verlangt eine korrekte Eingabe. Nach Durchfahrt durch die Kontrollstationen wird der Wochentag der Geburt ermittelt. Die verwendete Formel geht auf den Geistlichen Christoph Zeller zurück, der sie 1885 entwickelt hat. Der Zug stoppt daraufhin in einem von sieben Bahnhöfen, die den einzelnen Wochentagen zugeordnet sind. Zur Berechnung des Ergebnisses und zur Steuerung des Zuges haben Mitarbeiter des HNF einen speziellen Mikrocontroller entwickelt. Die Besucher können anhand von LCD-Anzeigen und der Fahrtstrecke des Zuges den Rechenvorgang nachvollziehen. Sie erhalten damit einen Eindruck, wie ein Computerprogramm abläuft und was ein Algorithmus ist. Eine Programmschleife wird ebenso sichtbar wie ein Programmsprung. Die Software-Eisenbahn ist ab sofort im Foyer des Heinz Nixdorf MuseumsForum bei freiem Eintritt zu besichtigen. Das HNF ist dienstags bis freitags von 9.00 bis 18.00 Uhr und am Wochenende von 10.00 bis 18.00 Uhr geöffnet. http://www.hnf.de/presse/pressemitteilungen/2005/maerz/1703_05_SoftwareEisenbahn.html Links zum Thema Kalender: http://www.computus.de/kalenderlinks/kalenderlinks.htm