Eighth preliminary edition

8. PROBABILITY RECREATIONS 699
8.A. Buffon's Needle Problem 699

8.B. Birthday Problem 699

8.C. Probability that a Triangle is Acute 701

8.D. Attempts to Modify Boy Girl Ratio 702

8.E. St. Petersburg Paradox 702

8.F. Problem of Points 703

8.G. Probability that Three Lengths Form a Triangle 704

8.H. Probability Paradoxes 705

8.H.1. Bertrand's Box Paradox 705

8.H.2. Bertrand's Chord Paradox 705

8.I. Taking the Next Train 706

8.J. Clock Patience or Solitaire 706

8.K. Sucker Bets 706

8.L. Nontransitive Games 707

9. LOGICAL RECREATIONS 708
9.A. All Cretans are Liars, etc. 708

9.B. Smith -- Jones -- Robinson Problem 712

9.C. Forty Unfaithful Wives 712

9.D. Spots on Foreheads 712

9.E. Strange Families 714

General Studies of Kinship Relations 714

Deceased Wife's Sister, etc. 715

General Family Riddles 716

9.E.1. That Man's Father is My Father's Son, etc. 722

9.E.2. Identical Siblings who are not Twins 726

9.F. The Unexpected Hanging 726

9.G. Truthtellers and Liars 726

9.H. Prisoner's Dilemma 729

9.I. Hempel's Raven Paradox 729

9.J. Use of a Fallen Signpost 729

9.K. Lewis Carroll's Barber Paradox 730

10. PHYSICAL RECREATIONS 730
10.A. Overtaking and Meeting Problems 731

10.A.1. Circling an Army 743

10.A.2. Number of Buses Met 745

10.A.3. Times from Meeting to Finish Given 745

10.A.4. The Early Commuter 746

10.A.5. Head Start Problems 747

10.A.6. Double Crossing Problems 748

10.A.7. Trains Passing 748

10.A.8. Too Slow, Too Fast 748

10.B. Fly Between Trains 748

10.C. Lewis Carroll's Monkey Problem 750

10.D. Mirror Problems 751

10.D.1. Mirror Reversal Paradox 751

10.D.2. Other Mirror Problems 753

10.D.3. Magic Mirrors 753

10.E. Wheel Paradoxes 754

10.E.1. Aristotle's Wheel Paradox 754

10.E.2. One Wheel Rolling Around Another 754

10.E.3. Hunter and Squirrel 755

10.E.4. Railway Wheels Paradox 756

10.F. Floating Body Problems 756

10.G. Motion in a Current or Wind 758

10.H. Snail Climbing out of Well 759

10.I. Limited Means of Transport -- Two Men and a Bike, etc. 763

10.J. Resistor Networks 764

10.K. Problem of the Date Line 765

10.L. Falling Down a Hole Through the Earth 767

10.M. Celts = Rattlebacks 770

10.M.1. Tippee Tops 771

10.N. Ship's Ladder in Rising Tide 771

10.O. Erroneous Averaging of Velocities 772

10.P. False Balance 772

10.Q. Push a Bicycle Pedal 773

10.R. Clock Hand Problems 774

10.S. Walking in the Rain 776

10.T. Centrifugal Puzzles 777

10.U. Shortest Route Via a Wall 777

10.V. Pick Up Puzzles = Pluck It 777

10.W. Puzzle Vessels 778

10.X. How Far Does a Phonograph Needle Travel? 784

10.Y. Double Cone Rolls Uphill 784

10.Z. The Wobbler 784

10.AA. Non-Regular Dice 785

10.AB. Bicycle Track Problems 788

10.AC. Roberval's Balance 790

10.AD. Pound of Feathers 790

10.AE. Juggling over a Bridge 790
11. TOPOLOGICAL RECREATIONS 791
11.A. Scissors on String 791

11.B. Two People Joined by Ropes at Wrists 792

11.C. Two Balls on String Through Leather Hole and Strap = Cherries Puzzle 793

11.D. Solomon's Seal 795

11.E. Loyd's Pencil Puzzle 797

11.F. The Imperial Scale 797

11.G. Trick Purses 798

11.H. Removing Waistcoat Without Removing Coat 799

11.H.1. Removing Loop from Arm 799

11.I. Heart and Ball Puzzle and Other Loop Puzzles 799

11.J. Möbius Strip 802

11.K. Wire Puzzles 805

11.K.1. Ring and Spring Puzzle 807

11.K.2. String and Spring Puzzle 807

11.K.3. Magic Chain = Tumble Rings 807

11.K.4. Puzzle Rings 808

11.K.5. Ring Mazes 808

11.K.6. Interlocked Nails, Hooks, Horns, etc. 809

11.K.7. Horseshoes Puzzle 809

11.K.8. The Caught Heart 810

11.L. Jacob's Ladder and Other Hinging Devices 810

11.M. Puzzle Boxes 811

11.N. Three Knives Make a Support 813

11.O. Borromean Rings 815

11.P. The Lonely Monk 816

11.Q. Turning an Inner Tube Inside Out 817

11.R String Figures 817

11.S. Puzzle Knives 818

ABBREVIATIONS
DIACRITICAL MARKS AND NOTATION
Before converting to LocoScript, I used various conventions, given below, to represent diacritical marks. Each symbol (except ') occurred after the letter it referred to. I have now converted these and all mathematical conventions into correct symbols, so far as possible, but I may have missed some, so I am keeping this information for the present.

Common entries using such marks are given later in this section and only the abbreviated or simplified form is used later -- e.g. I use Problemes for Bachet's work rather than Problèmes. (Though this may change??)

Initially, I did not record all diacritical marks, so some may be missing though I have checked almost all items. I may omit diacritical marks which are very peculiar.

Transliterations of Arabic, Sanskrit, Chinese, etc. are often given in very different forms. See Smith, History, vol. 1, pp. xvii-xxii for a discussion of the problems. The use of ^ and ˉ seems interchangeable and I have used ^ when different versions use both ^ and ˉ , except when quoting a title or passage when I use the author's form. [Smith, following Suter, uses ^  for Arabic, but uses ˉ for Indian. Murray uses ˉ for both. Wieber uses ˉ for Arabic. Van der Linde uses ´ for Arabic. Datta & Singh use ^ for Indian.]

There are two breathing marks in Arabic -- ayn ‘ and alif/hamzah ’ -- but originally I didn't have two forms easily available, so both were represented by '. I have now converted almost all of these to ‘ and ’. These don't seem to be as distinct in the printing as on my screen.

French practice in accenting capitals is variable and titles are often in capitals, so expected marks may be missing. Also, older printing may differ from modern usage -- e.g. I have seen: Liège and Liége; Problèmes, Problêmes and Problémes. When available, I have transcribed the material as printed without trying to insert marks, but many places insert the marks according to modern French spelling.

Greek and Cyrillic titles are now given in the original with an English transliteration (using the Amer. Math. Soc. transliteration for Cyrillic).

I usually ignore the older usage of v for u and i for j, so that I give mathematiqve as mathematique and xiij as xiii.

I used a1, a2, ..., ai, etc. for subscripted variables, though I also sometimes used a(1), a(2), ..., a(i), etc. Superscripts or exponents were indicated by use of ^, e.g. 2^3 is 8. These have been converted to ordinary sub- and superscript usage, but ^ may be used when the superscript is complicated -- e.g. for 2^a_i or 9^(9⁹).

Greek letters were generally spelled out in capitals or marked with square brackets, e.g. PI, [pi], PHI, but these have probably all been converted.

My word processor does not produce binomial coefficients easily, so I use BC(n, k) for n!/k!(n k)!

Many problems have solutions which are sets of fractions with the same denominator and I abbreviate a/z, b/z, c/z as (a, b, c)/z. Notations for particular problems are explained at the beginning of the topic.

Rather than attempting to italicise letters used as symbols, I generally set them off by double-spaces on each side -- see examples above. Other mathematical notations may be improvised as necessary and should be obvious.

Recall that the symbols below occurred after the letter they referred to, except for ' .

" denoted umlaut or diaeresis in general, e.g.: ä, ë, ï, ö, ü.

/ was used after a letter for accent acute, ́, after l for ł in Polish, and after o for ø in Scandinavian.

\ denoted accent grave, ̀.

^ denoted the circumflex, ^, in Czech, etc.; the overbar (macron) ˉ or ^ for a long vowel in Sanskrit, Hindu, etc.; and the overbar used to indicate omission in medieval MSS.

@ denoted the cedilla (French ç and Arabic ş) and the ogonek or Polish hook (Polish ą).

. denoted the underdot in ḥ, ḳ, ṇ, ṛ, ṣ, ṭ, in Sanskrit, Hindu, Arabic. These are sometimes written with a following h -- e.g. k may also be written kh and I may sometimes have used this. (It is difficult to search for ḥ. , etc., so not all of these may be converted.) This mark vanishes when converted to WordPerfect.

* denoted the overdot for ġ, ṁ, ṅ, in Sanskrit, Hindu, Arabic. This vanishes over m and n in WordPerfect.

~ denoted the Spanish tilde ~ and the caron or hachek ˇ, in ğ, š. The breve is a curved version, ˘, of the same symbol and is essentially indistinguishable from the caron. It occurs in Russian й, which is translitereated as ĭ.

_ denoted the underbar in ḏ , j, ṯ (I cannot find a j with an underbar in Arial). This mark vanishes in WordPerfect.

' denotes breathing marks in Arabic, etc. There are actually two forms of this -- ayn ’ and alif/hamzah ‘ -- but I didn't have two forms easily available and originally entered both as apostrophe ' . These normally occur between letters and I placed the ' in the same space. I have converted most of these.

Commonly occurring words with diacritical marks are: Académie, arithmétique, bibliothèque, Birkhäuser, café, carré, école, Erdös, für, géomètre, géométrie, Göttingen, Hanoï -- in French only,  ième, littéraire, mathématique, mémoire, ménage, misère, Möbius, moiré, numérique, Pétersbourg, probabilités, problème (I have seen problêmes??), Rätsel, récréation, Sändig, siècle, société, Thébault, théorie, über, umfüllung.
I have used ?? to indicate uncertainty and points where further work needs to be done. The following symbols after ?? indicate the action to be done.

* check for diacritical marks, etc.

NX no Xerox or other copy

NYS not yet seen

NYR not yet read

o/o on order

SP check spelling

Other comments may be given.

ABBREVIATIONS OF JOURNALS AND SERIES. See: AMM, CFF, CM, CMJ, Family Friend, G&P, G&PJ, HM, JRM, MG, MiS, MM, MS, MTg, MTr, M500, OPM, RMM, SA, SM, SSM in Common References below.
ABBREVIATIONS OF PUBLISHERS. See: AMS, C&W, CUP, Loeb Classical Library, MA, MAA, NCTM, OUP in Common References below.
ABBREVIATIONS OF MONTHS. All months are given by their first three letters in English: Jan, Feb, ....
PUBLISHERS' LOCATIONS. The following publisher's locations will not be cited each time. Other examples may occur and can be found in the file PUBLOC.
AMS (American Mathematical Society), Providence, Rhode Island, USA.

Chelsea Publishing, NY, USA.

CUP (Cambridge University Press), Cambridge, UK.

Dover, NY, USA.

Freeman, San Francisco, then NY, USA.

Harvard University Press, Cambridge, Massachusetts, USA.

MA (Mathematical Association), Leicester, UK.

MAA (Mathematical Association of America), Washington, DC, USA.

NCTM (National Council of Teachers of Mathematics), Reston, Virginia, USA.

Nelson, London, UK.

OUP (Oxford University Press), Oxford, UK (and also NY, USA).

Penguin, Harmondsworth, UK.

Simon & Schuster, NY, USA.


COMMON REFERENCES.
NOTES. When referring to items below, I will usually include the earliest reasonable date, even though the citation may be to a much later edition. For example, I would say "Canterbury Puzzles, 1907", even though I am citing problem numbers or pages from the 1958 Dover reprint of the 1919 edition. Sometimes the earlier editions are hard to come by and I have sometimes found that the earlier edition has different pagination -- in that case I will (eventually) make the necessary changes.

Edition information in parentheses indicates items or editions that I have not seen, though I don't always do this when the later version is a reprint or facsimile.

Abbaco. See: Pseudo-dell'Abbaco.

Abbot Albert. Abbot Albert von Stade. Annales Stadenses. c1240. Ed. by J. M. Lappenberg. In: Monumenta Germaniae Historica, ed. G. H. Pertz, Scriptorum t. XVI, Imp. Bibliopolii Aulici Hahniani, Hannover, 1859 (= Hiersemann, Leipzig, 1925), pp. 271 359. (There are 13 recreational problems on pp. 332 335.) [Vogel, on p. 22 of his edition of the Columbia Algorism, dates this as 1179, but Tropfke gives 1240, which is more in line with Lappenberg's notes on variants of the text. The material of interest, and several other miscellaneous sections, is inserted at the year 1152 of the Annales, so perhaps Vogel was misled by this.] I have prepared an annotated translation of this: The problems of Abbot Albert (c1240). I have numbered the problems and will cite this problem number.

Abraham. R. M. Abraham. Diversions and Pastimes. Constable, London, 1933 = Dover, 1964 (slightly amended and with different pagination, later retitled: Tricks and Amusements with Coins, Cards, String, Paper and Matches). I will cite the Constable pages (and the Dover pages in parentheses).

Ackermann. Alfred S. E. Ackermann. Scientific Paradoxes and Problems and Their Solutions. The Old Westminster Press, London, 1925.

D. Adams. New Arithmetic. 1835.

Daniel Adams (1773-1864). ADAMS NEW ARITHMETIC. Arithmetic, in which the principles of operating by numbers are analytically explained, and synthetically applied; thus combining the advantages to be derived both from the inductive and synthetic mode of instructing: The whole made familiar by a great variety of useful and interesting examples, calculated at once to engage the pupil in the study, and to give him a full knowledge of figures in their application to all the practical purposes of life. Designed for the use of schools and academies in the United States. J. Prentiss, Keene, New Hampshire, 1836, boarded. 1-262 pp + 2pp publisher's ads, apparently inserted backward. [Halwas 1-6 lists 1st ed as 1835, then has 1837, 1838, 1839, 1842, c1850.] This is a reworking of The Scholar's Arithmetic of 1801.

D. Adams. Scholar's Arithmetic. 1801.

Daniel Adams (1773-1864). The Scholar's Arithmetic; or, Federal Accountant: Containing. I. Common arithmetic, .... II. Examples and Answers with Blank Spaces, .... III. To each Rule, a Supplement, comprehending, 1. Questions .... 2. Exercises. IV. Federal Money, .... V. Interest cast in Federal Money, .... VI. Demonstration by engravings .... VII. Forms of Notes, .... The Whole in a Form and Method altogether New, for the Ease of the Master and the greater Progress of the Scholar. Adams & Wilder, Leominster, Massachusetts, 1801; 2nd ed, 1802. 3rd ed ??. 4th ed, by Prentiss, 1807; 6th ed, 1810; 10th ed, 1816; Stereotype Edition, Revised and Corrected, with Additions, 1819, 1820, 1824; John Prentiss, Keene, New Hampshire, 1825. [Halwas 8-14.] I have the 1825, whose Preface is for the 10th ed of 1816, so is probably identical to that ed. The Preface says he has now made some revisions. The only change of interest to us is that he has added answers to some problems. So I will cite this as 1801 though I will be giving pages from the 1825 ed. The book was thoroughly reworked as Adams New Arithmetic, 1835.

M. Adams. Indoor Games. 1912.

Morley Adams, ed. The Boy's Own Book of Indoor Games and Recreations. "The Boy's Own Paper" Office, London, 1912; 2nd ptg, The Religious Tract Society, London (same address), 1913. [This is a major revision of: G. A. Hutchison, ed.; Indoor Games and Recreations; The Boy's Own Bookshelf; New ed., Religious Tract Society, London, 1891 (possibly earlier) -- see 5.A.]

M. Adams. Puzzle Book. 1939.

Morley Adams. The Morley Adams Puzzle Book. Faber & Faber, London, 1939.

M. Adams. Puzzles That Everyone Can Do. 1931.

Morley Adams. Puzzles that Everyone Can Do. Grant Richards, London, 1931, boarded.

AGM. Abhandlungen zur Geschichte der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Begründet von Moritz Cantor. Teubner, Leipzig. The first ten volumes were Supplements to Zeitschrift für Math. u. Physik, had a slightly different title and are often bound in with the journal volume.

Ahrens, Wilhelm Ernst Martin Georg (1872-1927). See: A&N, MUS, 3.B, 7.N.

al Karkhi. Aboû Beqr Mohammed Ben Alhaçen Alkarkhî [= al Karagi = al Karajī]. Untitled MS called Kitāb al-Fakhrī (or just Alfakhrî) (The Book Dedicated to Fakhr al-Din). c1010. MS 952, Supp. Arabe de la Bibliothèque Impériale, Paris. Edited into French by Franz Woepcke as: Extrait du Fakhrî. L'Imprimerie Impériale, Paris, 1853; reprinted by Georg Olms Verlag, Hildesheim, 1982. My page citations will be to Woepcke. Woepcke often refers to Diophantos, but his numbering gets ahead of Heath's.

Alberti. 1747. Giuseppe Antonio (or Giusepp-Antonio) Alberti (1715-1768). I Giochi Numerici Fatti Arcani Palesati da Giuseppe Antonio Alberti. Bartolomeo Borghi, Bologna, 1747, 1749. Venice, 1780, 1788(?). 4th ed., adornata di figure, Giuseppe Orlandelli for Francesco di Niccolo' Pezzana, Venice, 1795 (reprinted: Arnaud, Florence, 1979), 1813. Adornata di 16 figure, Michele Morelli, Naples, 1814. As: Li Giuochi Numerici Manifestati, Edizione adorna di Figure in rame, Giuseppe Molinari, Venice, 1815.

The editions have almost identical content, but different paginations. I have compared several editions and seen little difference. The 1747 ed. has a dedication which is dropped in the 2nd ed. which also omits the last paragraph of the Prefazione. I only saw one other point where a few words were changed. I will give pages of 1747 (followed by 1795 in parenthesis). Much of Alberti, including almost all the material of interest to us and many of the plates, is translated from vol. 4 of the 1723 ed. of Ozanam.

(Serge Plantureux's 1993 catalogue describes a 1747-1749 ed. with Appendice al Trattato de' Giochi Numerici (1749, 72 pp) & Osservazioni all'Appendice de' Giochi Numerici (38 pp), ??NYS. The copy in the Honeyman Collection had the Appendice. Christopher 3 has the Osservazioni. The Appendice is described by Riccardi as a severe criticism of Alberti, attributed to Giovanni Antonio Andrea Castelvetri and published by Lelio dall Volpe, Bologna, 1749. The Osservazioni are Alberti's response.)

Alcuin (c735-804).

Propositiones Alcuini doctoris Caroli Magni Imperatoris ad acuendos juvenes. 9C.

IN: B. Flacci Albini seu Alcuini, Abbatis et Caroli Magni Imperatoris Magistri. Opera Omnia: Operum pars octava: Opera dubia. Ed. D. Frobenius, Ratisbon, 1777, Tomus secundus, volumen secundum, pp. 440 448. ??NYS. Revised and republished by J. P. Migne as: Patrologiae Cursus Completus: Patrologiae Latinae, Tomus 101, Paris, 1863, columns 1143 1160.

A different version appears in: Venerabilis Bedae, Anglo Saxonis Presbyteri. Opera Omnia: Pars Prima, Sectio II -- Dubia et Spuria: De Arithmeticus propositionibus. Tomus 1, Basel, 1563. (Rara, 131, says there were earlier editions: Paris, 1521 (part), 1544 1545 (all), 1554, all ??NYS.) Revised and republished by J. P. Migne as: Patrologiae Cursus Completus: Patrologiae Latinae, Tomus 90, Paris, 1904, columns 665 672. Incipiunt aliae propositiones ad acuendos juvenes is col. 667 672. A version of this occurs in Ens' Thaumaturgus Mathematicus of 1636 -- cf under Etten.

The Alcuin has 53 numbered problems with answers. The Bede has 3 extra problems, but the problems are not numbered, there are only 31 1/2 answers and there are several transcription errors. The editor has used the Bede to rectify the Alcuin.

There is a recent critical edition of the text by Folkerts -- Die älteste mathematische Aufgabensammlung in lateinischer Spräche: Die Alkuin zugeschriebenen Propositiones ad Acuendos Iuvenes; Denkschriften der Österreichischen Akademie der Wissenschaften, Mathematische naturwissenschaftliche Klasse 116:6 (1978) 13 80. (Also separately published by Springer, Vienna, 1978. The critical part is somewhat revised as: Die Alkuin zugeschriebenen "Propositiones ad Acuendos Iuvenes"; IN: Science in Western and Eastern Civilization in Carolingian Times, ed. by P. L. Butzer & D. Lohrmann; Birkhäuser, Basel, 1993, pp. 273-281.) He finds that the earliest text is late 9C and is quite close to the first edition cited above. He uses the same numbers for the problems as above and numbers the extra Bede problems as 11a, 11b, 33a. I use Folkerts for the numbering and the titles of problems.

John Hadley kindly translated Alcuin for me some years ago and made some amendments when Folkerts' edition appeared. I annotated it and it appeared as: Problems to Sharpen the Young, MG 76 (No. 475) (Mar 1992) 102-126. A slightly corrected and updated edition, containing some material omitted from the MG version, is available as Technical Report SBU-CISM-95-18, School of Computing, Information Systems, and Mathematics, South Bank University, Oct 1995, 28pp.

Menso Folkerts and Helmuth Gericke have produced a German edition: Die Alkuin zugeschriebenen Propositiones ad Acuendos Juvenes (Aufgabe zur Schärfung des Geistes der Jugend); IN: Science in Western and Eastern Civilization in Carolingian Times, ed. by P. L. Butzer & D. Lohrmann; Birkhäuser, Basel, 1993, pp. 283-362.

See also: David Singmaster. The history of some of Alcuin's Propositiones. IN: Charlemagne and his Heritage 1200 Years of Civilization and Science in Europe: Vol. 2 Mathematical Arts; ed. by P. L. Butzer, H. Th. Jongen & W. Oberschelp; Brepols, Turnhout, 1998, pp. 11 29.
AM. 1917. H. E. Dudeney. Amusements in Mathematics. Nelson, 1917. (There were reprintings in 1919, 1920, 1924, 1925, 1927, 1928, 1930, 1932, 1935, 1938, 1939, 1941, 1943, 1946, 1947, 1949, 1951, but it seems that the date wasn't given before 1941?) = Dover, 1958.

AMM. American Mathematical Monthly.

AMS. American Mathematical Society.

Les Amusemens. 1749.

Les Amusemens Mathématiques Precedés Des Elémens d'Arithmétique, d'Algébre & de Géométrie nécessaires pour l'intelligence des Problêmes. André Joseph Panckoucke, Lille, 1749. Often listed with Panckoucke as author (e.g. by the NUC, the BNC and Poggendorff), but the book gives no such indication. Sometimes spelled Amusements. There were 1769 and 1799 editions.

Apianus. Kauffmanss Rechnung. 1527.

Petrus Apianus (= Peter Apian or Bienewitz or Bennewitz) (1495 1552). Eyn Newe Unnd wolgegründte underweysung aller Kauffmanss Rechnung in dreyen Büchern / mit schönen Regeln uň [NOTE: ň denotes an n with an overbar.] fragstucken begriffen. Sunderlich was fortl unnd behendigkait in der Welschē Practica uň Tolletn gebraucht wirdt / des gleychen fürmalss wider in Teützscher noch in Welscher sprach nie gedrückt. durch Petrum Apianū von Leyssnick / d Astronomei zů Ingolstat Ordinariū / verfertiget. Georgius Apianus, Ingolstadt, (1527), facsimile, with the TP of the 1544 ed. and 2pp of publication details added at the end, Polygon-Verlag, Buxheim-Eichstätt, 1995, with 8pp commentary leaflet by Wolfgang Kaunzner. (The TP of this has the first known printed version of Pascal's Triangle.) Smith, Rara, pp. 155-157. (The d is an odd symbol, a bit like a δ or an 8, which is used regularly for der both as a single word and as the ending of a word, e.g. and for ander.) Smith notes that Apianus follows Rudolff (1526) very closely.

AR. c1450. Frater Friedrich Gerhart (attrib.). Latin & German MSS, c1450, known as Algorismus Ratisbonensis. Transcribed and edited from 6 MSS by Kurt Vogel as: Die Practica des Algorismus Ratisbonensis; C. H. Beck'sche Verlagsbuchhandlung, Munich, 1954. (Kindly sent by Prof. Vogel.) Vogel classifies the problems and gives general comments on the mathematics on pp. 155 189. He gives detailed historical notes on pp. 203 232. When appropriate, I will cite these pages before the specific problems. He says (on p. 206) that almost all of Munich 14684 (see below) is included in AR.