Sources in recreational mathematics

Bakhshali MS also has it - this was formerly dated c4C, but is more recently dated c7C. (The BLLD doesn't have G. R. Kaye's Indian Mathematics, Calcutta & Simla, 1915.) Libbrecht quotes a number of nonsensical commentaries on Chang. I have examples from Alcuin, Mahavira, Sridhara, Abu Kamil, Bhaskara II, Fibonacci, etc.

7.P.5. Selling Different Amounts 'At Same Prices' Yielding the Same. Mahavira (c850) has 6 examples, but the results are not clearly found. Sridhara (c900) and Bhaskara II (c1150) have comprehensible examples. These all specify the relative amounts, by giving the capitals, and one price. The European versions specify the amounts, and sometimes the yield. The first European versions are Fibonacci (1202), Munich 14684 & dell'Abbaco. Ozanam (1694?) seems to be the first to give a general approach?? Ozanam Riddle (1840) finds all 10 integral solutions to the case where the amounts are 10, 25, 30. I have analysed both the Indian and the western versions and found a new simple rule for the number of solutions in the western case.

7.P.7. Robbing and Restoring. First appears in al-Karkhi, c1010. This is said to be problem 5 in the contest between Fibonacci and John of Palermo in 1225. I haven't seen a description of this contest, but the problem is extensively treated in Fibonacci.

7.Q. Blind Abbess and Her Nuns. That is, how to arrange objects along edges of a 3 x 3 square so edge sums are constant. This is in van Etten (1653). A simpler type of problem is in De Viribus. Murray's History of Chess says this appears in At-Tilimsani, c1370 - can anyone provide details?

7.Q.1. Rearrangement on a Cross. This is in De Viribus, c1500, then in Les Amusemens, 1749, then numerous 19C examples. I have seen only one example of rearrangement on a Y, from 1912.

7.Q.2. Rearrange a Cross of Six to Make Two Lines of Four. New section   my examples are 1749, 1921, 1930s?, 1939.

7.R. "If I Had One from You, I'd Have Twice You". This is in Heiberg & Menge's edition of Euclidis Opera. It is also in the Greek Anthology (c510). Diophantos (c250) gives a general formulation of this and a version for 3 and 4 people. Fibonacci gives many versions, including one which is inconsistent which he says has no solutions.

7.R.1. Men Find a Purse and 'Bloom' of Thymarides. Diophantos has similar problems. Iamblichus and Mahavira have several examples. Fibonacci has many examples, including some with negative solutions. He is the first to consider the case where the i-th says he'd have a_i times the i+1st person.

7.R.2. "If I Had 1/3 of Your Money, I could buy the Horse". There are several variants of this in the Chiu Chang Suan Ching. Diophantos gives a general formulation for 3 and 4 people.

7.R.3. Sisters and Brothers. I have this only back to 1924?

7.R.4. "If I Sold Your Eggs at My Price, I'd Get ...." I find this in Simpson's Algebra, 1745, then in 1940.

7.S. Dilution and Mixing Problems. I have a reference to Recorde, The Grounde of Artes, 1579 ed., but it is not in my facsimile of the 1542 ed., though perhaps this was done from an imperfect version??

7.S.1. Dishonest Butler Drinking Some and Replacing with Water. There is a simple version in the Rhind Papyrus. There is a version in the Bakhshali MS. Cardan (1539) has a version, as do Tartaglia, Quesiti, ..., (1546); Buteo (1559) and Trenchant (1566, NYS).

7.S.2. Water in Wine versus Wine in Water. What is the origin of this problem? It was a favourite of Lewis Carroll. I've found it in Ball (MRE, 3rd ed., 1896) and Pearson (1907). I also find more direct related versions c1900.

7.T. Four Number Game. That is: (a, b, c, d) goes to (a b, b-c, c d, d a). The first reference seems to be Ciamberlini and Marengoni (1937).

7.U. Frobenius's Postage Stamp Problem. That is, find the largest value not obtainable from a set of values. I can't locate anything that relates this to Frobenius until 1962. Various authors cite Sylvester (1884), NYS, for the two value case, but Dickson is surprisingly obscure on this subject. The second reference on the subject appears to be a two part paper of A. Brauer in 1942 & 1954.

7.V. x^y = y^x and Iterated Exponentials. Archibald (AMM 28 (1921) 141 143) and Knoebel (AMM 88 (1981) 235 252) survey the history. Any other sources than those cited?

7.W. Card Piling Over a Cliff. This seems to first appear as Problem 3009, AMM 30 (1923) 76 and the Otto Dunkel Problem Book lists this as unsolved in 1957 and I believe it is still unsolved, though at least one author has claimed that the harmonic series gives the maximum overhang. Ramsey's Statics (1934, pp 47 48, NYS - I have seen the 2nd ed. of 1941) gives a related problem referred to a Tripos. The problem appears several times in the 1950s. Gamow & Stern give it in 1958. MG (Oct 1980) discusses the problem, but their earliest reference is Barnard's column in The Observer (1962).

7.X. How Old Is Ann? (Or Mary??). The name varies, though the problem is usually the same. Loyd refers to How Old is Ann and gives a How Old is Mary in the Cyclopedia. A. C. White, Sam Loyd and His Chess Problems (1913), says Loyd invented How Old Was Mary. G. G. Bain, "An Interview with Sam Loyd" (1907), refers to How Old Was Mary. Dudeney, AM, prob. 51, attributes it to Loyd. But Clark Kinnaird (1946) says it is due to Loyd Jr., and made him famous. He goes on to say that such problems go back to 1789, but gives no references. W. R. Ransom says it is much older than the early 1900s. I suspect these are referring to the following type of problem: X is now a times as old as Y; after b years, X  is c times as old as Y, which I have traced back to 1745..

7.Y. Combining Amounts and Prices Incorrectly. Also called the appleseller's problem. This is in Alcuin, ibn Ezra, Fibonacci, etc.

7.Y.1. Reversal of Averages Paradox. Sometimes known as Simpson's Paradox. I have an example from 1944 and a reference to 1934, then Simpson's paper of 1951.

7.Y.2. Unfair Division. Farmer is to give 2/5 of his yield to the landlord, but the farmer uses 45 bushels of the harvest before they can divide it. He then proposes to give 18 bushels to the landlord and then divide up the rest. Is this correct? I have three essentially identical versions from 1857-1860. Does this problem ever occur anywhere else?

7.Z. Missing Dollar Paradox. This is the one where one counts in different ways and comes out with $27 and $28. The earliest I have is 1939, but a related form confusing the amount withdrawn from a bank goes back to 1933 and there is an ancestral from in 1751.

7.AC.1. Cryptarithms: SEND + MORE = MONEY, etc. Shortz reports an example from 1864, NYR. Versions are in Loyd's Cyclopedia. SEND + ... is in Dudeney's column in Strand Mag. (1924).

7.AC.2. Skeleton Arithmetic: Solitary Seven, Etc. Berwick's seven sevens division appeared in 1906. Ackermann says Berwick was inspired by similar problems in Workman's Tutorial Arithmetic. Fred Schuh (Nieuw Tijds. voor Wisk. 8 (1920 21) 64) gives a skeleton division with no figures shown, but there is no answer in this or the succeeding volumes. The problem and answer do appear in AMM (1921 & 1922) and in Schuh's book (1943).

In 1992, an anonymous postcard to The Science Correspondent, "The Glasgow Herald", 8 May 1963, was found in Prof. Lenihan's copy of Gardner's More Mathematical Puzzles and Diversions giving the full skeleton of 1062 / 16 = 66.375 with no digits specified - the solution is unique. I thought this might be from Tom O'Beirne, but Mrs O'Beirne says it is not in his handwriting.

7.AC.3. Pan-digital Sums. The earliest versions are tricks: 15 + 36 + 47  =  98 + 2  =  100. These start in 1857. Dudeney (1897-98) has examples like 235 + 746 = 981.

7.AC.4. Pan Digital Products. Loyd (1897) has examples like 3907 * 4 = 15628. Dudeney (1902) has 2 * 78  =  4 * 39  =  156.

7.AC.5. Pan Digital Fractions. I have versions from Pearson (1907).

7.AD. Selling, Buying and Selling the Same Item. A version is in Loyd's Cyclopedia, but he avoids giving an answer.

7.AD.1. Pawning Money. Carroll used to give this.

7.AE. Use of Counterfeit Bill. Versions begin in 1857.

7.AG. 2592. That is, 2592 = 2⁵ * 9². This is in Dudeney, AM (1917), Phillips (1937) and RMM (1962). Greenblatt, Mathematical Entertainments (1968), asserts that this was discovered by his officemate.

7.AH. Multiplying by Reversing. I have just noted a curious connection of this with 1089, because 9 * 1089 = 9801; 4 * 2178 = 8712 and 2178 = 2 * 1089!! This follows since 1089 = 1100 - 11 so that k * 1089  =  kk00   kk  =  k,k 1,9 k,10-k in base 10. From this we see that k*1089 is the reverse of (10-k)*1089. Now 10-k is a multiple of k for k = 1, 2, but we get some new types of solution for k = 3, 4, namely: 7 * 3267 = 3 * 7623; 6 * 4356 = 4 * 6534. I cannot see a proof that this gives all solutions of this problem.

7.AI. Impossible Exchange Rates. Phillips, Week End, 1932, describes two currencies, each valued at 90% of the other and cites New Statesman and Nation, late 1931, NYS. The Week End Book, 1924 [no relation to the Phillips book], says the US & Mexico each valued the other's currency at 29/30.

7.AJ. Multiplying by Shifting. E.g. find abc...mn such that nabc...m is twice as big. I've recently seen this when the second was 3/2 times the first, given by Bronowski in 1949. Dickson's first entry for any form of the problem is Hausted, 1878, but special cases occur in Babbage's MSS, c1820, and in 1854. A version is in Hoffmann (1893).

7.AK. Lazy Worker. This is in Al-Buzajani (NYS), al-Karkhi, At-Tabari (NYS), Fibonacci.

7.AL. If A is B, What is C? Fibonacci discusses the simplest form of this in some detail. More complex forms appear in Hoffmann (1893), Pearson (1907), etc.

7.AM. Crossnumber Puzzles. When do these originate? Dudeney gives examples in 1926 and 1932. I also have a 1927 version. Hubert Phillips gives three simple ones in Brush (1936). The 'Dog's Mead Puzzle' or 'Little Pigley' or 'Little Pigsby' has various dates involved in it   I have seen 1935, 1936, 1939 and an attribution to Michael N. Dorey, but my earliest source is 1940 and I have no reference to an original location.

7.AN. Three Odds Make an Even, etc. Alcuin describes this as a fable. Numerous forms appear from 1694 onward

7.AP. Knowing Sum vs Knowing Product. I have a version from 1940, but then 1987.

7.AQ. Numbers in Alphabetic Order. When does this first appear? I learned it in college, c1957.

7.AR. 1089. The earliest version seems to be 1890, but all examples used money until 1898.

7.AS. Cigarette Butts. New section - I have just realised that if b butts are needed to make a smoke, then B butts produce [(B-1)/(b-1)] smokes.

7.AT. Bookworm's Distance. New section - me earliest version is 1914.

7.AV. How Long to Strike Twelve? New section - my earliest version is 1927.

8. PROBABILITY RECREATIONS

8.B. Birthday Paradox. Feller cites von Mises (1938 39), but von Mises gets the expected number of repetitions, not the usual result. Ball, MRE (11th ed., 1939) cites Davenport, but Coxeter says that Davenport did not publish anything on it and others, including Mrs Davenport, say that Davenport explicitly denied originality for it. However, George Tyson, who was a student at Manchester a year ahead of Davenport, says Davenport showed it to him in 1927 and with an attitude that made Tyson believe it was due to Davenport.

8.C. Probability That a Triangle is Acute. Lewis Carroll gives a version in Pillow Problems (1893). Sylvester (1865) attributes a version to Woolhouse, NYS.

8.D. Attempts to Modify Boy/Girl Ratio. Laplace discusses this in his Essai Philosophique, c1819.

8.E. St. Petersburg Paradox. Euler says this originates in a letter from N. Bernoulli to Montmort on 9 Sep 1713. I have a reference to Montmort's publishing it in his book, but I had trouble finding it. Tissandier (1880?) and Dan Pedoe give vague references to D. Bernoulli publishing it in the Transactions [sic] of the St. Petersburg Academy or other unknown journals and to Buffon playing the game 2048 times. I recall hearing something on this in one of Tom Stoppard's plays, but I can't find it in Rosencrantz & Guildenstern are Dead nor in Jumpers?

8.F. Problem of Points. This is in Pacioli's Summa and Tartaglia. Ore says he has seen it in Italian MSS back to 1380 and believes it may be Arabic.

8.G. Probability that Three Lengths Form a Triangle. Fourrey, Curiosities Geom. (1907), cites Lemoine (1872 73). There were a number of following papers, some of which I have seen.

8.H. Probability Paradoxes. I have examples deriving from Bertrand (1889), including vague reference to Weaver and Whitehead.

8.I. Taking the Next Train. My first example is 1940.

9. LOGICAL RECREATIONS
9.A. All Cretans are Liars, Etc. Bochenski's History_of_Formal_Logic'>Ancient Formal Logic and History of Formal Logic give details of the classical versions. Cervantes gives the 'I will be hanged' version in Don Quixote (1605). What are the origins of other versions, e.g. the barber? Whitehead and Russell discuss several of the paradoxes in Principia Math, vol. 1, pp. 63 64. I have a reference to Descremps' Les Petits Aventures de Jerome Sharp, Brussels, 1769 for the sentinel paradox. Can anyone help with this?

An oriental version is the following: The Chief Warden swallows an elixir of immortality which he was supposed to convey to the Prince. The Prince orders the Warden's execution, but the Warden argues that if the execution succeeds, then the elixir was false and he is innocent of crime. The Prince pardons him. There are two versions which may be -2C.

9.B. Smith Jones Robinson. This is in Dudeney's Puzzles and Curious Problems, 1932. Gardner's note in 536 says Dudeney invented it. Schaaf recalls it appearing about 1926, so it probably was in Dudeney's Strand column.

9.C. Forty Unfaithful Wives. This occurs in Gamow and Stern, 'communicated by V. Ambarzuminian'. I now realise that it is a form of the 'Spots on foreheads' problem (9.D).

9.D. Spots on Foreheads. This appears in Kraitchik's Mathematical Recreations, but not in his earlier Math. des Jeux. There is a predecessor of the spots problem which has several persons in a railway carriage getting smudged in a tunnel and then laughing at each other, until they realise. I find this in the AMM 42 (1935) 256 and the solution says it was originated by 'Dr. Church of Princeton' and refers to a three person version in SSM 35 (1935) 429. Phillips, Week End, 1932, gives the problem of two boys, one getting smudged and the other going to wash his own face. There is an ancestral version from 1903 (or 1883). Phillips, Brush, 1936, gives the three spots problem. A book of Arabic stories gives the three spots problem with the simpler solution based on symmetry. The author says he heard it in Palestine before 1948.

Littlewood (1953) seems to be the first to note that the problem can be extended to n people, i.e. section 9.C.

9.E. Strange Families. Widower and son marry widow and daughter; widows and sons marry; are in the Bede version of Alcuin. Ripley's Believe It Or Not, 6th Series, reports a widower and son marrying a girl and her widowed mother. Bombaugh (1905) reports two widowers marrying each other's daughter. More real examples would be welcome.

9.E.1. That Man's Father is My Father's Son. New section   my earliest example is 1883, but similar problems occur back to 1844.

9.F. Unexpected Hanging. N. Falletta (1983) relates that this occurred as a Swedish wartime radio announcement and the Lennart Ekbom noticed its paradoxical nature and discussed it with students. Did he ever publish anything on it??

9.G. Truthtellers and Liars. This is in Kraitchik's Mathematical Recreations (1942), but not in Math. des Jeux (1930). Phillips gives problems with truthtellers, liars and alternators in 1937 and 1945. In the mid to late 19C, there were complex problems with several persons with known probabilities of lying.

9.H. Prisoner's Dilemma. This is attributed to Merrill M. Flood and Melvin Dresher, c1950, but I have no reference. When was it first published? A. W. Tucker is said to have formalised it or invented it. I find anticipations of it in Babbage (1832) and F. W. Taylor (1911).

9.J. Fallen Signpost. I have this from 1927, where it is considered one of the best 100 puzzles.

10. PHYSICAL RECREATIONS
10.A. Overtaking and Meeting Problems. There are many versions of these. Smith, History, II, gives some. There are several in the Chiu Chang Suan Shu (c 1C), some of considerable complexity. Several versions occur in the Bakhshali MS.

10.A.2. Number of Buses Met. New section. Mittenzwey, 1879, has a version. Proctor does it in 1883. Carroll (1885) gives some complex examples. A version is in Pearson (1907).

10.A.4. The Early Commuter. This is in a collection of best puzzles in 1943.

10.B. Fly Between Trains. This appears, as moderately well known, in a letter of G. H. Hardy in 1924. It appears in Dudeney, PCP (1932), and he has several versions in Strand Mag. for 1919, 1924? and 1925. In the MAA film John von Neumann (1966), Eugene Wigner relates this as being posed by Max Born to von Neumann, involving a swallow flying between two bicyclists. Wigner says it was popular in the 1920s. H&S (1927) says it is 'modern'. However, I have found it in Laisant (1906).

10.D. Mirror Reversal Paradox. That is, why does it reverse right and left but not up and down? This interested Carroll, who used to pose it to his girl friends. This is in Dudeney, PCP (1932). Gardner, Ambidextrous Universe, doesn't give an origin.

10.E.2. One Wheel Rolling Around Another. My earliest puzzle version in 1907, but there must be earlier examples since the phenomenon has been known in astronomy for many centuries.

10.E.3. Hunter and Squirrel. I have material from 1883.

10.F. Floating Body Problems. Gamow & Stern give the barge full of iron in a lock. Dudeney gives it in Strand Mag. (1931).

What happens to a glass full of ice water as the ice melts   I have found this in the Daily Mail (3 Feb 1905).

Ackermann has a barge going over a canal bridge   is there more weight on the bridge? I may add the question of the heavy lorry load of birds   can anyone supply early references?

Perelman has a version: which is heavier, a bucket full of water or a bucket full to the same level with a piece of wood floating in it? Again, I don't know the original dates of Perelman's works.

10.G. Motion in a Current or Wind. Loyd's Cyclopedia has an against and with the wind problem. Sanford quotes Nicholas Pike (1788) and says it is the earliest version of the problem. There are many versions of this problem and I have few references.

What is the origin of up & down a stream versus across & back? Did it first occur as a recreational or arithmetic problem or did it first arise in studying light and the ether?

10.H. Snail Climbing Out of Well. Kraitchik, Math. des Jeux, says this is a Hindu problem, but the Hindu versions don't really involve alternation of rates nor the end effect. Mahavira has a snake crawling into a hole while growing and other problems of multiple rates, but doesn't deal with the end effect due to alternation. Dell'Abbaco and the Columbia Algorism have the first proper versions. Chuquet has several proper versions. It is in Calandri's Arithmetic of 1491, but he doesn't note the end effect.

10.I. Two Men and a Bike, etc. Loyd gives a version with three men and a tandem. Dudeney, 1932, has 12 soldiers and a taxi which can take 4 of them. I have a 1939 version of two men and a bike. I have now found this in Laisant, 1906. I have developed a symmetric approach which solves the problem neatly.

10.J. Resistor Networks. AMM solution E2620 (1978) says Coxeter, Regular Polytopes, was first to give the cubical case, but I can't find it in the 2nd ed. This problem, E2620, appears to be the first to consider the regular polyhedra, though it only does between furthest points. It also says Gardner (SA Dec 1958 = 2nd Book, p. 22) gives the other polyhedral cases, but Gardner only considers the cube and cites Trigg, MM (Nov Dec 1960) who cites Brooks & Poyser, Magnetism and Electricity, 1920, NYS. Any idea of the source of the infinite square lattice problem? It appears in Eureka (1951) where the resistances from (0,0) to (0,1) and (1,1) are wanted. Two solvers of SIAM Review prob. 79 16, 21 (1979) 559, obtained the resistance between points distance i apart in the n dimensional cubical lattice, in terms of an n fold integral. Van der Pol & Bremmer's book on the Laplace transform obtains a general solution.

10.K. Problem of the Date Line. Oresme proposed this c1355. This is in an almanac of 1493, then in Cardan (1539).

10.L. Falling Down a Hole Through the Earth. Ozanam (1725) discusses this. Ozanam Riddle (1840) computes the time to reach the centre. Gardner's Annotated Alice says this problem interested Plutarch, Bacon and Voltaire (references??) and that it was resolved by Galileo. I have recently seen an argument that g increases with depth for a while due to the non uniform density of the earth and this is said to be a late 19C result.

10.M. Celts = Rattlebacks. I have just seen a paper by G. T. Walker (Proc. Camb. Phil. Soc. 8 (1892/5) 305 306) and he refers to 'celts' as though the term and the object were well known. Celt was a common term for a hand axe or chisel by early 19C, but when did it get associated with the rattleback?

10.M.1. Tippee Tops. This was patented in Germany in 1891, but appears to have been a fad in 1953 - can anyone provide information on this?

10.N. Ship's Ladder in Rising Tide. Phillips gives this in Brush, 1936.

10.O. Erroneous Averaging of Rates. My earliest example is 1941, but this is closely related to the older 7.Y.

10.P. False Balance. Cardan has a version, 1550, and there are versions from 1582, 1624, 1694. Determining the true weight by weighing on each side is in Ozanam-Hutton (1803) and Jackson, 1821..

10.Q. Push a Bicycle Pedal. My first example is Pearson 1907).

10.R. Clock Problems. There are many 20C versions. My earliest version is Ozanam (1725), who asks when the hands meet.

10.W. Puzzle Vessels. I'd like details of ancient examples.

10.X. How Far Does a Phonograph Needle Travel? New Section - I only have a 1984 example.

11. TOPOLOGICAL RECREATIONS

11.A,B,C,D. Scissors on String, Two People Joined at Wrists, String Puzzles. These appear in Ozanam (1725), though the 1692 ed. and the 1708 English ed. do not have these plates. Alberti (1747) copies the plates. A number of these puzzles appear in the Catel and Bestelmeier Catalogues of 1790 & 1801 1803.

11.E. Loyd's Trick Pencil (on a short loop of string). Any idea of the date of this? Gardner (SA (Nov 1971) = Wheels, Chap. 12) says Loyd invented it for the New York Life Insurance Company. Loyd's account of this is given in the Delineator (April 1911). I have a 1910 description.

11.G. Trick Purses. There are examples in van Etten and Ozanam (1725).

11.H. Removing Waistcoat without Removing Coat. This is in Cassell's Book, ..., 1881.

11.H.1. Removing Loop from Arm. I have this from 1850, then 1930s?

11.I. Loop Puzzles. Cardan has two versions of the Alliance or Victoria puzzle.

11.J. Möbius Strip. I haven't yet got references to Möbius and Listing. My earliest usage as an amusement or puzzle is Tissandier (1888). I am interested in modern applications - in ink ribbon cartridges; as a low inductance resistor; as molecules. Can anyone provide details?

11.K. Wire Puzzles. The first books that I have seen which show many of these are Hoffmann (1893) and The Boy's Own Book of Indoor Games and Recreations, c1912. These are supposed to have been popular in China, but I have no specific sources.

11.K.4. Puzzle Rings. Any information on the origins would be interesting.

11.L. Jacob's Ladder and Other Hinging Devices. See also 6.X. I have a picture of a c1850 example

11.M. Puzzle Boxes. I have examples from the Catel & Bestelmeier catalogues, 1790 & 1801.