Sources page biographical material

5.M. SIX PEOPLE AT A PARTY -- RAMSEY THEORY
In a group of six people, there is a triple who all know each other or there is a triple who are all strangers. I.e., the Ramsey number R(3,3) = 6. I will not go into the more complex aspects of this -- see Graham & Spencer for a survey.
P. Erdös & G. Szekeres. A combinatorial problem in geometry. Compositio Math. 2 (1935) 463 470. [= Paul Erdös; The Art of Counting; Ed. by Joel Spencer, MIT Press, 1973, pp. 5 12.] They prove that if n  BC(a+b-2, a-1), then any two colouring of K_a contains a monochromatic K_a or K_b.

William Lowell Putnam Examination, 1953, part I, problem 2. In: L. E. Bush; The William Lowell Putnam Mathematical Competition; AMM 60 (1953) 539-542. Reprinted in: A. M. Gleason, R. E. Greenwood & L. M. Kelly; The William Lowell Putnam Mathematical Competition Problems and Solutions -- 1938 1964; MAA, 1980; pp. 38 & 365 366. The classic six people at a party problem.

R. E. Greenwood & A. M. Gleason. Combinatorial relations and chromatic graphs. Canadian J. Math. 7 (1955) 1-7. Considers n = n(a,b,...) such that a two colouring of K_n contains a K_a of the first colour or a K_b of the second colour or .... Thus n(3,3) = 6. They find the bound and many other results of Erdös & Szekeres.

C. W. Bostwick, proposer; John Rainwater & J. D. Baum, solvers. Problem E1321 -- A gathering of six people. AMM 65 (1958) 446 & 66 (1959) 141 142.

Gamow & Stern. 1958. Diagonal strings. Pp. 93 95.

G. J. Simmons. The Game of Sim. JRM 2 (1969) 66.

M. Gardner. SA (Jan 1973) c= Knotted, chap. 9. Exposits Sim. Reports Simmons' result that it is second person (determined after his 1969 article above). The Addendum in Knotted reports that several people have shown that Sim on five points is a draw. Numerous references.

Ronald L. Graham & Joel H. Spencer. Ramsey theory. SA 263:1 (Jul 1990) 80 85. Popular survey of Ramsey theory beginning from Ramsey and Erdös & Szekeres.

5.N. JEEP OR EXPLORER'S PROBLEM
See Ball for some general discussion and notation.
Alcuin. 9C. Prob. 52: Propositio de homine patrefamilias. Wants to get 90 measures over a distance of 30 leagues. He is trying to get the most to the other side, so this is different than the 20C versions. Solution is confusing, but Folkerts rectifies a misprint and this makes it less confusing. Alcuin's camels only eat when loaded!! (Or else they perish when their carrying is done??) The camel take a load to a point 20 leagues away and leaves 10 there, then returns. This results in getting 20 to the destination.

The optimum solution is for the camel to make two return trips and a single trip to 10 leucas, so he will have consumed 30 measures and he has 60 measures to carry on. He now makes one return and a single trip of another 15 leucas, so he will have consumed another 30 measures, leaving 30 to carry on the last 5 leucas, so he reaches home with 25 measures.

Pacioli. De Viribus. c1500. Probs. 49 52. Agostini only describes Prob. 49 in some detail.

Ff. 94r - 95v. XLIX. (Capitolo) de doi aportare pome ch' piu navanza (Of two ways to transport as many apples as possible). = Peirani 134 135. One has 90 apples to transport 30 miles from Borgo [San Sepolcro] to Perosia [Perugia], but one eats one apple per mile and one can carry at most 30 apples. He carries 30 apples 20 miles and leaves 10 there and returns, without eating on the return trip! (So this = Alcuin.) Pacioli continues and gives the optimum solution!

F. 95v. L. C(apitolo). de .3. navi per .30. gabelle 90. mesure (Of three ships holding 90 measures, passing 30 customs points). Each ship has to pay one measure at each customs point. Mathematically the same as the previous.

F. 96r. LI. C(apitolo). de portar .100. perle .10. miglia lontano 10. per volta et ogni miglio lascia 1^a (To carry 100 pearls 10 miles, 10 at a time, leaving one every mile). = Peirani 136-137. Takes them 2 miles in ten trips, giving 80 there. Then takes them to the destination in 8 trips, getting 16 to the destination.

Ff. 96v - 97r. LII. C(apitolo). el medesimo con piu avanzo per altro modo (The same with more carried by another method). Continues the previous problem and takes them 5 miles in ten trips, giving 50 there. Then takes them to the destination in 5 trips, getting 25 to the destination.

[This is optimal for a single stop -- if one makes the stop at distance a, then one gets a(10-a) to the destination. One can make more stops, but this is restricted by the fact that pearls cannot be divided. Assuming that the amount of pearls accumulated at each depot is a multiple of ten, one can get 28 to the destination by using depots at 2 and 7 or 5 and 7. One can get 27 to the destination with depots at 4 and 9 or 5 and 9. These are all the ways one can put in two depots with integral multiples of 10 at each depot and none of these can be extended to three such depots. If the material being transported was a continuous material like grain, then I think the optimal method is to first move 1 mile to get 90 there, then move another 10/9 to get 80 there, then another 10/8 to get 70 there, ..., continuing until we get 40 at 8.4563..., and then make four trips to the destination. This gets 33.8254 to the destination. Is this the best method??]

Cardan. Practica Arithmetice. 1539. Chap. 66, section 57, ff. EE.vi.v - EE.vii.v (pp. 152 153). Complicated problem involving carrying food and material up the Tower of Babel! Tower is assumed 36 miles high and seems to require 15625 porters.

Mittenzwey. 1880. Prob. 135, pp. 28-29; 1895?: 153, p. 32; 1917: 153, pp. 29. If eight porters can carry eight full loads from A to B in an hour, how long will it take four porters? The obvious answer is two hours, but he observes that the porters have to return from B to A and it will take three hours. [Probably a little less as they should return in less time than they go.]

Pearson. 1907. Part II, pp. 139 & 216. Two explorers who can carry food for 12 days. (No depots, i.e. form A of Ball, below.)

Loyd. A dash for the South Pole. Ladies' Home Journal (15 Dec 1910). ??NYS -- source?? -- WS??

Ball. MRE, 5th ed., 1911. Exploration problems, pp. 23 24. He distinguishes two forms of the problem, with n explorers who can carry food for d days.

A. Without depots, they can get one man nd/(n+1) days into the desert and back.

B. With depots permitted, they can get a man d/2 (1/1 + 1/2 + ... + 1/n) into the desert and back. This is the more common form.

Dudeney. Problem 744: Exploring the desert. Strand Mag. (1925). ??NX. (??= MP 49)

Dudeney. MP. 1926. Prob. 49: Exploring the desert, pp. 21 & 111 (= 536, prob. 76, pp. 22 & 240). A version of Ball's form A, with n = 9, d = 10, but replacing days by stages of length 40 miles.

Abraham. 1933. Prob. 34 -- The explorers, pp. 13 & 25 (9 10 & 112). 4 explorers, each carrying food for 5 days. Mentions general case. This is Ball's form A.

Haldeman-Julius. 1937. No. 10: The four explorers, pp. 4 & 21. Ball's form A, with n = 4, d = 5.

Olaf Helmer. Problem in logistics: The Jeep problem. Project Rand Report RA 15015 (1 Dec 1946) 7pp.

N. J. Fine. The jeep problem. AMM 54 (1947) 24 31.

C. G. Phipps. The jeep problem: a more general solution. AMM 54 (1947) 458 462.

G. G. Alway. Note 2707: Crossing the desert. MG 41 (No. 337) (1957) 209. If a jeep can carry enough fuel to get halfway across, how much fuel is needed to get across? For a desert of width 2, this leads to the series 1 + 1/3 + 1/5 + 1/7 + .... See Lehmann and Pyle below.

G. C. S[hephard, ed.] The problems drive. Eureka 11 (Jan 1949) 10-11 & 30. No. 2. Four explorers, starting from a supply base. Each can carry food for 100 miles and goes 25 miles per day. Two men do the returning to base and bringing out more supplies. the third man does ferrying to the fourth man. How far can the fourth man get into the desert and return? Answer is 100 miles. Ball's form B would give 104 1/6.

Gamow & Stern. 1958. Refueling. Pp. 114 115.

Pyle, I. C. The explorer's problem. Eureka 21 (Oct 1958) 5-7. Considers a lorry whose load of fuel takes it a distance which we assume as the unit. What is the widest desert one can cross? And how do you do it? This is similar to Alway, above. He starts at the far side and sees you have to have a load at distance 1 from the far side, then two loads at distance 1 + 1/3 from the far side, then three loads at 1 + 1/3 + 1/5, .... This diverges, so any width can be crossed. Does examples with given widths of 2, 3 and 4 units. Editor notes that he is not convinced the method is optimal.

Martin Gardner, SA (May & June 1959) c= 2nd Book, chap. 14, prob. 1. (The book gives extensive references which were not in SA.)

R. L. Goodstein. Letter: Explorer's problem. Eureka 22 (Oct 1959) 23. Says Alway shows that Pyle's method is optimal. Editor notes Gardner's article and that Eureka was cited in the solutions in Jun.

David Gale. The jeep once more or jeeper by the dozen & Correction to "The Jeep once more or jeeper by the dozen". AMM 77:5 (May 1970) 493-501 & 78:6 (Jun-Jul 1971) 644-645. Gives an elaborate approach via a formula of Banach for path lengths in one dimension. This formally proves that the various methods used are actually optimal and that a continuous string of depots cannot help, etc. Notes that the cost for a round trip is only slightly more than for a one-way trip -- but the Correction points out that this is wrong and indeed the round trip is nearly four times as expensive as a one-way trip. Considers sending several jeeps. Says he hasn't been able to do the round trip problem when there is fuel on both sides of the desert. Comments on use of dynamic programming, noting that R. E. Bellman [Dynamic Programming; Princeton Univ. Press, 1955, p. 103, ex. 54-55] gives the problem as exercises without solution and that he cannot see how to do it!

Birtwistle. Math. Puzzles & Perplexities. 1971.

The expedition, pp. 124-125, 183 & 194. Ball's form A, first with n = 5, d = 6, then in general.

Second expedition, pp. 125-126. Ball's form B, done in general.

Third expedition, pp. 126, 183-184 & 194. Three men want to cross a 180 mile wide desert. They can travel 20 miles per day and can carry food for six days, which can be stored at depots. Minimize the total distance travelled. Solution seems erroneous to me.

A. K. Austin. Jeep trips and card stacks. MTg 58 (1972) 24 25. There are n flags located at distances a₁, a₁ + a₂, a₁ + a₂ + a₃, .... Jeep has to begin at the origin, go to the first flag, return to the origin, go to the second flag, return, .... He can unload and load fuel at the flags. Can he do this with F fuel? Author shows this is equivalent to successfully stacking cards over a cliff with successive overhangs being a₁, a₂, a₃, ....

Doubleday - 3. 1972. Traveller's Tale, pp. 63-64. d = 8 and we want one man to get across the desert of width 12. How many porters, who return to base, are needed? The solution implies that no depots are used. Reasoning as in Ball's case A, we see that n men can support one man crossing a desert of width 2nd/(n+1). If depots are permitted, this is essentially the jeep problem and n men can support a man getting across a desert of width d [1 + 1/3 + 1/5 + ... + 1/(2n-1)]

Johannes Lehmann. Kurzweil durch Mathe. Urania Verlag, Leipzig, 1980. No. 13, pp. 27 & 129. d = 4 and we want to get a man across a desert of width 6. Similar to Doubleday - 3.

Pierre Berloquin. [Le Jardin du Sphinx. Dunod, Paris, 1981.] Translated by Charles Scribner Jr as: The Garden of the Sphinx. Scribner's, NY, 1985.

Prob. 1: Water in the desert, pp. 3 & 85.

Prob. 40: Less water in the desert, pp. 26 & 111.

Prob. 80: Beyond thirst, pp. 48 & 140.

Prob. 141: The barrier of thirst, pp. 79 & 181.

Prob. 150: No holds barred, pp. 82 & 150.

In all of these, d = 5 and we want to get a man across a desert of width 4, and sometimes back, which is slightly different than the problem of getting to the maximum distance and back.

Prob. 1 is Ball's form A, with n = 4 men, using 20 days' water.

Prob. 40 is Ball's form B, but using only whole day trips, using 14 days' water.

Prob. 80 is Ball's form B, optimized for width 4, using 11½ days' water.

Prob. 141 uses depots and bearers who don't return, as in Alcuin?? You can get one man, who is the only one to return, a distance d (1/2 + 1/3 + ... + 1/(n+1)) into the desert this way. He gives the optimum form for width 4, using 9½ days' water.

Prob. 150 is like prob. 141, except that no one returns! You can get him d (1 + 1/2 + ... + 1/n) into the desert this way. The optimum here uses 4 days' water.

D. R. Westbrook. Note 74.7: The desert fox, a variation of the jeep problem. MG 74 (No. 467) (1990) 49 50. A more complex version, posed by A. K. Dewdney in SA (Jan 1987), is solved here.

Liz Allen. Brain Sharpeners. Op. cit. in 5.B. 1991. Round-trip, pp. 96-97 & 140. Plane wants to circle the earth, but can only carry fuel to go half-way. Other planes can accompany and transfer fuel, but must return to base.

Dylan Gow. Flyaway. MS 25:3 (1992/3) 84-86. Considers the standard problem without return as in Alway, Pyle and Lehmann -- but finds a non-optimal solution.

Wolfram Hinderer. Optimal crossing of a desert. MS 26:4 (1993/4) 100-102. Finds optimal solutions for Gow's problem and for the case with return -- i.e. Ball's B. Also considers use of extra jeeps that do not return, i.e. Berloquin's 141 & 150. Notes that extra jeeps that must return to base do not change the distance that one jeep can reach. [But it changes the time required.]

Harold Boas. Letter: Crossing deserts. MS 26:4 (1993/4) 122. Notes the problem has a long history and cites Fine, Phipps, Gale (and correction), Alway.

David Singmaster. Letter: Crossing deserts. MS 27:3 (1994/5) 63. Points out that the history is far older and sketches the history given above.

Günter Rote & Guochuan Zhang. Optimal Logistics for Expeditions: The Jeep Problem with Complete Refilling. Karl-Franzens-Universität Graz & Technische Universität Graz. Bericht 71 (24 Jun 1996). This deals with a variant. "We have n cans of fuel on the edge of a desert and a jeep with an empty tank whose capacity is just one can. The jeep can carry one can in addition to the fuel in its tank. Moreover, when a can is opened, the fuel must immediately be filled into the jeep's tank. The goal is to find the farthest point in the desert which the jeep can reach by consuming the n cans of fuel. Derick Wood [1984] treated this problem similarly to the classical problem and gave the first solution. Ute and Wilfried Brauer [1989] presented a new strategy and got a better solution than Wood's. They also conjectured that their solution was optimal for infinitely many values of n. We give an algorithm which produces a better solution than Brauers' for all n > 6, and we use a linear programming formulation to derive an upper bound which shows that our solution is optimal." 14 references, several not given above.
5.O. TAIT'S COUNTER PUZZLE: BBBBWWWW TO WBWBWBWB
See S&B 125.

The rules are that one can move two counters as an ordered pair, e.g. from BBBBWWWW to BBB..WWWBW, but not to BBB..WWWWB -- except in Lucas (1895) and AM prob. 237, where such reversal must be done. Also, moving to BBB..WWW.BW is sometimes explicitly prohibited, but it is not always clear just where one can move to. It is also not always specified where the blank spaces are at the beginning and end positions.

Gardner, 1961, requires that the two counters must be BW or WB.

Barbeau, 1995, notes that moving to BWBWBWBW is a different problem, requiring an extra move. I had not noticed this difference before -- indeed I previously had it the wrong way round in the heading of this section. I must check to see if this occurs earlier. See Achugbue & Chin, 1979-80, for this version.

Genjun Nakane (= Hōjiku Nakane). Kanja otogi soshi (Book of amusing problems for the entertainment of thinkers). 1743. ??NYS. (See: T. Hayashi; Tait's problem with counters in the Japanese mathematics; Bibl. Mathem. (3) 6 (1905) 323, for this and other Japanese references of 1844 and 1879, ??NYS.)

P. G. Tait. Listing's Topologie. Philosophical Mag. (Ser. 5) 17 (No. 103) (Jan 1884) 30 46 & plate opp. p. 80. Section 12, pp. 39 40. He says he recently saw it being played on a train.

George Hope Verney (= Lloyd Verney). Chess eccentricities. Longmans, 1885. P. 193: The pawn puzzle. ??NX With 4 & 4.

Lucas. Amusements par les jetons. La Nature 15 (1887, 2nd sem.) 10-11. ??NYS -- cited by Ahrens, title obtained from Harkin. Probably c= the material in RM3, below.

Ball. MRE, 1st ed., 1892, pp. 48 49.

Berkeley & Rowland. Card Tricks and Puzzles. 1892. Card Puzzles No. XIV: The eight-card puzzle, pp. 14-15. Uses cards: BRBRBRBR and asks to bring the colours together, explicitly requiring the moved cards to be placed in contact with the unmoved cards.

Hoffmann. 1893. Chap. VI, pp. 270 271 & 284 285 = Hoffmann-Hordern, pp. 184-186, with photo.

No. 19: The "Four and Four" puzzle. Photo on p. 184 shows a version named Monkey Puzzle advertising Brooke's Soap to go from BBBBBWWWW.. to ..WBWBWBWB .

No. 20: The "Five and Five" puzzle.

No. 21: The "Six and Six" puzzle.

Lucas. RM3. 1893. Amusements par les jetons, pp. 145 151. He gives Delannoy's general solution for n of each colour in n moves. Remarks that one can reverse the moved pair.

Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co., NY), nd [1895]. P. 11: The Egyptian disc puzzle. 4 & 4. "Two discs adjoining each other to be moved at a time; no gaps to be left in the line." -- this seems to prevent one from making any moves at all!! No solution.

Lucas. L'Arithmétique Amusante. 1895. Pp. 84-108.

Prob. XXI - XXIV and Méthode générale, pp. 84-97. Gives solution for 4, 5, 6, 7 and the general solution for n & n in n moves due to Delannoy.

Rouges et noires, avec interversion, prob. XXV - XXVIII and Méthode générale, pp. 97 108. Interversion means that the two pieces being moved are reversed or turned over, e.g. from BBBBWWWW to BBB..WWWWB, but not to BBB..WWWBW. Gives solutions for 4, 5, 6, 7, 8 pairs and in general in n moves, but he ends with a gap, e.g. ....BB..BB and it takes an extra move to close up the gap.

Ball. MRE, 3rd ed., 1896, pp. 65 66. Cites Delannoy's solution as being in La Nature (Jun 1887) 10. ??NYS.

Ahrens. MUS I. 1910. Pp. 14-15 & 19-25. Cites Tait and gives Delannoy's general solution, from Lucas.

Ball. MRE, 5th ed., 1911, pp. 75-77. Adds a citation to Hayashi, but incorrectly gives the date as 1896.

Loyd. Cyclopedia. 1914. After dinner tricks, pp. 41 & 344. 4 & 4.

Williams. Home Entertainments. 1914. The eight counters puzzle, pp. 116-117. Standard version, but with black and white reversed, in four moves. Says the moved counters must be placed in line with and touching the others.

Dudeney. AM. 1917.

Prob. 236: The hat puzzle, pp. 67 & 196-197. BWBWBWBWBW.. to have the Bs and Ws together and two blanks at an end. Uses 5 moves to get to ..WWWWWBBBBB.

Prob. 237: Boys and girls, pp. 67-68 & 197. ..BWBWBWBW to have the Bs and Ws together with two blanks at an end, but pairs must be reversed as they are moved. Solution in 5 moves to WWWWBBBB... = Putnam, no. 2. Cf Lucas, 1895.

Blyth. Match-Stick Magic. 1921. Transferring in twos, pp. 80-81. WBWBWBWB.. to ..BBBBWWWW in four moves.

King. Best 100. 1927. No. 66, pp. 27 & 55. = Foulsham's, no. 9, pp. 9 & 13. BWBWBWBW.. to ..WWWWBBBB, specifically prescribed.

Rohrbough. Brain Resters and Testers. c1935. Alternate in Four Moves, p. 4. ..BBBBWWWW to WBWBWBWB.. , but he doesn't specify the blanks, showing all stages as closed up to 8 spaces, except the first two stages have a gap in the middle.

McKay. At Home Tonight. 1940.

Prob. 43: Arranging counters, pp. 73 & 87-88. RBRBRB.... to ....BBBRRR in three moves. Sketches general solution.

Prob. 45: Triplets, pp. 74 & 88. YRBYRBYRB.. to BBBYYYRRR.. in 5 moves.

McKay. Party Night. 1940. Heads and tails again, p. 151. RBRBR.. to ..BBRRR in three moves. RBRBRB.. to ..BBBRRR in four moves. RBRBRBRB.. to ..BBBBRRRR in four moves. Notes that the first move takes coins 2 & 3 to the end and thereafter one is always filling the spaces just vacated.

Gardner. SA (Jun & Jul 1961) = New MD, chap. 19, no. 1: Collating the coins. BWBWB to BBBWW, moving pairs of BW or WB only, but the final position may be shifted. Gardner thanks H. S. Percival for the idea. Solution in 4 moves, using gaps and with the solution shifted by six spaces to the right. Thanks to Heinrich Hemme for this reference.

Joseph S. Madachy. Mathematics on Vacation. (Scribners, NY, 1966, ??NYS); c= Madachy's Mathematical Recreations. Dover, 1979. Prob. 3: Nine-coin move, pp. 115 & 128-129 (where the solution is headed Eight-coin move). This uses three types of coin, which I will denote by B, R, W. BRWBRWBRW  WWWRRRBBB by moving two adjacent unlike coins at a time and not placing the two coins away from the rest. Eight move solution leaves the coins in the same places, but uses two extra cells at each end. From the discussion of Bergerson's problem, see below, it is clear that the earlier book omitted the word unlike and had a nine move solution, which has been replaced by Bergerson's eight move solution.

Yeong Wen Hwang. An interlacing transformation problem. AMM 67 (1967) 974 976. Shows the problem with 2n pieces, n > 2, can be solved in n moves and this is minimal.

Doubleday - 1. 1969. Prob. 70: Oranges and lemons, pp. 86 & 170. = Doubleday - 4, pp. 95 96. BWBWBWBWBW.. considered as a cycle. There are two solutions in five moves: to ..WWWWWBBBBB, which never uses the cycle; and to: BBWWWWWW..BBB.

Howard W. Bergerson, proposer; Editorial discussion; D. Dobrev, further solver; R. H. Jones, further solver. JRM 2:2 (Apr 1969) 97; 3:1 (Jan 1970) 47-48; 3:4 (Oct 1970) 233-234; 6:2 (Spring 1973) 158. Gives Madachy's 1966 problem and says there is a shorter solution. The editor points out that Madachy's book and Bergerson have omitted unlike. Bergerson has an eight move solution of the intended problem, using two extra cells at each end, and Leigh James gives a six move solution of the stated problem, also using two extra cells at each end. Dobrev gives solutions in six and five steps, using only two extra cells at the right. Jones notes that the problem does not state that the coins have to be adjacent and produces a four move solution of the stated problem, going from ....BRWBRWBRW.... to WWW..R..R..R..BBB.

Jan M. Gombert. Coin strings. MM 42:5 (Nov 1969) 244-247. Notes that BWBWB......  ......BBBWW can be done in four moves. In general, BWB...BWB, with n Ws and n+1 Bs alternating can be transformed to BB...BWW...W in n² moves and this is minimal. This requires shifting the whole string n(n+1) to the right and a move can go to places separated from the rest of the pieces. By symmetry, ......BWBWB  WWBBB...... in the same number of moves.

Doubleday - 2. 1971. Two by two, pp. 107-108. ..BWBWBWBW to WWWWBBBB... He doesn't specify where the extra spaces are, but says the first two must move to the end of the row, then two more into the space, and so on. The solution always has two moving into an internal space after the first move.

Wayne A. Wickelgren. How to Solve Problems. Freeman, 1974. Checker-rearrangement problem, pp. 144 146. BWBWB to BBBWW by moving two adjacent checkers, of different colours, at a time. Solves in four moves, but the pattern moves six places to the left.

Putnam. Puzzle Fun. 1978.

No. 1: Nickles [sic] & dimes, pp. 1 & 25. Usual version with 8 coins. Solution has blanks at the opposite end to where they began.

No. 2: Nickles [sic] & dimes variation, pp. 1 & 25. Same, except the order of each pair must be reversed as it moves. Solution in five moves with blanks at opposite end to where they started. = AM 237. Cf Lucas, 1895.

James O. Achugbue & Francis Y. Chin. Some new results on a shuffling problem. JRM 12:2 (1979-80) 125-129. They demonstrate that any pattern of n & n occupying 2n consecutive cells can be transformed into any other pattern in the same cells, using only two extra cells at the right, except for the case n = 3 where 10 cells are used. They then find an optimal solution for BB...BW...WW  BWBW...BW in n+1 moves using two extra cells. They seem to leave open the question of whether the number of moves could be shortened by using more cells.

Walter Gibson. Big Book of Magic for All Ages. Kaye & Ward, Kingswood, Surrey, 1982.

Six cents at a time, p. 117. Uses pennies and nickels. .....PNPNP to NNPPP..... in four moves.

Tricky turnover, p. 137. HTHTHT to HHHTTT in two moves. This requires turning over one of the two coins on each move.

Ed Barbeau. After Math. Wall & Emerson, Toronto, 1995. Pp. 117, 119 & 123-126. He asks to move BBBWWW to WBWBWB and to BWBWBW and notes that the latter takes an extra move. He sketches the general solutions.

5.P. GENERAL MOVING PIECE PUZZLES
See also under 5.A.
5.P.1. SHUNTING PUZZLES
See Hordern, op. cit. in 5.A, pp. 167 177, for a survey of these puzzles. The Chifu Chemulpo (or Russo Jap Railway) Puzzle of 1903 is actually not of this type since all the pieces can move by themselves -- Hordern, pp. 124 125 & plate VIII.

See S&B 124 125.

A 'spur' is a dead end line. A 'side line' is a line or siding joined to another at both ends.
Mittenzwey. 1880. Prob. 219-221, pp. 39-40 & 91; 1895?: 244-246, pp. 43-44 & 93; 1917: 244-246, pp. 40 & 89. First two have a canal too narrow to permit boats to pass, with a 'bight', or widening, big enough to hold one boat while another passes. First problem has two boats meeting one boat; second problem has two boats meeting two boats. The third problem has a single track railway with a side-line big enough to hold an engine and 16 wagons on the side-line or on the main line between the switches. Two trains consisting of an engine and 20 wagons meet.

Lucas. RM2, 1883, pp. 131 133. Passing with a spur and with a side line.

Alexander Henry Reed. UK Patent 15,051 -- Improvements in Puzzles. Complete specification: 8 Dec 1885. 4pp + 1p diagrams. Reverse a train using a small turntable on the line. This has forms with one line and with two crossing lines. One object is to spell 'Humpty Dumptie'. He also has a circular line with three turntables (equivalent to the recent Top-Spin Puzzle of F. Lammertinck).

Pryse Protheroe. US Patent 332,211 -- Puzzle. Applied: 18 Sep 1885; patented: 8 Dec 1885. 3pp + 1p diagrams. Described in Hordern, p. 167. Identical to the Reed patent above! Both Reed and Protheroe are described as residents of suburban London. The Reed patent says it was communicated from abroad by an Israel J. Merritt Jr of New York and it doesn't assert that Reed is the inventor, so perhaps Reed and Merritt were agents for Protheroe.

Jeffrey & Son (Syracuse, NY). Great Railroad Puzzle. Postcard puzzle produced in 1888. ??NYS. Described in Hordern, pp. 175 176. Passing with a turntable that holds two wagons.

Arthur G. Farwell. US Patent 437,186 -- Toy or Puzzle. Applied: 20 May 1889; patented: 30 Sep 1890. 1p + 1p diagrams. Described in Hordern, pp. 167 169. Great Northern Puzzle. This requires interchanging two cars on the legs of a 'delta' switch which is too short to allow the engine through, but will let the cars through. Hordern lists 6 later patents on the same basic idea.

Ball. MRE, 1st ed., 1892, pp. 43 44. Great Northern Puzzle "which I bought some eight or nine years ago." (Hordern, p. 167, erroneously attributes this quote to Ahrens.)

Loyd. Problem 28: A railway puzzle. Tit Bits 32 (10 Apr & 1 May 1897) 23 & 79. Engine and 3 cars need to pass 4 cars by means of a 'delta' switch whose branches and tail hold only one car. Solution with 28 reversals.

Loyd. Problem 31: The turn table puzzle. Tit Bits 32 (1 & 22 May 1897) 79 & 135. Reverse an engine and 9 cars with an 8 track turntable whose lines hold 3 cars. The turntable is a double curved connection which connects, e.g. track 1 to tracks 4 or 6.

E. Fourrey. Récréations Arithmétiques. Op. cit. in 4.A.1. 1899. Art. 239: Problèmes de Chemin de fer, pp. 184-189.

I. Three parallel tracks with two switched crossing tracks. Train of 21 wagons on the first track must leave wagons 9 & 12 on third track.

II. Delta shape with a turntable at the point of the delta, which can only hold the wagons and not the engine, so this is isomorphic to Farwell.

III. This is a more complex railway problem involving timetables on a circular line.

J. W. B. Shunting! c1900. ??NYS. Described in Hordern, pp. 176 177 & plate XII. Reversing a train with a turntable that holds three wagons.

Orril L. Hubbard. US Patent 753,266 -- Puzzle. Applied: 21 Apr 1902; patented: 1 Mar 1904. 3pp + 1p diagrams. Great Railroad Puzzle, described in Hordern, pp. 175 176. Improved version of the Jeffrey & Son puzzle of 1888. Engine & 2 cars to pass engine & 3 cars, using a turntable that holds two cars, preserving order of each train.

Harry Lionel Hook & George Frederick White. UK Patent 26,645 -- An Improved Puzzle or Game. Applied: 3 Dec 1902; accepted: 11 Jun 1903. 2pp + 1p diagrams. This is very cryptic, but appears to be a kind of sliding piece Puzzle using turntables.

Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 10:1 (May 1903) 50-51 & 10:2 (Jun 1903) 140-141 & 10:4 (Aug 1903) 336-337. A railway puzzle. One north-south line with a spur heading north which is holding 7 trucks, but cannot hold the engine as well, so the engine is on the main line heading south. An engine pulling seven trucks arrives from the north and wants to get past. First solution uses 17 stages; second uses 12 stages.

Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 10:5 (Sep 1903) 426-427 & 10:6 (Oct 1903) 530 531. A shunting problem. Same as Fourrey - II, hence isomorphic to Farwell. Solution in 17 stages.

Celluloid Starch Puzzle. c1905. Described in Hordern, pp. 169 170. Cars on the three parts of a 'delta' switch with an engine approaching. Reverse the engine, leaving all cars on their original places. More complexly, suppose the tail of the 'delta' only holds one car or the engine.

Livingston B. Pennell. US Patent 783,589 -- Game Apparatus. Applied: 20 Mar 1902; patented: 28 Feb 1905. 3pp + 1p diagrams. Described in Hordern, p. 173. Passing with a side line -- engine & 3 cars to pass engine & 3 cars using a siding which already contains 3 cars, without couplings, so these three can only be pushed. Also the engines can move at most three cars at a time.

William Rich & Harry Pritchard. UK Patent 7647 -- Railway Game and Puzzle. Applied: 11 Apr 1905; complete specification: 11 Oct 1905; accepted: 14 Dec 1905. 2pp + 1p diagrams. Main line with two short and two long spurs.

Ball. MRE, 4th ed., 1905, pp. 61-63, adds a problem with a side-line, "on sale in the streets in 1905“. The 5th ed., 1911, pp. 69-71 & 82, adds the name "Chifu-Chemulpo Puzzle" and that the minimum number of moves is 26, in more than one way. P. 82 gives solutions of both problems.

Dudeney. The world's best puzzles. Op. cit. in 2. 1908. Great Northern Puzzle. He says the "Railway puzzle" was very popular "about twenty years ago".

Ahrens. MUS I. 1910. Pp. 3-4. Great Northern. Says it is apparently modern and cites Fourrey for other examples.

Anon. Prob. 6. Hobbies 32 (No. 814) (20 May 1911) 145 & (No. 817) (10 Jun 1911) 208. Great Northern Puzzle. Solution asks if readers know any other railway puzzles.

Loyd. The switch problem & Primitive railroading problem. Cyclopedia, 1914, pp. 167 & 361; 89 & 350 (= MPSL2, prob. 24, pp. 18 19; MPSL1, prob. 95, pp. 92 & 155). Passing with a 'delta' switch & passing with a spur. The first is like Tit-Bits Problem 28, but the engine and 3 cars have to pass 5 cars. Solution in 32 moves. See Hordern, pp. 170 171.

Hummerston. Fun, Mirth & Mystery. 1924. The Chinese railways, pp. 103 & 188. Imagine a line of positions: ABCEHGJLMN with single positions D, I, F, K attached to positions C, H, G, L. You have eight engines at ABCD and KLMN and the object is to exchange them, preserving the order. He does it in 18 moves, where a move can be of any length.

King. Best 100. 1927. No. 14, pp. 12 & 41. Side line with a bridge over it too low for the engine. Must interchange two wagons on the side line which are on opposite sides of the bridge.

B. M. Fairbanks. Railroad switching problems. IN: S. Loyd Jr., ed.; Tricks and Puzzles; op. cit. in 5.D.1 under Chapin; 1927. P. 85 & Answers p. 7. Three realistic problems with several spurs and sidelines.

Loyd Jr. SLAHP. 1928. Switching cars, pp. 54 & 106. Great Northern puzzle. See Hordern, pp. 168 169.

Doubleday - 2. 1971. Traffic jam, pp. 85-86. Version with cars in a narrow lane and a lay-by. Two cars going each way. Though the lay-by is three cars wide and just over a car long, he restricts its use so that it acts like it is two cars wide.
5.P.2. TAQUIN
Jacques Haubrich has kindly enlightened me that 'taquin' simply means 'teaser'. So these items should be re-categorised.
Lucas. RM3. 1893. 3ème Récréation -- Le jeu du caméléon et le jeu des jonctions de points, pp. 89 103. Pp. 91 97 -- Le taquin de neuf cases avec un seul port. I thought that taquin was the French generic term for such puzzles, but I find no other usage than that below, except in referring to the 15 Puzzle -- see references to taquin in 5.A.

Au Bon Marché (the Paris department store). Catalogue of 1907, p. 13. Reproduced in Mary Hillier; Automata and Mechanical Toys; An Illustrated History; Jupiter Books, London, 1976, p. 179. This shows Le Taquin Japonais Jeu de Patience Casse-tete. This comprises 16 hexagonal pieces, looking like a corner view of a die, so each has three rhombic parts containing a pattern of pips. They are to be placed as the corners of four interlocked hexagons with the numbers on adjacent rhombi matching.

5.Q. NUMBER OF REGIONS DETERMINED BY N LINES OR PLANES
Mittenzwey. 1880. Prob. 200, pp. 37 & 89; 1895?: 225, pp. 41 & 91; 1917: 225, pp. 38 & 88. Family of 4 adults and 4 children. With three cuts, divide a cake so the adults and the children get equal pieces. He makes two perpendicular diametrical cuts and then a circular cut around the middle. He seems to mean the adults get equal pieces and the children get equal pieces, not necessarily the same. But if the circular cut is at 2/2 of the radius, then the areas are all equal. Not clear where this should go -- also entered in 5.T.

Jakob Steiner. Einige Gesetze über die Theilung der Ebene und des Raumes. (J. reine u. angew. Math. 1 (1826) 349 364) = Gesam. Werke, 1881, vol. 1, pp. 77 94. Says the plane problem has been raised before, even in a Pestalozzi school book, but believes he is first to consider 3 space. Considers division by lines and circles (planes and spheres) and allows parallel families, but no three coincident.

Richard A. Proctor. Some puzzles; Knowledge 9 (Aug 1886) 305-306 & Three puzzles; Knowledge 9 (Sep 1886) 336-337. "3. A man marks 6 straight lines on a field in such a way as to enclose 10 spaces. How does he manage this?" Solution begins: "III. To inclose ten spaces by six ropes fastened to nine pegs." Take (0,0), (1,0), ..., (n,0), (0,n), ..., (0,1), as 2n+1 points, using n+2 ropes from (0,0) to (n,0) and to (0,n) and from (i,0) to (0,n+1-i) to enclose n(n+1)/2 areas.

Richard A. Proctor. Our puzzles. Knowledge 10 (Nov 1886) 9 & (Dec 1886) 39-40. Describes several ways of solving previous problem and asks for a symmetric version.

G. Chrystal. Algebra -- An Elementary Text-Book. Vol. 2, A. & C. Black, Edinburgh, 1889. [Note -- the 1889 version of vol. 1 is a 2nd ed.] Chap. 23, Exercises IV, p. 34. Several similar problems and the following.

No. 7 -- find number of interior and of exterior intersections of the diagonals of a convex n-gon.

No. 8 -- n points in general position in space, draw planes through every three and find number of lines and of points of intersection.

L. Schläfli. Theorie der vielfachen Kontinuität. Neue Denkschriften der allgemeinen schweizerischen Gesellschaft für die Naturwissenschaften 38:IV, Zürich, 1901, 239 pp. = Ges. Math. Abh., Birkhäuser, Basel, 1950 1956, vol. 1, pp. 167 392. (Pp. 388 392 are a Nachwort by J. J. Burckhardt.) Material of interest is Art. 16: Über die Zahl der Teile, ..., pp. 209 212. Obtains formula for k hyperplanes in n space.

Loyd, Dudeney, Pearson & Loyd Jr. give various puzzles based on this topic.

Howard D. Grossman. Plane- and space-dissection. SM 11 (1945) 189-190. Notes Schläfli's result and observes that the number of regions determined by k+1 hyperspheres in n space is twice the number of regions determined by k hyperplanes and gives a two to one correspondence for the case n = 2.

Leo Moser, solver. MM 26 (Mar 1953) 226. ??NYS. Given in: Charles W. Trigg; Mathematical Quickies; (McGraw Hill, NY, 1967); corrected ed., Dover, 1985. Quickie 32: Triangles in a circle, pp. 11 & 90 91. N points on a circle with all diagonals drawn. Assume no three diagonals are concurrent. How many triangles are formed whose vertices are internal intersections?

Timothy Murphy. The dissection of a circle by chords. MG 56 (No. 396) (May 1972) 113 115 + Correction (No. 397) (Oct 1972) 235 236. N points on a circle, in a plane or on a sphere; or N lines in a plane or on a sphere, all simply done, using Euler's formula.

Rowan Barnes-Murphy. Monstrous Mysteries. Piccolo, 1982. Slicing cakes, pp. 33 & 61. Cut a circular cake into 12 equal pieces with 4 cuts. [From this, we see that N full cuts can yield either 2N or 4(N-1) equal pieces. Further, if we make k circular cuts producing k+1 regions of equal area and then make N-k diametric cuts equally spaced, we get 2(k+1)(N-k) pieces of the same size.]

Looking at this problem, I see that one can obtain any number of pieces from N+1 up through the maximum.

5.Q.1. NUMBER OF INTERSECTIONS DETERMINED BY N LINES
Chrystal. Text Book of Algebra. 2nd ed., vol. 2, 1889, p. 34, ex. 7. See above.

Loyd Jr. SLAHP. 1928. When drummers meet, pp. 74 & 115. Six straight railroads can meet in 15 points.

Paul Erdös, proposer; Norbert Kaufman & R. H. Koch and Arthur Rosenthal, solvers. Problem E750. AMM 53 (1946) 591 & 54 (1947) 344. The first solution is given in Trigg, op. cit. in 5.Q, Quickie 191: Intersections of diagonals, pp. 53 & 166 167. In a convex n gon, how many intersections of diagonals are there? This counts a triple intersection as three ordinary (i.e. double) intersections or assumes no three diagonals are concurrent. Editorial notes add some extra results and cite Chrystal.
5.R. JUMPING PIECE GAMES
See also 5.O. Some of these are puzzles, but some are games and are described in the standard works on games -- see the beginning of 4.B.
5.R.1. PEG SOLITAIRE
See MUS I 182-210.
Ahrens, MUS I 182 183, gives legend associating this with American Indians. Bergholt, below, and Beasley, below, find this legend in the 1799 Encyclopédie Méthodique: Dictionnaire des Jeux Mathématique (??*), ??NYS. Ahrens also cites some early 19C material which has not been located. Bergholt says some maintain the game comes from China.
Thomas Hyde. Historia Nerdiludii, hoc est dicere, Trunculorum; .... (= Vol. 2 of De Ludis Orientalibus, see 7.B for vol. 1.) From the Sheldonian Theatre (i.e. OUP), Oxford, 1694. De Ludo dicto Ufuba wa Hulana, p. 233. This has a 5 x 5 board with each side having 12 men, but the description is extremely brief. It seems to have two players, but this may simply refer to the two types of piece. I'm not clear whether it's played like solitaire (with the jumped pieces being removed) or like frogs & toads. I would be grateful if someone could read the Latin carefully. The name of the puzzle is clearly Arabic and Hyde cites an Arabic source, Hanzoanitas (not further identified on the pages I have) -- I would be grateful to anyone who can track down and translate Arabic sources.

G. W. Leibniz. Le Jeu du Solitaire. Unpublished MS LH XXXV 3 A 10 f. 1-2, of c1678. Transcribed in: S. de Mora-Charles; Quelques jeux de hazard selon Leibniz; HM 19 (1992) 125-157. Text is on pp. 152-154. 37 hole board. Says the Germans call it 'Die Melancholy' and that it is now the mode at the French court.

Claude Auguste Berey. Engraving: Madame la Princesse de Soubize jouant au Jeu de Solitaire. 1697(?). Beasley (below) discovered and added this while his book was in proof. It shows the 37 hole French board. Reproduced in: Pieter van Delft & Jack Botermans; Creative Puzzles of the World; op. cit. in 5.E.2.a, p. 170.

G. W. Leibniz. Jeu des Productions. Unpublished MS LH XXXV 8,30 f. 4, of 1698. Transcribed in: de Mora-Charles, loc. cit. above. Text is on pp. 154-155. 37 hole board. Considers the game in reverse.

Trouvain. Engraving: Dame de Qualité Jouant au Solitaire. 1698(?).

Claude Auguste Berey. Engraving: Nouveau Jeu de Solitaire. Undated, but Berey was active c1690 c1730. Reproduced in: R. C. Bell; The Board Game Book; Marshall Cavendish, London, 1979, pp. 54 55 and in: Jasia Reichardt, ed.; Play Orbit [catalogue of an exhibition at the ICA, London, and elsewhere in 1969-1970]; Studio International, 1969, p. 38. Beasley's additional notes point out that this engraving is well known, but he had not realised its date until the earlier Berey engraving was discovered. This engraving includes the legend associating the game with the American indians -- "son origine vient de l'amerique ou les Peuples vont seuls à la chasse, et au retour plantent leurs flèches en des trous de leur cases, ce qui donna idée a un françois de composer ce jeu ...." Reichardt says the original is in the Bibliothèque Nationale.

The three engravings above are reproduced in: Henri d'Allemagne; Musée rétrospectif de la classe 100, Jeux, à l'exposition universelle international de 1900 à Paris, Tome II, pp. 152 158. D'Allemagne says the originals are in the Bibliothèque Nationale, Paris. He (and de Mora-Charles) also cites Rémond de Montmort, 2nd ed., 1713 -- see below.

G. W. Leibniz. Annotatio de quibusdam Ludis; inprimis de Ludo quodam Sinico, differentiaque Scachici et Latrunculorum & novo genere Ludi Navalis. Misc. Berolinensia (= Misc. Soc. Reg., Berlin) 1 (1710) 24. Last para. on p. 24 relates to solitaire. (English translation on p. xii of Beasley, below.)

Pierre Rémond de Montmort. Essai d'analyse sur les jeux de hazards. (1708); Seconde edition revue & augmentee de plusieurs lettres, (Quillau, Paris, 1713 (reprinted by Chelsea, NY, 1980)); 2nd issue, Jombert & Quillau, 1714. Avertissement (to the 2nd ed.), xli-xl. "J'ai trouvé dans le premier volume de l'Academie Royale de Berlin, ...; il propose ensuite des Problèmes sur un jeu qui a été à la mode en France il y a douze ou quinze ans, qui se nomme Le Solitaire."

Edward Hordern's collection has a wooden 37 hole board on the back of which is inscribed "Invented by Lord Derwentwater when Imprisoned in the Tower". The writing is old, at least 19C, possibly earlier. However the Encyclopedia Britannica article on Derwentwater and the DNB article on Radcliffe, James, shows that the relevant Lord was most likely to have been James Radcliffe (1689-1716), the 3rd Earl from 1705, who joined the Stuart rising in 1715, was captured at Preston, was imprisoned in the Tower and was beheaded on 24 Feb 1716, so the implied date of invention is 1715 or 1716. The third Earl became a figure of romance and many stories and books appeared about him, so the invention of solitaire could well have been attributed to him.

Though the title was attainted and hence legally extinct, it was claimed by relatives. Both James's brother Charles (1693 1746), the claimed 5th Earl from 1731, and Charles's son James Bartholomew (1725-1786), the claimed 6th Earl from 1746, spent time in prison for their Stuart sympathies. Charles escaped from Newgate Prison after the 1715 rising, but both were captured on their way to the 1745 rising and taken to the Tower where Charles was beheaded. If either of these is the Lord Derwentwater referred to, then the date must be 1745 or 1746. A guide book to Northumberland, where the family lived at Dyvelston (or Dilston) Castle, near Hexham, asserts the last Derwentwater was executed in 1745, while the [Blue Guide] says the last was executed for his part in the 1715 uprising.

In any case, the claim seems unlikely.

G. W. Leibniz. Letter to de Montmort (17 Jan 1716). In: C. J. Gerhardt, ed.; Die Philosophischen Schriften von Gottfried Wilhelm Leibniz; (Berlin, 1887) = Olms, Hildesheim, 1960; Vol. 3, pp. 667 669. Relevant passage is on pp. 668 669. (Poinsot, op. cit. in 5.E, p. 17, quotes this as letter VIII in Leibn. Opera philologica.)

J. C. Wiegleb. Unterricht in der natürlichen Magie. Nicolai, Berlin & Stettin, 1779. Anhang von dreyen Solitärspielen, pp. 413 416, ??NYS -- cited by Beasley. First known diagram of the 33 hole board.

Catel. Kunst-Cabinet. 1790. Das Grillenspiel (Solitaire), p. 50 & fig. 167 on plate VI. 33 hole board. (Das Schaaf- und Wolfspiel, p. 52 & fig. 169 on plate VI, is a game on the 33-hole board.)

Bestelmeier. 1801. Item 511: Ein Solitair, oder Nonnenspiel. 33 hole board.

Strutt. Op. cit. in 4.B.1. The Solitary Game. (1801: Book IV, p. 238. ??NYS -- cited by Beasley -- may be actually 1791??) 1833: Book IV, chap. II, art. XV, p. 319. c= Strutt-Cox, p. 259. Beasley says this is the first attribution to a prisoner in the Bastille. The description is vague: "fifty or sixty" holes and "a certain number of pegs". Strutt-Cox adds a note that "The game of Solitaire, reimported from France, ..., came again into Fashion in England in the late" 1850s and early 1860s.

Ada Lovelace. Letter of 16 Feb 1840 to Charles Babbage. BM MSS 37191, f. 331. ??NYS -- reproduced in Teri Perl; Math Equals; Addison-Wesley, Menlo Park, California, 1978, pp. 109-110. Discusses the 37 hole board and wonders if there is a mathematical formula for it.

M. Reiss. Beiträge zur Theorie des Solitär Spiels. J. reine angew. Math. 54 (1857) 376 379.

St. v. Kosiński & Louis Wolfsberg. German Patent 42919 -- Geduldspiel. Patented: 25 Sep 1877. 1p + 1p diagrams. 33 hole version.

The Sociable. 1858. The game of solitaire, pp. 282-284. 37 hole board. "It is supposed to have been invented in America, by a Frenchman, to beguile the wearisomeness attendant upon forest life, and for the amusement of the Indians, who pass much of their time alone at the chase, ...."

Anonymous. Enquire Within upon Everything. 66th ed., 862nd thousand, Houlston and Sons, London, 1883, HB. Section 135: Solitaire, p. 49. Mentions a 37 hole board but shows a 33 hole board. This material presumably goes back some time before this edition. It later shows Fox and Geese on the 33 hole board.

Hoffmann. 1893. Chap. X, no. 11: Solitaire problems, pp. 339-340 & 376-377 = Hoffmann Hordern, pp. 232-233, with photo on p. 235. Three problems. Photo on p. 235 shows a 33-hole board in a square frame, 1820-1840, and a 37-hole board with a holding handle, 1840-1890.

Ernest Bergholt. Complete Handbook to the Game of Solitaire on the English Board of Thirty-three Holes. Routledge, London, nd [Preface dated Nov 1920] -- facsimile produced by Naoaki Takashima, 1993. This is the best general survey of the game prior to Beasley.

King. Best 100. 1927. No. 68, pp. 28 & 55. = Foulsham's no. 24, pp. 9 & 13. 3 x 3 array of men in the middle of a 5 x 5 board. Men can jump diagonally as well as orthogonally. Object is to leave one man in the centre.

Rohrbough. Puzzle Craft. 1932. Note on Solitaire & French Solitaire, pp. 14-15 (= pp. 6-7 of 1940s?). 33 hole board, despite being called French.

B. M. Stewart. Solitaire on a checkerboard. AMM 48 (1941) 228-233. This surveys the history and then considers the game on the 32 cell board comprising the squares of one colour on a chessboard. He tilts this by 45^o to get a board with 7 rows, having 2, 4, 6, 8, 6, 4, 2 cells in each row. He shows that each beginning-ending problem which is permitted by the parity rules is actually solvable, but he gives examples to show this need not happen on other boards.

Gardner. SA (Jun 1962). Much amended as: Unexpected, chap. 11, citing results of Beasley, Conway, et al. Cites Leibniz and mentions Bastille story.

J. D. Beasley. Some notes on solitaire. Eureka 25 (Oct 1962) 13-18. No history of the game.

Jeanine Cabrera & René Houot. Traité Pratique du Solitaire. Librairie Saint Germain, Paris, 1977. On p. 2, they give the story that it was invented by a prisoner in the Bastille, late 18C, and they even give the name of the reputed inventor: "Comte"(?) Pellisson. They say that a Paul Pellisson Fontanier was in the Bastille in 1661 1666 and was a man of some note, but history records no connection between him and the game.

The Diagram Group. Baffle Puzzles -- 3: Practical Puzzles. Sphere, 1983. No. 12. On the 33-hole board place 16 markers: 1 in row 2; 3 in row 3; 5 in row 4; 7 in row 5; making a triangle centred on the mid-line. Can you remove all the men, except for one in the central square? Gives a solution in 15 jumps.

J. D. Beasley. The Ins and Outs of Peg Solitaire. OUP, 1985. History, pp. 3 7; Selected Bibliography, pp. 253 261. PLUS Additional notes, from the author, 1p, Aug 1985. 57 references and 5 patents, including everything known before 1850.

Franco Agostini & Nicola Alberto De Carlo. Intelligence Games. (As: Giochi della Intelligenza; Mondadori, Milan, 1985.) Simon & Schuster, NY, 1987. This gives the legend of the nobleman in the Bastille. Then says that "it would appear that a very similar game" is mentioned by Ovid "and again, it was widely played in ancient China -- hence its still frequent alternative name, "Chinese checkers."" I have included this as an excellent example of how unreferenced statements are made in popular literature. I have never seen either of these latter statements made elsewhere. The connection with Ovid is pretty tenuous -- he mentions a game involving three in a row and otherwise is pretty cryptic and I haven't seen anyone else claiming Ovid is referring to a solitaire game -- cf 4.B.5. The connection with Chinese checkers is so far off that I wonder if there is a translation problem -- i.e. does the Italian name refer to some game other than what is known as Chinese checkers in English??

Nob Yoshigahara. Puzzlart. Tokyo, 1992. Coin solitaire, pp. 5 & 90. Four problems on a 4 x 4 board.

Marc Wellens, et al. Speelgoed Museum Vlaanderen -- Musée du Jouet Flandre -- Spielzeug Museum Flandern -- Flanders Toy Museum. Speelgoedmuseum Mechelen, Belgium, 1996, p. 90 (in English), asserts 'It was invented by the French nobleman Palissen, who had been imprisoned in the Bastille by Louis XIV' in the early 18C.
5.R.1.a. TRIANGULAR VERSION
The triangular version of the game has only recently been investigated. The triangular board is generally numbered as below.
1

2 3

4 5 6

7 8 9 10

11 12 13 14 15
Herbert M. Smith. US Patent 462,170 -- Puzzle. Filed: 13 Mar 1891; issued: 27 Oct 1891. 2pp + 1p diagrams. A board based on a triangular lattice.

Rohrbough. Puzzle Craft. 1932. Triangle Puzzle, p. 5 (= p. 6 in 1940s?). Remove peg 13 and leave last peg in hole 13.

Maxey Brooke. (Fun for the Money, Scribner's, 1963); reprinted as: Coin Games and Puzzles; Dover, 1973. All the following are on the 15 hole board.

Prob. 1: Triangular jump, pp. 10-11 & 75. Remove one man and jump to leave one man on the board. Says Wesley Edwards asserts there are just six solutions. He removes the middle man of an edge and leaves the last man there.

Prob. 2: Triangular jump, Ltd., pp. 12-13 & 75. Removes some of the possible jumps.

Prob. 3: Headless triangle, pp. 14 & 75. Remove a corner man and leave last man there.

M. Gardner. SA (Feb 1966) c= Carnival, 1975, chap. 2. Says a 15 hole version has been on sale as Ke Puzzle Game by S. S. Adams for some years. Addendum cites Brooke and Hentzel and says much unpublished work has been done.

Irvin Roy Hentzel. Triangular puzzle peg. JRM 6:4 (1973) 280-283. Gives basic theory for the triangular version. Cites Gardner.

[Henry] Joseph & Lenore Scott. Quiz Bizz. Puzzles for Everyone -- Vol. 6. Ace Books (Charter Communications), NY, 1975. Pennies for your thoughts, pp. 179-182. Remove a coin and solve. Hint says to remove the coin at 13 and that you should be able to have the last coin at 13. The solution has this property.

Alan G. Henney & Dagmar R. Henney. Computer oriented solutions. CM 4:8 (1978) 212 216. Considers the 'Canadian I. Q. Problem', which is the 15 hole board, but they also permit such jumps as 1 to 13, removing 5. They find solutions from each initial removal by random trial and error on a computer.

Putnam. Puzzle Fun. 1978. No. 15: Jumping coins, pp. 5 & 28. 15 hole version, remove peg 1 and leave last man there.

Benjamin L. Schwarz & Hayo Ahlburg. Triangular peg solitaire -- A new result. JRM 16:2 (1983-84) 97-101. General study of the 15 hole board showing that starting and ending with 5 is impossible.

J. D. Beasley. The Ins and Outs of Peg Solitaire. Op. cit. above, 1985. Pp. 229-232 discusses the triangular version, citing Smith, Gardner and Hentzel, saying that little has been published on it.

Irvin Roy Hentzel & Robert Roy Hentzel. Triangular puzzle peg. JRM 18:4 (1985-86) 253 256. Develops theory.

John Duncan & Donald Hayes. Triangular solitaire. JRM 23:1 (1991) 26-37. Extended analysis. Studies army advancement problem.

William A. Miller. Triangular peg solitaire on a microcomputer. JRM 23:2 (1991) 109-115 & 24:1 (1992) 11. Summarises and extends previous work. On the 10 hole triangular board, the classic problem has essentially a unique solution -- the removed man must be an edge man (e.g. 2) and the last man must be on the adjacent edge and a neighbour of the starting hole (i.e. 3 if one starts with 2). On the 15 hole board, the removed man can be anywhere and there are many solutions in each case.

Remove man from hole: 1 2 4 5

Number of solutions: 29760 14880 85258 1550

Considers the 'tree' formed by the first four rows and hole 13.
5.R.1.b. OTHER SHAPES
New section. See also King and Stewart in 5.R.1 for some forms based on a square board.

A