Sources page biographical material

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.