Sources page biographical material

5.R.2. FROGS AND TOADS: BBB_WWW TO WWW_BBB
In the simplest version, one has n black men at the left and n white men at the right of a strip of 2n+1 cells, e.g. BBB_WWW. One can slide a piece forward (i.e. blacks go left and whites go right) into an adjacent place or one can jump forward over one man of the other colour into an empty place. The object is to reverse the colours, i.e. to get WWW_BBB. S&B 121 & 125, shows versions.

One finds that the solution never has a man moving backward nor a man jumping another man of the same colour. Some authors have considered relaxing these restrictions, particularly if one has more blank spaces, when these unusual moves permit shorter solutions. Perhaps the most general form of the one-dimensional problem would be the following. Suppose we have m men at the left of the board, n men at the right and b blank spaces in the middle. The usual case has b = 1, but when b > 1, the kinds of move permitted do change the number of moves in a minimal solution. First, considering slides, can a piece slide backward? Can a piece slide more than one space? If so, is there a maximum distance, s, that it is allowed to slide? (The usual problem has s = 1.) Of course s  b. Second, considering jumps, can a piece jump backward? Can a piece jump over a piece (or pieces) of its own colour and/or a blank space (or spaces) and/or a mixture of these? If so, is there a maximum number of pieces, p, that it can jump over? (The usual problem has p = 1.) It is not hard to construct simple examples with s > 1 such that shorter solutions exist when unusual moves are permitted. Are there situations where one can show that backward moves are not needed?

The game is sometimes played on a 2-dimensional board, where one colour can move down or right and the other can move up or left. See: Hyde ??; Lucas (1883); Ball; Hoffmann and 5.R.3. Chinese checkers is a later variation of this same idea. On these more complex boards, one is usually allowed to make multiple jumps and the object is usually to minimize the number of moves to accomplish the interchange of pieces.

There is a trick version to convert full and empty glasses: FFFEEE to FEFEFE in one move, which is done by pouring. I've just noted this in a 1992 book and I'll look for earlier examples.

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.

American Agriculturist (Jun 1867). Spanish Puzzle. ??NYR -- copy sent by Will Shortz.

Anonymous. Every Little Boy's Book A Complete Cyclopædia of in and outdoor games with and without toys, domestic pets, conjuring, shows, riddles, etc. With two hundred and fifty illustrations. Routledge, London, nd. HPL gives c1850, but the material is clearly derived from Every Boy's Book, whose first edition was 1856. But the text considered here is not in the 1856 ed of Every Boy's Book (with J. G. Wood as unnamed editor), nor in the 8th ed of 1868 (published for Christmas 1867), which was the first seriously revised edition, with Edmund Routledge as editor, nor in the 13th ed. of 1878. So this material is hard to date, though in 4.A.1, I've guessed this book may be c1868.

P. 12: Frogs and toads. "A new and fascinating game of skill for two players; played on a leather board with twelve reptiles; the toads crawling, and the frogs hopping, according to certain laws laid down in the rules. The game occupies but a few minutes, but in playing it there is scarcely any limit to the skill that can be exhibited, thus forming a lasting amusement. (Published by Jaques, Hatton Garden.)" This does not sound like our puzzle, but perhaps it is related. Unfortunately Jaques' records were destroyed in WW2, so it is unlikely they can shed any light on what the game was. Does anyone know what it was?

Hanky Panky. 1872. Checker puzzle, p. 124. Three and three, with solution.

Mittenzwey. 1880. Prob. 239, pp. 44 & 94; 1895?: 267-268, pp. 48 & 96; 1917: 267-268, pp. 44 & 91 92. Problem with 3 & 3 brown and white horses in stalls. 1895? adds a version with 4 & 4.

Bazemore Bros. (Chattanooga, Tennessee). The Great "13" Puzzle! Copyright No. 1033   O   1883. Hammond & Jones Printers. Advertising puzzle consisting of two 3 and 3 versions arranged in an X pattern.

Lucas. RM2. 1883.

Pp. 141 143. Finds number of moves for n and n.

Pp. 144 145. Considers game on 5 x 5, 7 x 7, ..., boards and gives number of moves.

Edward Hordern's collection has an example called Sphinxes and Pyramids from the 1880s.

Sophus Tromholt. Streichholzspiele. (1889; 5th ed, 1892.) Revised from 14th ed. of 1909 by R. Thiele; Zentralantiquariat der DDR, Leipzig, 1986. Prob. 11, 41, 81 are the game for 4 & 4, 2 & 2, 3 & 3.

Ball. MRE, 1st ed., 1892, pp. 49 51. 3 & 3 case, citing Lucas, with generalization to n & n; 7 x 7 board, citing Lucas, with generalization to 2n+1 x 2n+1.

Berkeley & Rowland. Card Tricks and Puzzles. 1892. German counter puzzle, p. 112. 3 & 3 case.

Hoffmann. 1893. Chap. VI, pp. 269 270 & 282 284 = Hoffmann-Hordern, pp. 182-185, with photo.

No. 17: The "Right and Left" puzzle. Three and three. Photo on p. 184 shows: a cartoon from Punch (18 Dec 1880): The Irish Frog Puzzle -- with a Deal of Croaking; and an example of a handsome carved board with square pieces with black and white frogs on the tops, registered 1880. Hordern Collection, p. 77, shows the latter board and two further versions: Combat Sino-Japonais (1894 1895) and Anglais & Boers (1899-1902).

No. 18. Extends to a 7 x 5 board.

Puzzles with draughtsmen. The Boy's Own Paper 17 or 18?? (1894??) 751. 3 and 3.

Lucas. L'Arithmétique Amusante. 1895. Prob. XXXV: Le bal des crapauds et des grenouilles, pp. 117-124. Does 2 and 2, 3 and 3, 4 and 4 and the general case of n and n, showing it can be done in n(n+2) moves -- n² jumps and 2n steps. The general solution is attributed to M. Van den Berg. M. Schoute notes that each move should make as little change as possible from the previous with respect to the two aspects of changing type of piece and changing type of move.

Clark. Mental Nuts. 1904, no. 72; 1916, no. 62. A good study. 3 and 3.

Burren Loughlin & L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY, 1909. Doola's Game, pp. 42-43 & 61-62. 3 and 3.

Anon. Prob. 47: The monkey's dilemma. Hobbies 30 (No. 762) (21 May 1910) 168 & 182 & (No. 765) (11 Jun 1910) 228. Basically 3 & 3, but there are eight posts for crossing a river, with the monkeys on 1,2,3 and 6,7,8. The monkeys can jump onto the bank and we want the monkeys to all get to the bank they are headed for, so this is not the same as BBB..WWW to WWW..BBB. The solution doesn't spell out all the steps, so it's not clear what the minimum number of moves is -- could we have a monkey jumping another of the same colour?

Ahrens. MUS I. 1910. Pp. 17-19. Basically repeats some of Lucas's work from 1883 & 1895.

Williams. Home Entertainments. 1914. The cross-over puzzle, pp. 119-120. 3 and 3 with red and white counters. Doesn't say how many moves are required.

Dudeney. AM. 1917. Prob. 216: The educated frogs, pp. 59-60 & 194. _WWWBBB to BBBWWW_ with frogs able to jump either way over one or two men of either colour. Solution in 10 jumps.

Ball. MRE, 9th ed., 1920, pp. 77-79, considers the m & n case, giving the number of steps in the solution.

Blyth. Match-Stick Magic. 1921. Matchstick circle transfer, pp. 81 82. 3 and 3 in 15 moves.

Hummerston. Fun, Mirth & Mystery. 1924. The frolicsome frogs, Puzzle no. 2, pp. 17 & 172. Two 3 & 3 problems with the boards crossing at the centre cell. He notes that the easiest solution is to solve the boards one at a a time. He says: "It is not good play to jump a counter over another of the same colour."

Lynn Rohrbough, ed. Socializers. Handy Series, Kit G, Cooperative Recreation Service, Delaware, Ohio, 1925. Six Frogs, p. 5. Dudeney's 1917 problem done in 11 moves.

Botermans et al. The World of Games. Op. cit. in 4.B.5. 1989. P. 235 describes this as The Sphinx Puzzle, "very popular around the turn of the century, particularly in the United States and France" and they show an example of the period labelled The Sphinx and Pyramid Puzzle -- An Egyptian Novelty.

Haldeman-Julius. 1937. No. 162: Checker problem, pp. 18 & 29. 3 & 3.

See Harbin in 5.R.4 for a 1963 example.

Doubleday - 1. 1969.

Prob. 77: Square dance, pp. 93 & 171. = Doubleday - 5, pp. 103-104. Start with _WWWBBB. He says they must change places, with a piece able to move into the vacant space by sliding (either way) or by jumping one or two pieces of any colour. Asks for a solution in 10 moves. His solution gets to BBBWWW_, which does not seem to be 'changing places' to me.

Prob. 79: All change, pp. 95 & 171. = Doubleday - 5, pp. 105-106. BB_WW

Start with the pattern at the right and change the whites and BB_WW

blacks in 10 moves, where a piece can slide one place into an

adjacent vacant square or jump one or two pieces into a vacant square. However, the solution simply does each row separately.

Katharina Zechlin. (Dekorative Spiele zum Selbermachen; Verlag Frech, WWWWW

Stuttgart-Botnang, 1973.) Translated as: Making Games in Wood Games BWWWW

you can build yourself. Sterling, 1975, pp. 24-27: The chess knight game. BBOWW

5 x 5 board with 12 knights of each of two colours, arranged as at the right. BBBBW

The object is to reverse them by knight's moves. Says it can be done within BBBBB

50 moves and 'is almost impossible to do it in less than 45'.

Wickelgren. How to Solve Problems. Op. cit. in 5.O. 1974. Discrimination reversal problem, pp. 78 81. _WWWBBB to BBBWWW with the extra place not specified in the goal, with pieces allowed to move into the vacant space by sliding or by hopping over one or two pieces. Gets to BBBWW_W in 9 moves. [I find it takes 10 moves to get to BBBWWW_ .]

Joe Celko. Jumping frogs and the Dutch national flag. Abacus 4:1 (Fall 1986) 71-73. Same as Wickelgren. Celko attributes this to Dudeney. Gives a solution to BBBWWW_ in 10 moves and asks for results for higher numbers.

Johnston Anderson. Seeing induction at work. MG 75 (No. 474) (Dec 1991) 406-414. Example 2: Frogs, pp. 408-411. Careful proof that BB...BB_WW...WW to WW...WW_BB...BB, with n counters of each colour, requires n² + 2n moves.