M. G. Tarry has suggested an extension of the problem,

which still further complicates its solution. He supposes that
* Lucas, vol. I, pp. 15—18, 237—238.
CH. IV] GEOMETRICAL RECREATIONS 73
each husband travels with a harem of m wives or concubines ; moreover, as Moliammedan women are brought up in seclusion, it is reasonable to suppose that they would be unable to row a boat by themselves without the aid of a man. But perhaps the difficulties attendant on the travels of one wife may be deemed sufficient for Christians, and I content myself with merely mentioning the increased anxieties experienced by Moham- medans in similar circumstances.
Geodesies. Geometrical problems connected with finding the shortest routes from one point to another on a curved surface are often difficult, but geodesies on a flat surface or flat surfaces are in general readily determinable.
I append one instance*, though I should have hesitated to do so, had not experience shown that some readers do not readily see the solution. It is as follows. A room is 30 feet long, 12 feet wide, and 12 feet high. On the middle line of one of the smaller side walls and one foot from the ceiling is a wasp. On the middle line of the opposite wall and 11 feet from the ceiling is a fly. The wasp catches the fly by crawling all the way to it: the fly, paralysed by fear, remaining still. The problem is to find the shortest route that the wasp can follow.
To obtain a solution we observe that we can cut a sheet of paper so that, when folded properly, it will make a model to scale of the room. This can be done in several ways. If, when the paper is again spread out flat, we can join the points repre- senting the wasp and the fly by a straight line lying wholly on the paper we shall obtain a geodesic route between them. Thus the problem is reduced to finding the way of cutting out the paper which gives the shortest route of the kind.
Here is the diagram corresponding to a solution of the above question, where A represents the floor, B and D the longer side-walls, G the ceiling, and W and F the positions on the two smaller side-walls occupied initially by the wasp and fly.
• This is due to Mr H. E. Dudeney. I heard a similar question propounded at Cambridge in 1903, but I first saw it in print in the Daily Hail, London, February 1, 1905.
74
GEOMETRICAL RECREATIONS
[CH. IV
In the diagram the square of the distance between W and F is (32)2 ^ (24)2; iiej^ce the distance is 40 feet.
w
«K
N.
Frohlems with Counters placed in a row. Numerous dynamical problems and puzzles may be illustrated with a box of counters, especially if there are counters of two colours. Of course coins or pawns or cards will serve equally well. I proceed to enumerate a few of these played with counters placed in a row.
First Problem with Counters. The following problem must be familiar to many of my readers. Ten counters (or coins) are placed in a row. Any counter may be moved over two of those adjacent to it on the counter next be3^ond them. It is required to move the counters according to the above rule so that they shall be arranged in five equidistant couples.
If we denote the counters in their initial positions by the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, we proceed as follows. Put 7 on 10, then 5 on 2, then 3 on 8, then 1 on 4, and lastly 9 on 6. Thus they are arranged in pairs on the places originally occupied by the counters 2, 4, 6, 8, 10.
Similarly by putting 4 on 1, then 6 on 9, then 8 on 3, then 10 on 7, and lastly 2 on 5, they are arranged in pairs on the places originally occupied by the counters 1, 3, 5, 7, 9.
If two superposed counters are reckoned as only one, solutions analogous to those given above will be obtained by
CH. IV] GEOMETRICAL RECREATIONS 75
putting 7 on 10, then 5 on 2, then .*> on 8, then 1 on 6, and lastly 9 on 4 ; or by putting 4 on 1, then 6 on 9, theu 8 on 3, then 10 on 5, and lastly 2 on 7 *.
There is a somewhat similar game played with eight counters, but in this ease the four couples finally formed are not equi- distant. Here the transformation will be effected if we move 5 on 2, then 3 on 7, then 4 on 1, and lastly 6 on 8. This form of the game is applicable equally to (8 + 2n) counters, for if we move 4 on 1 we have left on one side of this couple a row of (8 + 2n — 2) counters. This again can be reduced to one of (8 + 271 — 4) counters, and in this way finally we have left eight counters which can be moved in the way explained above.
A more complete generalization would be the case of n counters, where each counter might be moved over the m counters adjacent to it on to the one beyond them. For instance we may place twelve counters in a row and allow the moving a counter over three adjacent counters. By such movements we can obtain four piles, each pile containing three counters. Thus, if the counters be numbered consecutively, one solution can be obtained by moving 7 on 3, then 5 on 10, then 9 on 7, then 12 on 8, then 4 on 5, then 11 on 12, then 2 on 6, and then 1 on 2. Or again we may place sixteen counters in a row and allow the moving a counter over four adjacent counters on to the next counter available. By such movements we can get four piles, each pile containing four counters. Thus, if the counters be numbered consecutively, one solution can be obtained by moving 8 on 3, then 9 on 14, then 1 on 5, then 16 on 12, then 7 on 8, then 10 on 7, then 6 on 9, then 15 on 16, then 13 on 1, then 4 on 15, then 2 on 13, and then 11 on 6.
Second Problem with Counters. Another problem f, oi a somewhat similar kind, is of Japanese origin. Place four florins (or white counters) and four halfpence (or black counteis) alternately in a line in contact with one another. It is rec^uired
* Note by J. Fitzpatrick to a French translation of the third edition of tliis work, Paris, 1898.
t Bibliotheca Mxthematica, 1896, series 3, vol. vi, p. 323; P. G. Tait, Philo- Bophical Magazine, London, January, 1884, series 5, vol. xvii, p. 39; or Collected Scientijic i'ajjers^ Cambridge, vol. ii, 1690, p. 93.
76 GEOMETRICAL RECREATIONS [CH. IV
in four moves, each of a pair of two contiguous pieces, without altering the relative position of the pair, to form a continuous line of four halfpence followed by four florins.
This can be solved as follows. Let a florin be denoted by a and a halfpenny by h, and let x x denote two contiguous blank spaces. Then the successive positions of the pieces may be represented thus :
Initially . . . . xxabahabah.
After the first move . . baababaxxb.
After the second move . 6 a abxx a abb.
After the third move . . bxxbaaaabb.
After the fourth move . bbbb aaaax x.
The operation is conducted according to the following rule. Suppose the pieces to be arranged originally in circular order, with two contiguous blank spaces, then we always move to the blank space for the time being that pair of coins which occupies the places next but one and uext but two to the blank space on one assigned side of it.
A similar problem with 2n counters — n of them being white and n black — will at once suggest itself, and, if n is greater than 4, it can be solved in n moves. I have however failed to find a simple rule which covers all cases alike, but solutions, due to M. Delannoy, have been given* for the four cases where n is of the form 4m, 4?7i + 2, 47n + 1, or 4m + 3 ; in the first two cases the first ^n moves are of pairs of dissimilar counters and the last ^n moves are of pairs of similar counters; in the last two cases, the first move is similar to that given above, namely, of the penultimate and antepenultimate counters to the be- ginning of the row, the next J (w — 1) moves are of pairs of dissimilar counters, and the final J (n — 1) moves are of similar counters.
The problem is also capable of solution if we substitute the restriction that at each move the pair of counters taken up must be moved to one of the two ends of the row instead of the condition that the final arrangement is to be continuous.
* La Nature^ June, 1887, p. 10,
CH. IV] GEOMETRICAL KECKEATlONS 77
Tait suggested a varir-.tion of the problem by making it a condition that the two coins to be moved shall also be made to interchange places ; in this form it would seem that five moves are required ; or, in the general case, ti + 1 moves are reijuired.
Problems on a Cliess-hoard with Counters or Pawns. The following three problems require the use of a chess-board as well as of counters or pieces of two colours. It is more convenient to move a pawn than a counter, and if therefore I describe them as played with pawns it is only as a matter of convenience and not that they have any connection with chess. The first is characterized by the fact that in every position not more than two moves are possible ; in the second and third problems not more than four moves are possible in any position. With these limitations, analysis is possible. I shall not discuss the similar problems in which more moves are possible.
First Problem with Pawns*. On a row of seven squares on a chess-board 3 white pawns (or counters), denoted in the diagram by "a"s, are placed on the 3 squares at one end, and 3 black pawns (or counters), denoted by " b "s, are placed on the 3 squares at the other end — the middle square being left vacant. Each piece can move only in one direction ; the " a " pieces can move from left to right, and the *' b " pieces from right to left. If the square next to a piece is unoccupied, it can move on
a
a
a
b
b
b
to that ; or if the square next to it is occupied by a piece of the opposite colour and the square beyond that is unoccupied, thon it can, like a queen in draughts, leap over that piece on to the unoccupied square beyond it. The object is to get all the white pawns in the places occupied initially by the black pawns and vice versa.
The solution requires 15 moves. It may be effected by moving first a w^hite pawn, then successively two black pawns, then three white pawns, then three black pawns, then three white pawns, then two black pawns, and then one white pawn. We can express this solution by saying that if we number the
* Lucas, vol. n, part 5, pp. Ill — 143,
78
GEOMETRICAL RECREATIONS
[CH. IV
cells (a term used to describe each of the small squares on a chess-board) consecutively, then initially the vacant space occupies the cell 4 and in the successive moves it will occupy the cells 3, 5, 6, 4, 2, 1, 3, 5, 7, 6, 4, 2, 3, 5, 4. Of these moves, six are simple and nine are leaps.
Similarly if we have n white pawns at one end of a row of 2w + l cells, and n black pawns at the other end, they can be interchanged in n (m + 2) moves, by moving in succession 1 pawn, 2 pawns, 3 pawns, ... n — 1 pawns, n pawns, n pawns, n pawns, n — 1 pawns, ... 2 pawns, and 1 pawn — all the pawns in each group being of the same colour and different from that of the pawns in the group preceding it. Of these moves 2n are simple and n^ are leaps.
Second Problem with Pawns*. A similar game may be played on a rectangular or square board. The case of a square board containing 49 cells, or small squares, will illustrate this sufficiently: in this case the initial position is shown in the annexed diagram where the " a "s denote the pawns or pieces
a a
a
a
a a a a a a a
a a a
b b b b b
b
b
b b b
b b b
a
a
a a
a a
a
a a a
b b
b
b
b
b
b
b
b
b
of one colour, and the " b "s those of the other colour. The '* a" pieces can move horizontally from left to right or vertically down, and the " h " pieces can move horizontally from right to left or vertically up, according to the same rules as before.
The solution reduces to the preceding case. The pieces in the middle column can be interchanged in 15 moves. In the course of these moves every one of the seven cells in that column is at some time or other vacant, and whenever that
* Lucas, vol. H, part 5, p. 144.
CH. IV]
GEOMETRICAL RECREATIONS
79
is the case the pieces in the row containing the vacant cell can be interchanged. To interchange the pieces in each of the seven rows will require 15 moves. Hence to interchange all the pieces will require 15 + (7 x 15) moves, that is, 120 moves.
If we place 2?i(n+l) white pawns and 2n{7i+l) black pawns in a similar way on a square board of (2/1+1)^ cells, we can transpose them in 2n (n + 1) (?z + 2) moves: of these 4?i(7i4-l) are simple and 2n^ (^+1) are leaps.
Third Problem with Fawns. The followinsf analoorous problem is somewhat more complicated. On a square board of 25 cells, place eight white pawns or counters on the cells
a
b
c
—
d 9
e h
f
*
E B
G D
A
F
C
denoted by small letters in the annexed diagram, and eight black pawns or counters on the cells denoted by capital letters : the cell marked with an asterisk (*) being left blank. Each pawn can move according to the laws already explained — the white pawns being able to move only horizontally from left to right or vertically downwards, and the black pawns being able to move only horizontally from right to left or vertically upwards. The object is to get all the white pawns in the places initially occupied by the black pawns and vice versa. No moves outside the dark line are permitted.
Since there is only one cell on the board which is unoccupied, and since no diagonal moves and no backward moves are permitted, it follows that at each move not more than two pieces of either colour are capable of moving. There are how- ever a very large number of empirical solutions. In previous editions I have given a symmetrical solution in 48 moves, but the following, due to Mr H. E. Dudeney, is efiected in 46 moves :
nhg*F/c*CBHh*GDFjehbacf'GABHEFjdg*Hhbc*CFj''GHh*
80 GEOMETRICAL RECKEATIONS [CH. IV
the letters indicating the cells from which the pieces are suc- cessively moved. It will be noticed that the first twenty-three moves lead to a symmetrical position, and that the next twenty- two moves can be at once obtained by writing the first twenty- two moves in reverse order and interchanging small and capital letters. Similar problems with boards of various shapes can be easily constructed.
Probably, were it worth the trouble, the mathematical theory of games such as that just described might be worked out by the use of Vandermonde's notation, described later in chapter vi, or by the analogous method employed in the theory of the game of solitaire*.
Problems on a Chess-hoard with Chess-pieces. There are several mathematical recreations with chess-pieces, other than pawns. Some of these are given later in chapter VI.
Geometrical Puzzles with Rods, etc. Another species of geometrical puzzles, to which here I will do no more than allude, are made of steel rods, or of wire, or of wire and string. Numbers of these are often sold in the streets of London for a penny each, and some of them afford ingenious problems in the geometry of position. Most of them could hardly be discussed without the aid of diagrams, but they are inexpensive to construct, and in fact innumerable puzzles on geometry of position can be made with a couple of stout sticks and a ball of string, or with only a box of matches : several examples are given in various recent English works. Most of them exemplify the difiiculty of mentally realizing the effect of geometrical altera- tions in a figure unless they are of the simplest character.
Paradromic Rings. The fact just stated is illustrated by the familiar experiment of making paradromic rings by cutting a paper ring prepared in the following manner.
* On the theory of the solitaire, see Reiss, 'Beitrage zur Theorie des Solitdr- Spiels,^ Crelle's Joiirnal, Berlin, 1858, vol. uv, pp. oil — 379 ; and Lucas, vol. i,