CHAPTER IV.

GEOMETRICAL RECREATIONS CONTINUED.
Leaving now the question of formal geometrical proposi- tions, I proceed to enumerate a few games or puzzles which depend mainly on the relative position of things, but I post- pone to cliapter x the discussion of such amusements of this kind as necessitate any considerable use of arithmetic or algebra. Some writers regard draughts, solitaire, chess, and such like games as subjects for geometrical treatment in the same way as they treat dominoes, backgammon, and games with dice in connection with arithmetic: but these discussions require too many artificial assumptions to correspond with the games as actually played or to be interesting.
The amusements to which I refer are of a more trivial description, and it is possible that a mathematician may like to omit this chapter. In some cases it is difficult to say whether they should be classified as mainly arithmetical or geometrical, but the point is of no importance.
Statical Games of Position. Of the innumerable statical games involving geometry of position I shall mention only three or four.
Three-in-a-row. First, I may mention the game of three- in-a-row, of which noughts and crosses, one form of merrilees, and go-bang are well-known examples. These games are played on a board — generally in the form of a square con- taining n" small squares or cells. The common practice is for one player to place a white counter or piece or to make a cross on each small square or cell which he occupies : his opponent
CH. IV] GEOMETRICAL RECREATIONS 63
similarly uses black counters or pieces or makes a nought on each cell which he occupies. Whoever first gets three (or any other assigned number) of his pieces in three adjacent cells and in a straight line wins. There is no difficulty in giving the complete analysis for boards of 9 cells and of 16 cells: but it is lengthy and not particularly interesting. M(^st of these games were known to the ancients*, and it is for that reason I mention them here.
Three-in-a-row. Extension. I may, however, add an elegant but difficult extension which has not previously found its way, so far as I am aware, into any book of mathematical recreations. The problem is to place n counters on a plane so as to form as many rows as possible, each of which shall contain three and only three counters f.
It is easy to arrange the counters in a number of rows equal to the integral part of {n — iy/8. This can be effected by the following construction. Let P be any point on a cubic. Let the tangent at P cut the curve again in Q. Let the tangent at Q cut the curve in A. Let PA cut the curve in B, QB cut it in G, PC cut it in D, QD cut it in E, and so on. Then the counters must be placed at the points P, Q, A, B, .,.. Thus 9 counters can be placed in 8 such rows ; 10 counters in 10 rows ; 15 counters in 24 rows ; 81 counters in 800 rows ; and so on.
If however the point P is a pluperfect point of the nth order on the cubic, then Sylvester proved that the above con- struction gives a number of rows equal to the integral part of (n — l){n— 2)16. Thus 9 counters can be arranged in 9 rows, which is a well-known and easy puzzle; 10 counters in 12 rows; 15 counters in 30 rows ; and so on.
Even this however is an inferior limit and may be ex- ceeded— for instance, Sylvester stated that 9 counters can be placed in 10 rows, each containing three counters ; I do not know how he placed them, but one way of so arranging them is
* Becq de Fouqui^res, Lc Jeiix dcs Anciens, second edition, Paris, 1873, chop. XVIII.
t Educational Times Reprints, 18G8, vol. viii, p. lOO; Ihid. 188G, vol. xt.v, pp. 127—128.
64 GEOMETRICAL RECREATIONS [CH. IV
by putting them at points whose coordinates are (2, 0), (2, 2), (2, 4), (4, 0), (4, 2), (4, 4), (0, 0), (3, 2), (6, 4) ; another way is by putting them at the points (0, 0), (0, 2), (0, 4), (2, 1), (2, 2), (2, 3), (4, 0), (4, 2), (4, 4) ; more generally, the angular points of a regular hexagon and the three points (at infinity) of intersection of opposite sides form such a group, and therefore any projection of that figure will give a solution. At present it is not possible to say what is the maximum number of rows of three which can be formed from n counters placed on a plane.
Extension to p-in-a-row. The problem mentioned above at once suggests the extension of placing n counters so as to form as many rows as possible, each of which shall contain p and only p counters. Such problems can be often solved immediately by placing at infinity the points of intersection of some of the lines, and (if it is so desired) subsequently projecting the diagram thus formed so as to bring these points to a finite distance. One instance of such a solution is given above.
As examples I may give the arrangement of 10 counters in 5 rows, each containing 4 counters ; the arrangement of 16 counters in 1 5 rows, each containing 4 counters ; the arrange- ment of 18 counters in 9 rows, each containing 5 counters ; and the arrangement of 19 counters in 10 rows, each containing 5 counters. These problems I leave to the ingenuity of my readers.
Tesselation. Another of these statical recreations is known as tesselatioD, and consists in the formation of geometrical designs or mosaics covering a plane area by the use of tiles of given geometrical forms.
If the tiles are regular polygons, the resulting forms can be found by analysis. For instance, if we confine ourselves to the use of like tiles each of which is a regular polygon of n sides, we are restricted to the use of equilateral triangles, squares, or hex- agons. For suppose that to fill the space round a point where one of the angles of the polygon is situated we require m poly- gons. Each interior angle of the polygon is equal to (n—2)7r/n. Hence m (n — 2) w/n = 27r. Therefore (m - 2) (n - 2) = 4. Now from the nature of the problems m is greater than 2, and so is n. If m = 3, ?i = 6. If m > 3, then n 2,
CH. IV] GEOMETRICAL KECREATIONS 65
we have in this case only to consider the vahies n = 3, 7i=4, and ?i = 5. If w = 3 we have m = 6. If n = 4 we have in = 4. If n= 5, m is non-integral, and this is impossible. Thus the only solutions are m — S and n = 6, m=4 and w = 4, ?/i=6 and n = 3 *.
If, however, we allow the use of unlike tiles (ex. gr. regular hexagons, squares, and triangles), we can construct numerous geometrical designs covering a plane area; though it is im- possible to make such designs by the use of starred concave
polygons f.
The use of colours not only adds variety to the patterns, but introduces new considerations. One formation of a pavement by the employment of square tiles of two colours is illustrated by the common chess-board. In this the cells are coloured alter- nately white and black, or sometimes white and red. Another variety of a pavement made with square tiles of two colours was invented by SylvesterJ, who termed it anallagmatic. In the ordinary chess-board, if any two rows or any two columns are placed in juxtaposition, cell to cell, the cells which are side by side are either all of the same colour or all of different colours. In an anallagmatic arrangement, the cells are so coloured (with two colours) that when any two columns or any two rows are placed together side by side, half the cells next to one another are of the same colour and half are of different colours
Anallagmatic pavements composed of m^ cells or square tiles can be easily constructed by the repeated use of the four elementary anallagmatic arrangements given in the angular spaces of the accompanying diagram. In these fundamental forms A represents one colour and B the other colour. The diamond-shaped figure in the middle of the diagram represents an anallagmatic pavement of 256 tiles which is symmetrical about its diagonals. In half the rows and half the columns
* Monsieur A. Hermann has proposed an analogous theorem for polygons covering the surface of a sphere.
t On this, see the second edition of the French translation of this work, Paris, 1908, vol. ii, pp. 26—37.
X See Mathematical Questions from the Educational Times, London, vol. x, 1868, pp. 74—76 ; vol. lvi, 1892, pp. 97—99. The results are closely connected with theorems in the theory of equaiions,
B. E. 6
66
GEOMETRICAL RECREATIONS
[CH. IV
each line has 10 white tiles and 6 black tiles, and in the re- maining rows and columns each line has 6 white tiles and 10 black tiles. Such an arrangement, where the difference between the number of white and black tiles used in each line is constant, and equal to Vm, is called isochromatic. If m is odd or oddly even, it is impossible to construct anallagmatic boards which are isochromatic.
A
B
B
B
B
A
B
B
B
B
A
B
B
B
B
A
An Anallagmatic Isochromatic Pavement.
Interesting problems can also be proposed when the tiles are triangular, whether equilateral or isosceles right-angled. Two
equal isosceles right-angled tiles of different colours can be put together so as to make a square tile as shown in the margin. We can arrange four such tiles in no less than 256 different ways, making 64 distinct designs. With the use of more tiles the number of
CH. IV]
GEOMETRICAL RECREATIONS
67
possible ' designs increases with startling rapidity*. I content myself with giving two illustrations of designs of pavements constructed with sixty-four such tiles, all exactly alike.
Examples of Tesselated Pavements.
If more than two colours are used, the problems become increasingly difficult. As a simple instance of this class of problems I may mention one, sent to me by a correspondent who termed it Cross-Fours, wherein sixteen cardboard squares or tiles are used, the upper half of each being yellow, red, pink, or blue, and the lower half being gold, green, black, or white, no two tiles being coloured alike. Such tiles can be arranged in the form of a square so that in each vertical, horizontal, and diagonal line there shall be 8 colours and no more : they can be also arranged so that in each of these ten lines there shall be 6 colours and no more, or 5 colours and no more, or 4 colours and no more. Puzzles of this kind are but little known ; they are however not uninstructive.
Colour-Cube Problem. As an example of a recreation analogous to tesselation I will mention the colour-cube problem f. Stripped of mathematical technicalities the problem may be enunciated as follows. A cube has six faces, and if six colours are chosen we can paint each face with a diHPeient colour. By permuting the order of the colours we can obtain
* On this, see Lucas, Recreations Mathematiques, Paris, 1882-3, vol. ii, part 4 : hereafter I shall refer to this work by the name of the author.
t By Major MacMahon ; an abstract of his paper, read before the London Mathematicid Society on Feb. 9, 1893, was given in Naiwe, Feb. 23, 1893, vol. xLvn, p. 406.
6-2
68 GEOMETRICAL RECREATIONS [OH. IV
thirty such cubes, no two of which are coloured alike. Take anv one of these cubes, K, then it is desired to select eight out of the remaining twenty-nine cubes, such that they can be arranged in the form of a cube (whose linear dimensions are double those of any of the separate cubes) coloured like the cube K, and placed so that where any two cubes touch each other the faces in contact are coloured alike.
Only one collection of eight cubes can be found to satisfy these conditions. These eight cubes can be determined by the following rule. Take any face of the cube K: it has four angles, and at each angle three colours meet. By permuting the colours cyclically we can obtain from each angle two other cubes, and the eight cubes so obtained are those required. A little consideration will show that these are the required cubes, and that the solution is unique.
For instance suppose that the six colours are indicated by the letters a, 6, c, d, e, f. Let the cube K be put on a table, and to fix our ideas suppose that the face coloured / is at the bottom, the face coloured a is at the top, and the faces coloured 6, c, d, and e front respectively the east, north, west, and south points of the compass. I may denote such an arrangement by (/; a; 6, c, d, e). One cyclical permutation of the colours which meet at the north-east corner of the top face gives the cube (/; c; a, 6, d, e\ and a second cyclical permutation gives the cube (/; h\ c, a, d, e). Similarly cyclical permutations of the colours which meet at the north- west corner of the top face of ^ give the cubes (/; d\ b,a, c, e) and (/; c ; h^d, a, e). Similarly from the top south-west corner of K we get the cubes (/; e ; 6, c, a, d) and (/; d\ b, c, e, a) : and from the top south-east corner we get the cubes (/; e; a, c, d, b) and (/; 6 ; e, c, d, a).
The eight cubes being thus determined it is not difficult to arrange them in the form of a cube coloured similarly to K, and subject to the condition that faces in contact are coloured alike; in fact they can be arranged in two ways to satisfy these conditions. One such way, taking the cubes in the numerical order given above, is to put the cubes 3, 6, 8, and 2
CH. IV] GEOMETRICAL RECREATIONS 69
at the SE, NE, NW, and SW corners of the bottom face ; of course each placed with the colour /at the bottom, while 3 and 6 have the colour b to the east, and 2 and 8 have the colour d to the west : the cubes 7, 1, 4, and 5 will then form the SE, NE, NW, and SW corners of the top face; of course each placed with the colour a at the top, while 7 and 1 have the colour 6 to the east, and 5 and 4 have the colour d to the west. If K is not given, the difficulty of the problem is increased. Similar puzzles in two dimensions can be made.
Tangrams. The formation of designs by means of seven pieces of wood, namely, a square, a rhombus, and five triangles, known as tans, of fixed traditional shapes, is one of the oldest amusements in the East. Many hundreds of figures represent- ing men, women, birds, beasts, fish, houses, boats, domestic objects, designs, &c. can be made, but the recreation is not mathematical, and I reluctantly content myself with a bare mention of it.
Dynamical Games of Position. Games which are played by moving pieces on boards of various shapes — such as merrilees, fox and geese, solitaire, backgammon, draughts, and chess — present more interest. In general, however, they permit of so many movements of the pieces that any mathematical analysis of them becomes too intricate to follow out completely. Games in which the possible movements are very limited may be susceptible of mathematical treatment. One or two of these are given later : here I shall confine myself mainly to puzzles and simple amusements.
Shunting Problems. The first I will mention is a little puzzle which I bought some years ago and which was described as the "Great Northern Puzzle." It is typical of a good many problems connected with the shunting of trains, and though it rests on a most improbable hypothesis, I give it as a specimen of its kind.
The puzzle shows a railway, DBF, with two sidings, DBA and FGA, connected at A. The portion of the rails at A which is common to the two sidings is long enough to permit of a single wagon, like P or Q, running in or out of it ; but is too short to contain the whole of an engine, like R. Hence, if
70
GEOMETRICAL RECREATIONS
[CH. IV
an engine rnns up one siding, such as DBA, it must come back the same way.
Initially a small blork of wood, P, coloured to represent a wagon, is placed at 5 ; a similar block, Q, is placed at G \ and a longer block of wood, R, representing an engine, is placed at E. The problem is to use the^ engine R to interchange the wagons P and Q, without allowing any flying shunts.
Another shunting puzzle, on sale in the streets in 1905, under the name of the " Chifu-Chemulpo Puzzle," is made as follows. A loop-line BGE connects two points B and ^ on a railway track AF, which is supposed blocked at both ends, as shown in the diagram. In the model, the track AF is 9 inches long, AB = EF= If inches, and AH = FK=^BG== DE^l inch.
H K
On the track and loop are eight wagons, numbered succes- sively 1 to 8, each one inch long and one-quarter of an inch broad, and an engine, e, of the same dimensions. Originally the wagons are on the track from A \>o F and in the order 1, 2, 3, 4, 5, 6, 7, 8, and the engine is on the loop. The construction and the initial arrangement ensure that at any one time there cannot be more than eight vehicles on the track. Also if eight vehicles are on it only the penultimate vehicle at either end can be moved on to the loop, but if less than eight are on the track then the last two vehicles at either end can be moved on to the loop. If the points at each end of the ioop- line are clear, it will hold four, but not more than four, vehicles. The object is to reverse the order of the wagons on the track.
CH. IV] GEOMETRICAL RECREATIONS 71
SO that from ^ to i^ they will be numbered successively 8 to 1 ; and to do this by means which will involve as few transferences of the engine or a wagon to or from the loop as is possible. Twenty-six moves are required, and there is more than one solution in 26 moves.
Other shunting problems are not uncommon, but these two examples will suffice.
Ferry -Boat Problems, Everybody is familiar with the story of the showman who was travelling with a wolf, a .s^oat, and a basket of cal>bages ; and for obvious reasons was unable to leave the wolf alone with the goat, or the goat alone with the cabbages. The only means of transporting them across a river was a boat 80 small that he could take in it only one of them at a time. The problem is to show how the passage could be effected*.
A somewhat similar problem is to arrange for the passage of a river by three men and three boys who have the use of a boat which will not carry at one time more than one man or two boys. Fifteen passages are required f.
Problems like these were proposed by Alcuin, Tartaglia, and other medieval writers. The following is a common type of such questions. Three J beautiful ladies have for husbands three men, who -are young, gallant, and jealous. The party are travelling, and find on the bank of a river, over which they have to pass, a small boat which can hold no more than two persons. How can they cross the river, it being agreed that, in order to avoid scandal, no woman shall be left in the society of a man unless her husband is present ? Eleven passages are required. With two married couples five passages are required. The similar problem with four married couples is insoluble.
Another similar problem is the case of n married couples who have to cross a river by means of a boat which can be rowed by one person and will carry ?i — 1 people, but not m^re, with the condition that no woman is to be in the society of a man unless her husband is present. Alcuin's problem given
* Ozanam, 1803 edition, vol. i, p. 171 ; 1840 edition, p. 77. t H. E. Dudeney, The Tribune, October 4, 1906. X Bachet, Appendix, problem iv, p. 212.
72 GEOMETRICAL RECREATIONS [CH. IV
above is the case of n=S. Let y denote the number of passages from one bank to the other which will be necessary. Then it has been shown that if 7i=3, 3/= 11; if w = 4, y = 9; and if w > 4, 2/ = 7.
The following analogous problem is due to the late E. Lucas*. To find the smallest number x of persons that a boat must be able to carry in order that n married couples may by its aid cross a river in such a manner that no woman shall remain in the company of any man unless her husband is present ; it being assumed that the boat can be rowed by one person only. Also to find the least number of passages, say y, from one bank to the other which will be required. M. Delannoy has shown that if 71 = 2, then a; = 2, and y= 5. If n = S, then a; = 2, and 3/ = 11. If n = 4, then a? =3, and y = 9. If n=o, then a) = S, and 3/ = 11. And finally if n > 5, then ^ = 4, and y = 2n—l.