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 b 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, possible movements of the 

 pieces are so numerous that mathematical analysis is not 

 practicable, but in a few games the possible movements are 

 sufficiently limited as to permit 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, DEF, 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 JR. Hence, if 



