80 GEOMETRICAL RECREATIONS [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-board 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 Kods, 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 

 difficulty 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 Solitar- 

 Spiels,' Crelle's Journal, Berlin, 1858, vol. liv, pp. 344 — 379 ; and Lucas, vol. i, 

 part v, pp. 89—141, 



