CH. IV] GEOMETRICAL RECREATIONS 65 



we have in this case only to consider the values n = 3, n = 4, 

 and n = 5. If n = 3 we have m = 6. If n = 4 we have m = 4. 

 If n = 5, m is non-integral, and this is impossible. Thus the 

 only solutions are m = 3 and n = 6, m = 4 and » = 4, m = 6 

 and w = 3*- 



If, however, we allow the use of unlike equilateral tiles 

 (triangles, squares, &c), we can construct numerous geometrical 

 designs covering a plane area; though it is impossible to 

 do so by the use of such starred concave polygonsf. If at 

 each point the same number and kind of polygons are used, 

 analysis similar to the above shows that we can get six possible 

 superposable arrangements, namely when the polygons are 

 (i) 3-sided, 12-sided, 12-sided; (ii) 4-sided, 6-sided, 12-sided; 

 (iii) 4-sided, 8-sided, 8-sided; (iv) 3-sided, 3-sided, 6-sided, 

 6-sided; (v) 3-sided, 4-sided, 4-sided, 6-sided; (vi) 3-sided, 

 3-sided, 3-sided, 4-sided, 4-sided. 



The use of colours 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 alternately white and black. 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 a cells or square 

 tiles can be easily constructed by the repeated use of the four 

 elementary anallagmatic arrangements given in the angular 



* 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. n, 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 equations. 



u. ii. 5 



