CH. IV] 



GEOMETEICAL RECREATIONS 



67 



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 possible designs increases with 

 startling rapidity*. I content myself with 

 giving two illustrations of designs of pave- 

 ments 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 take sixteen square 

 tiles, 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; or so that there 

 shall be 6 colours and no more; or 5 colours and no more; 

 or 4 colours and no more. 



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 different 

 colour. By permuting the order of the colours we can obtain 



* On this, see Lucas, Recreations Nathematiques, Paris, 1882-3, vol. n, part 4 ; 

 hereafter I shall refer to this work by the name of the author. 



t P. A. MacMahon, London Mathematical Society Proceedings, vol. xxrv, 

 1893, pp. 145 — 155; and New Mathematical Pastimes, Cambridge, 1921, 

 pp. 42—46. 



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