256 



MISCELLANEOUS PROBLEMS 



[CH. XII 



dominoes, n (n* + 1)/5 different pentagonal dominoes, and so on. 

 With each of these sets we can make up puzzles. 



A particular case will illustrate the kind of questions treated. 

 Take the case of equilateral triangular dominoes. If four colours 

 are used we get a set of twenty-four different dominoes, and 

 these can be put together (preferably in a shallow box cut to 

 the right size) to make a regular hexagon. Innumerable con- 

 ditions may be imposed. For example, we may suppose that it 

 is required to fit the dominoes into the box so that the colours 

 of adjoining faces of the dominoes shall be the same, as also 

 those of all the exterior faces. Here is the solution of this prob- 

 lem. A consideration of what conditions may be imposed 



naturally arises, and leads to various other problems. If our 

 dominoes are right-angled triangles we get a hexagon arrange- 

 ment of a different shape. Further we can play with sets selected 

 from a particular full set and arranged in other geometrical 

 figures. Similar problems arise from the use of square dominoes, 

 hexagonal dominoes, etc. 



Geometrical Dissections. Problems requiring the division 

 by straight lines of a given plane rectilinear figure into pieces 

 which can be put together in some other assigned form are 

 well known. A class of Geometrical Recreations is concerned 

 with such constructions. 



Pythagorecm Extension. A familiar instance is found in many 

 text-books in a dissection proof of the Pythagorean property of 



