CH. XII] MISCELLANEOUS PROBLEMS 259 



Taylor*, and he recognised the desirability of finding the smallest 

 necessary number of cuts. 



Henceforth the explicit determination of the number of pieces 

 in which a figure has to be cut to give a solution was regarded 

 as an essential feature of the problem, and it is accepted that 

 that solution is to be preferred in which the number of pieces 

 is smallest. I have no doubt that Taylor's solutions do in fact 

 give the minimum number of pieces required in those cases. 



Puzzle Dissections. Of late years H. E. Dudeney has pro- 

 pounded various ingenious puzzles of this kind, in all cases the 

 number of parts being specified f. His reputation has served to 

 attract attention to this class of problems. As illustrating his 

 results and as geometrical recreations of this type, I pick out 

 his problems, (a) to divide, by two straight lines, a Greek cross, 

 that is, one made up of five equal squares, into four pieces all of 

 the same shape and size which can be put together to make a 

 Greek cross; (b) to divide an isosceles right-angled triangle into 

 four pieces which can be put together to make a Greek cross ; 

 (c) to divide a regular pentagon into six pieces which can be put 

 together to make a square; (d) to divide an equilateral triangle 

 into four pieces which can be put together to make a square. 

 The reader interested in the subject will like to compare 

 Dudeney's solution of the last question with that given by 

 Taylor already mentioned, and that by Macaulay referred to 

 below. 



Dudeney has also dealt with a few problems (to me, personally, 

 less interesting) in which more than two figures are concerned, 

 such as to divide a square into four pieces that will make two 

 equal Greek crosses; and to divide a Greek cross into five pieces 

 which will make two equal Greek crosses. 



Macaulay's Four-part Dissections. Recently the theory of 

 four-part dissections of pairs of rectilinear figures of equal area 

 has been discussed by W. H. Macaulay J. He has treated four- 



* Messenger of Mathematics, vol. xxxv, pp. 81—101. 



t For instance, see his Amusements in Mathematics, London, 1917, p. 27 el seq. 

 + Mathematical Gazette, 1914, vol. vii, p. 381 ; vol. viii, 1915, pp. 72, 109 ; 

 Messenger of Mathematics, vol. xlviii, 1919, p. 159; vol. xlix, 1919, p. 111. 



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