ESSAY-REVIEWS 



TACTIC, by G. B. Mathews, M.A., F.R.S. : on Combinatory Analysis, 

 by Major P. A. MacMahon, F.R.S. , D.Sc, LL.D. [Vol. I (1915), 

 pp. XX + 300 ; Vol. II (1916), pp. XX + 340.] (Cambridge : at the 

 University Press.) 



The term " combinatorial analysis " hardly admits of exact definition, and 

 is not used in the International Schedule of pure mathematics. Broadly 

 speaking, it has come to • mean the discussion of problems which involve 

 selections from, or arrangements of, a finite number of objects ; or com- 

 binations of these two operations. For the purpose of this article it will 

 be convenient to use Sylvester's term " tactic " as a synonym for " com- 

 binatorial analysis." 



A few typical examples will illustrate these remarks. The number of 

 changes that can be rung on a peal of n bells of different pitches is the product 

 1.2.3. .. M, or, as it is usually written, n ! ; this is the same as the number 

 of arrangements of n different things in a straight line. If nr means the 

 number of ways of choosing r things from n different things, then 

 nr = n! I [n — r) ! r !, with the convention that o .' ■= i . (Of course r cannot 

 exceed n.) These two functions, n ! and «y, are fundamental in the theory. 



Another classical problem is that of the magic square. The digits i to 9 

 may be arranged in the square : 



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7 5 3 

 6 I 8 



where the sum of each row, column, and diagonal is the same, namely 15. 

 Such an arrangement is called a magic square ; and the general problem 

 is that of arranging the numbers i, 2, 3 ... w^ in a square satisfying the 

 above conditions, and of enumerating the possible solutions when n is 

 assigned. The literature of magic squares is extensive, but, so far as I 

 know, the theory has not been completed. For certain forms of n there 

 are rules for obtaining at least one solution ; the most intractable values 

 of n are those which are of the form 4W + 2, that is to say, n = 2., 6, 10, etc. 

 There is no solution for « = 2, and I do not remember seeing a solution 

 for any number of this class, or a proof that it is impossible, except for n =• 2. 

 Many problems of tactic were suggested by games of chance. With two 

 ordinary dice the number of different throws is 36, because each die, inde- 

 pendently, can fall in six different ways. But the sum of the pips on the 

 top faces can only range from 2 to 12 ; and if we reckon the number of 

 different throws giving the same sum, we can construct the table 



sum = 2, 3, 4, 3, 6, 7, 8, 9, 10, II, 12 

 number of throws = i, 2, 3, 4, 5, 6, 5, 4, 3, 2, i 



For instance, 5 = 1 + 4 = 2 + 3 = 3 + 2 = 4 + 1. This table enables us 

 to estimate the probability of a random throw leading to a given sum s ; 



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