e-i 



DISCOVERY 



is that the solution of niaiiy of its classical prohlt-ms is 

 intimately associated with the theory of functions of 

 continuous magnitude. One feature that distinguishes 

 it from most of the other branches of Mathematics is 

 the comparative simplicity with which its problems 

 can be stated ; not merely the easier problems, but even 

 those that have hitherto defied solution. For example : 

 What is the minimum number of squares which, when 

 added together, will equal any given number ? What 

 is the number of primes less than « ? Goldbach's 

 Theorem that every even number is the sum of two 

 primes. Waring's general problem and the theorj' of 

 Mersenne's numbers ; and the famous last theorem of 

 Fermat that x' + y' = z' has no solution in positive 

 integers if n is greater than 2. 



More detailed mention ma}- be made of one of these.' 

 In 1644 Mersenne published a book in which he asserted 

 that, in order that 2" — i may be a prime number, the 

 only values of p, not greater than 257, which are 

 possible are i, 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257 ; 

 61 has since been added to this list. To confirm 

 Mersenne's statement is an extremely laborious under- 

 taking, and it has been tested in a few cases only, but 

 so far as we know at present the statement is true. The 

 number 2" — i is the highest prime number at present 

 known. Evaluated it is 2,305,843,009,213,693,951. 

 The expression 2" ~ ' (2' — i) where 2" — i is a prime 

 number is considered to include all perfect numbers. 

 A number is said to be perfect if it is equal to the sum 

 of all its integral subdivisors. 28, for example, is 

 a perfect number because its divisors i, 2, 4, 7, and 14, 

 add up to 28, so is 6, so is 496. It is considered that 

 every perfect number must be expressible in the form 

 given above, because the expression has been shown 

 by Euler to include all even perfect numbers, and it is 

 believed that an odd number cannot be perfect. 



Now, the prime numbers represented by the expres- 

 sion 2" — I are known from Mersenne's work. Thus, 

 ii p = 2, the corresponding prime number is 2' — i, 

 i.e. 3, and the corresponding perfect number is 2""' 

 (2' — i), i.e. 4. \l p = 3, the corresponding prime 

 number is similarly 7, and the corresponding perfect 

 number is 28. If ^ = 5, the prime number is 31, the 

 perfect number 496. If /> = 7, the prime number is 

 127, and the perfect number 8,128. As the values of 

 p increase, the prime and perfect numbers increase 

 alarmingly. Thus, if /> = 61, the prime number corre- 

 sponding is 2,147,483,647, and the perfect number 

 2,305,843,008,139,952,128. It would be rather a long 

 and difficult task to factorise this last number and 

 show that the sum of the factors is equal to itself ! 



This simplicity in subject-matter makes it a fruitful 

 field for mathematical recreation. The problems are 

 numerous, and many of them may be found in the 

 ' This is taken frcm Mr. Rouse Ball's book. 



well-known work of Mr. W. W. Rouse Ball. The 

 questions that arise may be trivial or important ; they 

 may make problems of a day or problems for all time. 

 There is a game which may sometimes be seen played 

 for high stakes in the smoke-room of an Atlantic liner, 

 and which may serve as an illustration of the former 

 type. It has the appearance of being solely dependent 

 on the chance arrangements of the numbers concerned, 

 but in reality an expert can nearly always beat an 

 unskilful opponent. The contents of a box of matches 

 are divided at random into three heaps. The rule of 

 the game is that the one whose turn it is to play may 

 take one or all from any heap, but he must not touch 

 more than one heap. The loser is the one who is 

 forced to take the last match. If, for example, after 

 several moves, a player leaves his opponent i : i : i, 

 he must win, for his opponent is forced to take the last 

 match. Again, if he leaves i : 2 : 3, it is not difficult 

 to see that he can also force a win. The method of the 

 expert is to remember certain key-numbers which will 

 force a win from th'e initial outlay. The solution of the 

 problem of finding these key-numbers has been given 

 in an American journal,' and is very ingenious. It 

 consists in expressing the numbers in each heap in the 

 binar}^ scale. (On the binary scale, i is represented 

 by I, 2 by 10, 3 by 11, 4 by 100, 5 by loi, 6 by no, 

 7 by III, 8 by 1,000, 9 by 1,001, 10 by 1,010, 11 by 

 1,011, 12 by 1,100, 13 by 1,101, 14 by 1,110, 15 by 

 1,111, 16 by 10,000, and so on. See Scales oj Notation 

 in an Arithmetic Book.) If the sums of the correspond- 

 ing digits are all even, the outlay forms a key-number. 

 By corresponding digits are meant all the digits at the 

 right of the numbers, or all those second from the right ; 

 or all those third from the right, and so on. If some 

 are odd, it can be so altered according to the rules to 

 form the required combination. For example, if the 

 initial outlay were 8:7:9, these numbers in the binary 

 scale are 1,000: in: 1,001. The sums of the corre- 

 sponding digits are 2, i, i, 2. It is, therefore, not a 

 key-number, but can be made one by removing the 

 two I's, i.e. by changing the middle heap to i. A little 

 experience both in the theory of this problem and in 

 its practice will make one an expert in it. 



The above example illustrates the recreative aspect 

 of number, but gives no indication of the vast number 

 of ingenious problems and games of skill that have been 

 devised. The more serious types of problems may not 

 always be justly termed recreative, but they certainly 

 can be described as interesting. They are characterised 

 by their age and historical connections, by the simplicity 

 of their statements and by their intractability. This is 

 sufficient to command the attention and interest of 

 even the immature, and it would not be ridiculous to 

 imagine a schoolboy making a praiseworthy attempt 

 • W. W. Rouse Ball, Mathematical Recreations, p. 21. 



