CH. I] ARITHMETICAL RECREATIONS 15 



by the use of a watch, the last four may be exemplified by the 

 use of a pack of playing cards. 



First Example *. The first of these examples is connected with 

 the hours marked on the face of a watch. In this puzzle some 

 one is asked to think of some hour, say, m, and then to touch a 

 number that marks another hour, say, n. Then if, beginning 

 with the number touched, he taps each successive hour marked 

 on the face of the watch, going in the opposite direction to that 

 in which the hands of the watch move, and reckoning to him- 

 self the taps as m, (m + 1), &c, the (n + 12)th tap will be on 

 the hour he thought of. For example, if he thinks of v and 

 touches ix, then, if he taps successively IX, vm, vn, vi, ..., 

 going backwards and reckoning them respectively as 5, 6, 7, 8, . . . , 

 the tap which he reckons as 21 will be on the V. 



The reason of the rule is obvious, for he arrives finally at 

 the (n + 12 — m)th hour from which he started. Now, since he 

 goes in the opposite direction to that in which the hands of 

 the watch move, he has to go over (n — m) hours to reach the 

 hour m : also it will make no difference if in addition he goes 

 over 12 hours, since the only effect of this is to take him 

 once completely round the circle. Now (n + 12 — m) is always 

 positive, since n is positive and m is not greater than 12, and 

 therefore if we make him pass over (n + 12 — m) hours we can 

 give the rule in a form which is equally valid whether m is 

 greater or less than n. 



Second Example. The following is another well-known 

 watch-dial problem. If the hours on the face are tapped suc- 

 cessively, beginning at vii and proceeding backwards round the 

 dial to VI, v, &c, and if the person who selected the number 

 counts the taps, beginning to count from the number of the hour 

 selected (thus, if he selected X, he would reckon the first tap 

 as the 11th), then the 20th tap as reckoned by him will be on 

 the hour chosen. 



For suppose he selected the nth hour. Then the 8th tap 

 is on XII and is reckoned by him as the (n + 8)th ; and the tap 



* Bachet, problem xz, p. 155; Oughtred or Leake, Mathematicall Recrea- 

 tion*, London, 1653, p. 28. 



