CH. i] ARITHMETICAL RECREATIONS 19 



game, played by two people, say A and B. A begins by 

 mentioning some number not greater tban (say) six, B may 

 add to that any number not greater than six, A may add 

 to that again any number not greater than six, and so on. 

 He wins who is the first to reach (say) 50. Obviously, if A 

 calls 43, then whatever B adds to that, A can win next time. 

 Similarly, if A calls 36, B cannot prevent A's calling 43 the 

 next time. In this way it is clear that the key numbers are 

 those forming the arithmetical progression 43, 36, 29, 22, 15, 

 8, 1 ; and whoever plays first ought to win. 



Similarly, if no number greater than m may be added at 

 any one time, and n is the number to be called by the victor, 

 then the key numbers will be those forming the arithmetical pro- 

 gression whose common difference is m + 1 and whose smallest 

 term is the remainder obtained by dividing n by m + 1. 



The same game may be played in another form by placing 

 p coins, matches, or other objects on a table, and directing each 

 player in turn to take away not more than m of them. Who- 

 ever takes away the last coin wins. Obviously the key numbers 

 are multiples of m + 1, and the first player who is able to leave 

 an exact multiple of (m + 1) coins can win. Perhaps a better 

 form of the game is to make that player lose who takes away 

 the last coin, in which case each of the key numbers exceeds 

 by unity a multiple of m + 1. 



Another variety* consists in placing p counters in the form 

 of a circle, and allowing each player in succession to take away 

 not more than m of them which are in unbroken sequence: 

 m being less than p and greater than unity. In this case 

 the second of the two players can always win. 



These games are simple, but if we impose on the original 

 problem the restriction that each player may not add the 

 same number more than (say) three times, the analysis 

 becomes by no means easy. I have never seen this ex- 

 tension described in print, and I will enunciate it at length. 

 Suppose that each player is given eighteen cards, three 

 of them marked 6, three marked 5, three marked 4, three 

 * y. Loyd, T'u-Biti, London, July 17, Aug. 7, 1897. 



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