32 ABITHMETICAL RECREATIONS [CH. 11 



of times he has lost, has he gained or lost by the transaction ? 

 He has, in fact, lost. 



Here is another simple question to which not unfrequently 

 I have received incorrect answers. One tumbler is half-full 

 of wine, another is half-full of water : from the first tumbler 

 a teaspoonful of wine' is taken out and poured into the 

 tumbler containing the water: a teaspoonful of the mixture 

 in the second tumbler is then transferred to the first tumbler. 

 As the result of this double transaction, is the quantity of 

 wine removed from the first tumbler greater or less than the 

 quantity of water removed from the second tumbler ? In my 

 experience the majority of people will say it is greater, but 

 this is not the case. 



Here is another paradox dependent on the mathematical 

 theory of probability. Suppose that a player at bridge or whist 

 asserts that an ace is included among the thirteen cards dealt 

 to him, and let p be the probability that he has another ace 

 among the other cards in his hand. Suppose, however, that 

 he asserts that the ace of hearts is included in the thirteen 

 cards dealt to him, then the probability, q, that he has another 

 ace among the other cards in his hand is greater than was the 

 probability p in the first case. For, if r is the probability that 

 when he has one ace it is the ace of hearts, we have p = r . q, 

 and since p, q, r are proper fractions, we must have q greater 

 than p, which at first sight appears to be absurd. 



Permutation Problems. Many of the problems in per- 

 mutations and combinations are of considerable interest. As 

 a simple illustration of the very large number of ways in which 

 combinations of even a few things can be arranged, I may note 

 that there are 500,291833 different ways in which change for a 

 sovereign can be given in current coins*, including therein the 

 obsolescent double-florin, and crown; also that as many as 

 19,958400 distinct skeleton cubes can be formed with twelve 

 differently coloured rods of equal length f; while there are no less 

 than (52 !)/(13 !) 4 , that is, 53644,737765,488792,839237,440000 



* The Tribune, September 3, 1906. 



t Mathematical Tripos, Cambridge, Fart 1, 1894. 



