90 MONTMORT. 



157. Montmort's own solution of the problem respecting 

 Pharaon depends on the first mode of summation explained in Art. 

 156, which coincides with Montmort's own process. The fact that 

 in Montmort's result when q is odd, ^^ — 1 terms are to be taken, 

 and when q^ is even, q terms are to be taken and the last doubled, 

 depends on the different values we have to ascribe to 8^ (n, 0) ac- 

 cording as n is even or odd ; see Montmort's page 98. 



Montmort gives another form to his result on his page 99 ; 

 this he obtained, after the publication of his first edition, from 

 Nicolas Bernoulli. It appears however that a wrong date is here 

 assigned to the communication of Nicolas Bernoulli ; see Mont- 

 mort's page 299. This form depends on the second mode of sum- 

 mation explained in Art. 156. It happens that in applying this 

 second mode of summation to the problem of Pharaon ?i + r is 

 always odd ; so that in Nicolas Bernoulli's form for the result 

 we have only one case, and not two cases according as q is even 

 or odd. 



There is a memoir by Euler on the game of Pharaon in the 

 Hist de VAcad Berlin ioY 1764, in which he expresses the ad- 

 vantage of the Banker in the same manner as Nicolas Bernoulli. 



158. Montmort gives two tables of numerical results respect- 

 ing Pharaon. One of these tables purports to be an exact exhibi- 

 tion of the Banker's advantage at any stage of the game, supposing 

 it played with an ordinary pack of 52 cards ; the other table is an 

 approximate exhibition of the Banker's advantage. A remark may 

 be made with respect to the former table. The table consists of 

 four columns ; the first and third are correct. The second column 



w + 2 



should be calculated from the formula -r — -, -. , by puttino^ for n 



2n (n — 1) -^ ^ ^ 



in succession 50, 48, 46, ... 4. But in the two copies of the second 



edition of Montmort's book which I have seen the column is given 



3117 26 



incorrectly ; it begins with ' ^ instead of ^ , and of the re- 



maining entries some are correct, but not in their simplest forms, 



and others are incorrect. The fourth column should be calculated 



2n — 5 



from the formula ^w tv-/ i^ ? ^Y putting for n in succession 



z{n—l){n — 3) , 



50, 48, 46 ... 4 ; but there are errors and unreduced results in it; 



