CH. XI] MISCELLANEOUS PROBLEMS 239 



The first couple (1 and 2) are in the first row. Of the next 

 couple (3 and 4), put one in the first row and one in the 

 second. Of the next couple (5 and 6), put one in the first row 

 and one in the third, and so on, as indicated in the diagram. 

 After filling up the first row proceed similarly with the second 

 row, and so on. 



Enquire in which rows the two selected cards appear. If 

 only one line, the mth, is mentioned as containing the cards, 

 then the required pair of cards are the mth and (m + l)th cards 

 in that line. These occupy the clue squares of that line. 

 Next, if two lines are mentioned, then proceed as follows. 

 Let the two lines be the pth and the gth and suppose q > p. 

 Then that one of the required cards which is in the gth line 

 will be the (q — p)th. card which is below the first of the clue 

 squares in the pth. line. The other of the required cards is in 

 the pth line and is the (q — p)th card to the right of the second 

 of the clue squares. 



Bachet's rule, in the form in which I have given it, is 

 applicable to a pack of n (n + 1) cards divided into couples, 

 and dealt in n rows each containing n + 1 cards ; for there are 

 \n(n+ 1) such couples, also there are ^n(n + l) homogeneous 

 products of two dimensions which can be formed out of n 

 things. Bachet gave the diagrams for the cases of 20, 30, and 

 42 cards : these the reader will have no difficulty in constructing 

 for himself, and I have enunciated the rule for 20 cards in a 

 form which covers all the cases. 



I have seen the same trick performed by means of a 

 sentence and not by numbers. If we take the case of ten 

 couples, then after collecting the pairs the cards must be dealt 

 in four rows each containing five cards, in the order indicated 

 by the sentence Maias dedit nomen Gods. This sentence must 

 be imagined as written on the table, each word forming one 

 line. The first card is dealt on the M. The next card (which 

 is the pair of the first) is placed on the second m in the sentence, 

 that is, third in the third row. The third card is placed on 

 the a. The fourth card (which is the pair of the third) is 

 placed on the second a, that is, fourth in the first row. Each 



