240 MISCELLANEOUS PROBLEMS [CH. XI 



of the next two cards is placed on a t, and so on. Enquire 

 in which rows the two selected cards appear. If two rows are 

 mentioned, the two cards are on the letters common to the 

 words that make these rows. If only one row is mentioned, 

 the cards are on the two letters common to that row. 



The reason is obvious : let us denote each of the first pair 

 by an a, and similarly each of any of the other pairs by an 

 e, i, o, c, d, m, n, s, or t respectively. Now the sentence Matas 

 dedit nomen Gocis contains four words each of five letters ; ten 

 letters are used, and each letter is repeated only twice. Hence, 

 if two of the words are mentioned, they will have one letter in 

 common, or, if one word is mentioned, it will have two like 

 letters. 



To perform the same trick with any other number of cards 

 we should require a different sentence. 



The number of homogeneous products of three dimensions 

 which can be formed out of four things is 20, and of these the 

 number consisting of products in which three things are alike 

 and those in which three things are different is 8. This leads 

 to a trick with 8 trios of things, which is similar to that last 

 given — the cards being arranged in the order indicated by the 

 sentence La/nata levete livini novoto. 



I believe that these arrangements by sentences are well- 

 known, but I am not aware who invented them. 



Gergonne's Pile Problem. Before discussing Gergonne's 

 theorem I will describe the familiar three pile problem, the 

 theory of which is included in his results. 



The Three Pile Problem*. This trick is usually performed 

 as follows. Take 27 cards and deal them into three piles, face 

 upwards. By " dealing " is to be understood that the top card 

 is placed as the bottom card of the first pile, the second card in 

 the pack as the bottom card of the second pile, the third card 

 as the bottom card of the third pile, the fourth card on the top 

 of the first one, and so on : moreover I assume that throughout 



* The trick is mentioned by Bacbet, problem xvm, p. 143, but his analysis 

 of it is insufficient. 



