238 MISCELLANEOUS PROBLEMS [CH. XI 



case of 16 cards. The first and second arrangements may be 

 represented by figures i and ii. Suppose we are told that in 

 figure i the card is in the third row, it must be either 9, 10, 

 11, 12: hence, if we know in which row of figure ii it lies, it is 

 determined. If we allow the pack to be cut between the deals, 

 we must secure somehow that the top card is either 1, 2, 3, 

 or 4, since that will leave the cards in each row of figure ii 

 unaltered though the positions of the rows will be changed. 



Determination op a selected pair of cards out of 

 ^n(n + l) given pairs*. Another common trick is to throw 

 twenty cards on to a table in ten couples, and ask someone to 

 select one couple. The cards are then taken up, and dealt out 

 in a certain manner into four rows each containing five cards. 

 If the rows which contain the given cards are indicated, the 

 cards selected are known at once. 



This depends on the fact that the number of homogeneous 

 products of two dimensions which can be formed out of four 

 things is 10. Hence the homogeneous products of two dimen- 

 sions formed out of four things can be used to define ten things 



Suppose that ten pairs of cards are placed on a table and 

 someone selects one couple. Take up the cards in their 



00000 

 00000 

 000 



14 



18 19 20 



couples. Then the first two cards form the first couple, the 

 next two the second couple, and so on. Deal them out in 

 four rows each containing five cards according to the scheme 

 shown above. 



* Baohet, problem xvn, avertissement, p. 146 et seq. 



