18 ARITHMETICAL RECREATIONS [CH. I 



iirst card of line 2, below that the first card of line 3, and below 

 that the first card of line 4. Turn this pile face downwards. 

 Next take up the four cards in the second column in the same 

 way, turn them face downwards, and put them below the first 

 pile. Continue this process until all the cards are taken up. 

 Ask someone to mention any card. Suppose the number of 

 pips on it is n. Then if the suit is A, it will be the 4nth card 

 in the pack ; if the suit is B, it will be the (in + 3)th card ; if 

 the suit is C, it will be the (4n. + 6)th card ; and if the suit is B, 

 it will be the (4re + 9)th card. Hence by counting the cards, 

 cyclically if necessary, the card desired can be picked out. It is 

 easy to alter the form of presentation, and a full pack can be 

 used if desired. The explanation is obvious. 



Medieval Problems in Arithmetic. Before leaving the 

 subject of these elementary questions, I may mention a few 

 problems which for centuries have appeared in nearly every 

 collection of mathematical recreations, and therefore may claim 

 what is almost a prescriptive right to a place here. 



First Example. The following is a sample of one class of 

 these puzzles. A man goes to a tub of water with two jars, 

 of which one holds exactly 3 pints and the other 5 pints. How 

 can he bring back exactly 4 pints of water? The solution 

 presents no difficulty. 



Second Example*. Here is another problem of the same 

 kind. Three men robbed a gentleman of a vase, containing 

 24 ounces of balsam. Whilst running away they met a 

 glass-seller, of whom they purchased three vessels. On reaching 

 a place of safety they wished to divide the booty, but found 

 that their vessels contained 5, 11, and 13 ounces respectively. 

 How could they divide the balsam into equal portions ? 

 Problems like this can be worked out only by trial. 



Third Example^. The next of these is a not uncommon 



* Some similar problems were given by Backet, Appendix, problem m, 



p. 206; problem rx, p. 233: by Oughtred or Leake in the Mathematicall 



Recreation*, p. 22: and by Ozanam, 1803 edition, vol. i, p. 174; 1840 edition, 



p. 79. Earlier instances occur in Tartaglia's writings. 



t Bachet, problem xxn, p. 170. 



