224 



CHAPTER XL 

 MISCELLANEOUS PKOBLEMS. 



I propose to discuss in this chapter the mathematical theory 

 of a few common mathematical amusements and games. I 

 might have dealt with them in the first four chapters, but, since 

 most of them involve mixed geometry and algebra, it is rather 

 more convenient to deal with them apart from the problems 

 and puzzles which have been described already; the arrange- 

 ment is, however, based on convenience rather than on any 

 logical distinction. 



The majority of the questions here enumerated have no 

 connection one with another, and I jot them down almost at 

 random. 



I shall discuss in succession the Fifteen Puzzle, the Tower 

 of Hanoi, Chinese Rings, and some miscellaneous Problems 

 connected with a Pack of Cards. 



The Fifteen Puzzle*. Some years ago the so-called 

 Fifteen Puzzle was on sale in all toy-shops. It consists of a 

 shallow wooden box — one side being marked as the top — in the 

 form of a square, and contains fifteen square blocks or counters 

 numbered 1, 2, 3, ... up to 15. The box will hold just sixteen 

 such counters, and, as it contains only fifteen, they can be 

 moved about in the box relatively to one another. Initially 

 they are put in the box in any order, but leaving the sixteenth 



* There are two articles on the subject in the American Journal of 

 Mathematics, 1879, vol. n, by Professors Woolsey Johnson and Storey ; but 

 the whole theory is deducible immediately from the proposition I give in 

 the text. 



