OH. IV] GEOMETRICAL RECREATIONS 71 



so that from A to F they will be numbered successively 8 to 1 ; 

 and to do this by means which will involve as few transferences 

 of the engine or a wagon to or from the loop as is possible. 

 Twenty-six moves are required, and there is more than one 

 solution in 26 moves. 



Other shunting problems are not uncommon, but these two 

 examples will suffice. 



Ferry-Boat Problems. Everybody is familiar with the story 

 of the showman who was travelling with a wolf, a goat, and a 

 basket of cabbages ; and for obvious reasons was unable to leave 

 the wolf alone with the goat, or the goat alone with the cabbages. 

 The only means of transporting them across a river was a boat 

 so small that he could take in it only one of them at a time. The 

 problem is to show how the passage could be effected*. 



A somewhat similar problem is to arrange for the passage of 

 a river by three men and three boys who have the use of a boat 

 which will not carry at one time more than one man or two 

 boys. Fifteen passages are required f. 



Problems like these were proposed by Alcuin, Tartaglia, and 

 other medieval writers. The following is a common type of such 

 questions. Three J beautiful ladies have for husbands three 

 men, who are young, gallant, and jealous. The party are 

 travelling, and find on the bank of a river, over which they 

 have to pass, a small boat which can hold no more than two 

 persons. How can they cross the river, it being agreed that, in 

 order to avoid scandal, no woman shall be left in the society of 

 a man unless her husband is present? Eleven passages are 

 required. With two married couples five passages are required. 

 The similar problem with four married couples is insoluble. 



Another similar problem is the case of n married couples 

 who have to cross a river by means of a boat which can be 

 rowed by one person and will carry n — 1 people, but not more, 

 with the condition that no woman is to be in the society of a 

 man unless her husband is present. Alcuin's problem given 



* Ozanam, 1803 edition, vol. i, p. 171 ; 1840 edition, p. 77. 

 t H. E. Dudeney, The Tribune, October 4, 1906. 

 $ Baohet, Appendix, problem it, p. 212. 



