72 GEOMETRICAL RECREATIONS [CH. IV 



above is the case of n = 3. Let y denote the number of passages 

 from one bank to the other which will be necessary. Then it 

 has been shown that if n = 3, y = 11 ; if n = 4, y = 9 ; and if 

 n > 4, y = 7. 



The following analogous problem is due to the late E. 

 Lucas*. To find the smallest number so of persons that a boat 

 must be able to cany in order that n married couples may by 

 its aid cross a river in such a manner that no woman shall 

 remain in the company of any man unless her husband is 

 present ; it being assumed that the boat can be rowed by one 

 person only. Also to find the least number of passages, say y, 

 from one bank to the other which will be required. M. Delannoy 

 has shown that if n = 2, then x = 2, and y — 5. If n = 3, then 

 x = 2, and y = ll. If w = 4, then x=3, and y = 9. If n=5, 

 then x=B, and y = ll. And finally if n>5, then a; = 4, and 

 y = 2n-l. 



M. De Fonteney has remarked that, if there was an island 

 in the middle of the river, the passage might be always effected 

 by the aid of a boat which could carry only two persons. If there 

 are only two or only three couples the island is unnecessary, and 

 the case is covered by the preceding method. His solution, 

 involving 8n — 8 passages, is as follows. The first nine passages 

 will be the same, no matter how many couples there may be : 

 the result is to transfer one couple to the island and one couple 

 to the second bank. The result of the next eight passages is 

 to transfer one couple from the first bank to the second bank, 

 this series of eight operations must be repeated as often as 

 necessary until there is left only one couple on the first bank, 

 only one couple on the island, and all the rest on the second 

 bank. The result of the last seven passages is to transfer all the 

 couples to the second bank. It would however seem that if n 

 is greater than 3, we need not require more than Qn — 7 passages 

 from land to landf. 



M. G. Tarry has suggested an extension of the problem, 

 which still further complicates its solution. He supposes that 



* Lucas, vol. i, pp. 15—18, 237—238. 



■f See H. E. Dudeney, Amusements in Mathematics, London, 1917, p. 237. 



