There are six towns, from which only two roads issue. 
Thus 1 must lie between 9 and 12 in the circular route. 
Mark these two roads as settled. 
Similarly mark 9, 5, 14, and 4, 8, 14, and 10, 6, 15, and 10, 2, 13, and 3, 7, 13. 
All these roads must be taken. 
Then you will find that he must go from 4 to 15, as 13 is closed, and that he is compelled to take 3, 11, 16, and also 16, 12. 
Thus, there is only one route, as follows: 1, 9, 5, 14, 8, 4, 15, 6, 10, 2, 13, 7, 3, 11, 16, 12, 1, or its reverse :reading the line the other way. 
Seven roads are not used.