It was required to show how a resident at the town marked A might visit every one of the towns once, and only once, and finish up his tour at Z. 
This puzzle conceals a little trick. 
After the solver has demonstrated to his satisfaction that it cannot be done in accordance with the conditions as he at first understood them, he should carefully examine the wording in order to find some flaw. 
It was said, "This would be easy enough if he were able to cut across country by road, as well as by rail, but he is not."

Now, although he is prohibited from cutting across country by road, nothing is said about his going by sea! 
If, therefore, we carefully look again at the map, we shall find that two towns, and two only, lie on the sea coast. 
When he reaches one of these towns he takes his departure on board a coasting vessel and sails to the other port. 
The annexed illustration shows, by a dark line, the complete route.