Here is another puzzle by Henry Dudeney, from his book "Amusement in Mathematics":
"The diagram is supposed to represent the passages or galleries in a mine. We will assume that every passage, A to B, B, to C, C to H, H to I, and so on, is one furlong in length. It will be seen that there are thirty-one of these passages. Now, an official has to inspect all of them and he descends by the shaft to the point A. How far must he travel, and what route do you recommend? The reader may at first say, “As there are thirty-one passages, each a furlong in length a furlong in length, he will have to travel just thirty-one furlongs.” But this is assuming that he need never go along a passage more than once, which is not the case. Take your pencil and try to find the shortest route. You will soon discover that there is room for considerable judgment. In fact, it is a perplexing puzzle."
I will add another question to this puzzle. If the inspector could start at a different node (other than A), which node would allow you to find the shortest route?