Holmes was not giving me enough credit this time. It is true that his methods are often incomprehensible to me, but I had very solid reasons for my actions. Although I had erred in not specifying my thinking to him at the time, when I was hot on the trail of the elusive Moriarty, my movements were well-considered. I now carefully explained my rationale to him.
"Moriarty was traveling throughout the Continent while mindfully avoiding the province where I had laid my trap. Obviously, he was somehow aware of my plans and I was not going to capture him so easily. Once I became aware of this, I carefully perused the reports of his border crossings and hastened to the one border that he did not cross. I thought to beat him there and set up a new snare. I did have some trouble securing my passage and he must have crossed before I arrived."
Holmes sighed and directed his keen gaze towards me. "My dear Watson, while I agree that Moriarty would gain some twisted pleasure out of traversing every border except the ones that led to your snare, the perimeter reports and the particulars of the map should have made it clear to you that such a feat would be impossible even to one of his considerable talents."
"But Holmes," I responded, "you don't even know which border was uncrossed! I can assure you that it would not have been impossible for Moriarty to cross it. The border does not have the greatest security and Moriarty is a master of disguise after all. I realize that I forgot to include in my telegraph to you the location of the border as well as that of my original trap and so I can understand how you could reach a false conclusion."
At this, Holmes flashed me an annoyed look and for a moment I thought that he would be angry; however, after a brief instant the look faded and he let out a small chuckle. "Watson," said he, "your note was quite sufficient. I immediately was able to determine the locations of your original trap, the uncrossed border, and Moriarty himself."
I stood flabbergasted. "How can you expect me to believe such an outrageous claim? There must be fifty or so provinces and who knows how many borders! Not even you could have guessed the correct one."
"Forty-nine provinces and 106 borders to be exact and I did not guess; I deduced the proper locations."
I stuttered a bit, but could not come out with a reply to Holmes' claim. My confusion must have been evident, for he took a deep breath and began to elucidate.
"First, the number of borders each province has is significant. If we are to assume that the reports you received are accurate, then Moriarty did not pass through any border more than once."
"The reports were quite thorough," I interrupted. "I'm sure that all Moriarty's crossing were recorded."
"Quite so, Watson. Further, it should be evident that with each crossing, Moriarty either travels from inside the province to outside it or vice versa."
I nodded my agreement, though I did not see how something so patently obvious could help.
"It then follows that with every odd number of crossings involving a particular province, the professor's location relative to the province will change while with every even amount of crossings, his location will remain the same. Now let us consider a province with an odd number of borders. If Moriarty starts on the outside of this province and then crosses every one of its borders once and once only, he will end up inside the province with none of the borders left uncrossed! How is he to leave the territory and continue his journey without re-crossing one of the borders?"
I was having a difficult time following Holmes in this discourse and I answered his question before thinking. "So none of the provinces have an odd number of borders, then!"
"Tut, tut. A look at Portugal will show this conclusion to be incorrect. This province only has one border! We know that Moriarty starts his journey from Switzerland. How could he enter Portugal and then leave?"
"Oh, I see! He must have left by ship. The coasts are too extensive to be watched and the reports did not cover any of Moriarty's sea voyages."
"Good show, doctor. There are precisely twenty-six provinces with an odd number of land borders (Albania, Ankara, Apulia, Bohemia, Brest, Budapest, Constantinople, Finland, Galicia, Gascony, Greece, Holland, Kiel, Marseilles, Moscow, Norway, Piedmont, Portugal, Ruhr, Spain, Sweden, Switzerland, Tuscany, Tyrolia, Ukraine and Vienna). Of these, seventeen can be eliminated because they have a coastline. This leaves Bohemia, Budapest, Galicia, Moscow, Ruhr, Switzerland, Tyrolia, Ukraine, and Vienna that need to be accounted for. How could Moriarty travel through all of these provinces' borders?"
"He didn't. I already told you that he did not cross the borders leading to my trap and that there was still another uncrossed border."
"Yes, but which province didn't our prey enter? Clearly it was Vienna. By avoiding Vienna, this would remove a border from Tyrolia, Bohemia, Galicia, and Budapest leaving them with an even amount. It also has the effect of leaving Trieste with an odd amount of land borders, but Trieste has a coast so it is of no matter. Now these provinces have been accounted for and only Moscow, Ruhr, Switzerland, and Ukraine are left."
This was not very clear to me at all, but since I already knew that Vienna was indeed where my trap was laid, I did not press the issue.
"Since Switzerland is where he started, it having an odd number of borders poses no difficulty because he will end up outside the province after crossing them all. And since the Moscow-Ukraine border was not crossed, those provinces are eliminated. (I deduced that the Moscow-Ukraine border was left uncrossed because only this omission would "fix" two of our remaining three provinces.)"
Holmes' powers of deduction were astounding. However, there was something not quite right with his explanation. "That still leaves Ruhr!"
"Precisely, my good fellow. But you'll remember that this simply means that Moriarty could not have left Ruhr without crossing one of its borders a second time. This leads to the inevitable result that he was still there when you sent me your note. There I awaited your arrival, and it was from there that he evaded justice once again."
I felt quite humbled when Holmes had finished. His reasoning was flawless, and had led him to Moriarty's very hideout. Holmes explained later that no matter which of the vast number of possible routes Moriarty could have taken on his journey from Reichenbach, what I had learned about his having left one province unvisited and one additional border uncrossed had meant that the dastardly professor could only have completed the journey in Ruhr. Had I only the intellect of my friend, I would have been by his side and Moriarty in chains!
Despite this depressing thought, I couldn't help but feel that I had at least helped Holmes to track Moriarty, and I began to think that I was beginning to learn a few of the finer points of Diplomatic detection that my friend practiced. This feeling did not last long, however, for soon enough he was presented with another task that completely baffled me. To that task, which thankfully required no travel on our parts, I have given the name "The Dead-Letters Affair." The interested reader can find my account of it elsewhere in this publication.
-- Dr. John H. Watson
via Jonathan McCue
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