Sherlock Holmes,
Consulting Diplomat

by Eric Wagoner and Derek McLachlin,
Budding Aenigmatists

Solution to The Dead-Letters Affair

Having dispatched emissaries, the evening before, on Holmes' recommendation, Lord Reginald Fortescue, the Foreign Secretary, returned to Baker Street after a sleepless night, anxious to hear my friend explain his reasoning in the curious case I have described under the title "The Dead Letters Affair."

Fortescue and I sat at the table as Holmes took his time filling his pipe-bowl. Fortescue's lack of sleep was apparent as he fidgeted in his chair. I, too, was anticipating Holmes' explanation -- my efforts to match his reasoning the day before had proven utterly unsuccessful.

Once his pipe was full, Holmes began his tale. "It might be best for me to use Dr. Watson's notes, which he prepared last night. Though they took him a full half-hour to prepare, the information contained within is all to be found in the Sultan's letters." He passed my notes over to Fortescue, who quickly read them over.

"Knowing the Sultan's propensity for setting out puzzles to be solved on the Diplomacy board, all we need to do is relate the clues (summarised here by the good doctor) to that arena, and the locations of the Sultan and his ministers will come clear."

"The Sultan clearly stated that no mobilization of forces has occurred (item one). So, we must look at the map as it is in Spring of 1901.

"Second, the ministers and the Sultan each have a military escort (items two and four). Thus, we can limit our search to twenty-two of the forty-nine provinces while also eliminating the possibility that any of the Suwati are afloat in some water space.

"Third, the ministers' escorts are both of the same type, either fleets or armies (item three). We have narrowed the possible locations for these two men to either thirteen (in the case of armies) or nine (if fleets).

"The ministers will arrive at the Sultan's location simultaneously (item five), and it would take longer for one of them to make a journey to the other minister than to travel to the Sultan's location (item six). This is a crucial piece of information.

"Finally, Watson's seventh and final item tells us that the ministers could meet each other at one of their present locations quicker than they could meet with the Sultan at his site."

Wishing to show Lord Fortescue that I had thought these through, I quickly added, "The escorts must be armies!" Holmes looked annoyed at my interruption, but nodded his head just the same.

"You are indeed right, my dear Watson, but to make that conclusion we must move beyond your notes. The phrase, 'One of you could indeed reach me faster than he could reach the other minister,' makes this clear. So, yes, the escorts are both armies."

Fortescue shook his head. "You are making a leap in logic, Holmes, one that at least eluded me last night -- much like my sleep! Slow down, if you don't mind." (The same leap had escaped me as well until I talked things through with Holmes, but I chose not to share this with the Foreign Secretary.)

Holmes lit his pipe, took a few puffs, and began again. "I'm surprised at you, milord! I've used nothing but elementary deductions. Nonetheless, I will slow down. To make things easier, I will call the two ministers A and B. Watson's sixth item specifies that minister A can reach the Sultan quicker than he can reach minister B. However, the Sultan also stated (in Watson's seventh point) that 'It will take more time for the pair of you to join me here than would be necessary for you two to one of your offices.' Thus, minister B can obviously reach A's location quicker than he could reach the Sultan!"

The Great Detective paused for effect, and when Fortescue's eyes registered that he understood at least that much, Holmes continued. "Remembering that minister A can reach the Sultan quicker than he can reach minister B, and now knowing that minister B is in exactly the opposite situation, the ministers must be in two cities situated such that travel between them in one direction is quicker than travel the opposite way."

"Yes!" exclaimed Fortescue. "Now I see why the escorts must be armies! Any two fleets, taking the shortest route, can reach each other's position in the same number of moves!" As Holmes puffed on his pipe, the Foreign Minister had time to reflect on his statement and soon enough his brow curled. "Wait, Holmes! What about St. Petersburg? Our fleet in Edinburgh can reach that city in only three seasons, but the Russian fleet is docked in the south, leaving it a five season voyage to Scotland!"

"You must be fatigued, milord," spoke Holmes. "Recall item four, which tells us that both ministers can reach the Sultan at the same time. Knowing this, we are left to conclude that if the voyage from one minister takes (as you have described for Edinburgh to St. Petersburgh) three moves, and the reverse journey (again, as for your St. Petersburgh to Edinburgh) takes five moves, the Sultan must be in a city that is four moves distant from both ministers' locations. No city exists that is four moves away from both St. Petersburg and Edinburgh. The expression simply cannot be true for any pair of fleets. It is this that enables us to finally conclude that the ministers must be accompanied by armies."

Fortescue still looked puzzled. "But Holmes, I am afraid I still do not understand. Fleets can exist on one of two coasts, but armies have no such possible distinction. How can one army take longer to travel to another's position than vice-versa? The most direct route between them must be the same distance!"

Holmes shook his head. "No, that won't do, Lord Secretary, that won't do at all. The distance may be the same, but the rapidity of travel may differ! Watson, would you care to solve this little paradox that so confounds our friend?"

I was caught by surprise, but was heartened by Holmes' faith that I could provide the answer. I had, in fact, thought about this for some time last night. "It seems to me that what is needed is a convoy, constructed such that the fleets are achieving the required positions simultaneously with necessary overland movement by minister B. For example, in Spring of 1901, Budapest can reach Constantinople in two moves (with a convoy through the Black Sea) but it would take three moves for the Constantinople army to have made the reverse journey. This difference arises because Constantinople must wait for the fleet in Sevastopol to take up position in the Black Sea before the convoy, but Budapest can make progress while the fleet is in motion."

I was proud of my explanation, but eager not to be put on the spot again, for this is as far as I could go. I made clear that Holmes should take back the reins when I added: "With what you've told us, however, we must find cities with a difference in travel times from each to the other of at least two moves. I must admit I do not yet have an answer. It seems that may require a very long convoy."

"Excellent, Watson! I have always known that you possessed a fine mind, if only you used it to its fullest capabilities."

While pleased by Holmes' compliment, I was not fond of the way in which it was delivered. I was thinking to retort when Fortescue cut in.

"Forgive me once more, gentlemen," he said, "but why must it be the case that the travel time from one of the minister's sites to the other must be two turns longer than the travel time for the reverse journey?"

Holmes patiently explained. "Remember that one minister, earlier identified as B, can reach the Sultan quicker than he can reach minister A, and that minister A..."

"Ah yes, I see now!" exclaimed Fortescue. "Minister A is quicker to minister B's site than to the Sultan's, and yet we are told that the two ministers can reach the Sultan in the same number of moves!"

"Precisely," was Holmes' calm response.

The Foreign Secretary was only just beginning, however. "Letting the length of time for the ministers to reach the Sultan be n, we see that the travel time from minister B to minister A is at most n-1, and the travel time from minister A to minister B is n+1 or greater, making a difference of at least two turns!"

Holmes smiled. "I'm glad to see you are waking up, milord. To help you along further, consider that the shorter journey must be longer than two turns in length; if it required only two, and thus only one move to set up the convoying fleet or fleets (such as the example -- Budapest to Constantinople -- that Watson cited), then the longer journey would always be possible in three turns or fewer. Given that we know that the difference must be at least two turns, we now know that the quicker journey must take at least three turns.

"Consider also -- a point of common knowledge, surely -- that no army begins a Diplomacy game more than five turns distant from any other army's location by the fastest available route. Using this, we know that the longer journey can take no more than five turns. Now, since the shorter journey must take at least three turns, and since the two journeys must differ by at least two turns in their length, we can conclude that the longer journey must take exactly five, and the shorter trip must take exactly three."

That no army is more than five moves distant from any other at the gamestart is not at all (to my mind) "a point of common knowledge," and while it was by no means an obvious fact to me at the time, I did not interrupt Holmes for an explanation. Lord Fortescue was likewise willing to accept the point. (I did, however, check Holmes' assertion myself at a later time, and found his statement to be correct.)

"We are therefore seeking two cities, travel between which takes three turns in one direction and five in the other," Holmes continued. "As it happens there are a mere nine pairs of cities for which travel in at least one direction takes five turns." Holmes moved to the Diplomacy board he kept set up in the corner of the room. "They are Smyrna to Munich, Constantinople to Munich, Venice to Moscow, Rome to Warsaw, Rome to Moscow, Marseilles to Moscow, Paris to Budapest, Liverpool to Vienna, and Liverpool to Budapest.

"You will notice," Holmes went on, "that many of these five-turn routes require the use of a lengthy convoy chain. The skillful use of convoys is necessary to solve the puzzle. The longest convoy chain that can be set up in the first two moves of the game is created as follows. Kiel to the North Sea, Edinburgh to the English Channel, London to the Mid-Atlantic, Brest to the Western Mediterranean, Naples to the Tyrrhenian Sea, Trieste to the Ionian Sea, and Ankara to the Aegean Sea -- all in two moves. Observe that the Adriatic, Ionian, and Aegean Seas can also be connected in the first two moves, as can the Barents, Norwegian and North Seas. As an example, consider the movement of an army from Smyrna to Munich. The army must wait for two turns for the convoy chain to be assembled, but then can travel all the way to Belgium or even Denmark in a single season, from whence it is two moves to Munich. Alternatively, the army could be convoyed to Venice or Trieste via the Aegean, Ionian, and Adriatic, and could again reach Munich overland in two turns." Lord Fortescue and I took the time to verify that the shortest possible route from the first city Holmes listed in each of the nine pairs to the second city was indeed five turns.

"Of these nine pairs," Holmes went on, "Rome to Warsaw, Marseilles to Moscow, and Paris to Budapest can be eliminated because the reverse journey takes longer than three turns." (Fortescue and I again made Holmes pause while we verified this.) "Furthermore, we know that the journey from each minister to the Sultan must take four turns, and thus we can eliminate the pairs involving Munich and Berlin, since an army in either of these locations can reach any city on the map in three turns or less."

Once more, Lord Fortescue and I again checked Holmes' statement by scrutinizing the board while Holmes stood puffing on his pipe with a bemused expression on his face. "Very good, Holmes," the Foreign Secretary said. We are now left with four possible combinations: Venice to Moscow, Rome to Moscow, Liverpool to Vienna, and Liverpool to Budapest. How are we to decide from among them?"

"We have established that the ministers must each take four turns to reach the Sultan. We must now look for cities that are four turns away from both cities in each possible pair. Care to take a stab at it, Watson?"

I studied the board. "Well, Paris is four turns away from both Budapest and from Vienna."

"Very good, Watson. Paris is in fact the only city four turns away from Vienna; the only other city equally as far from Budapest is Marseilles."

"Aha!" exclaimed the Foreign Secretary. "But Liverpool can reach Paris or Marseilles in three turns! Therefore, both pairs involving Liverpool are ruled out, and the ministers cannot be in Liverpool, Vienna, or Budapest!"

I finished Fortescue's thought. "...And one of the ministers must be in Moscow!" My heart raced at the deduction. "But is the other in Rome or Venice?"

Holmes face beamed with the air of a man enjoying himself immensely. "Yes, that is the key question, Watson. But let us leave it for the moment, for we now have enough information to determine the location of the Sultan!"

"Well, he must be four moves away from Moscow...." I pored over the board. "By moving to Norway in two turns, and then convoying to Picardy, Moscow can reach Paris in four turns."

"Also," Lord Fortescue pointed out, "One move to Sevastopol, then a convoy to Constantinople, followed by a long convoy to Spain, and Moscow can reach Marseilles in four turns. I do believe that once again, Paris and Marseilles are the only two options." He studied the board a moment longer. "But both Venice and Rome can reach Marseilles in fewer than four turns!" Lord Fortescue exclaimed. "The Sultan must be in Paris!" He was beside himself with excitement at this deduction.

"You have an excellent mind, once aroused from fatigue, milord!" Holmes cried with a chuckle. These 'discoveries' were, of course, old news to him. "Now we return to Watson's question. Is the second minister in Venice or Rome?"

We examined the board closely, and then Lord Fortescue became quite alarmed. "Holmes," he exclaimed, "surely there is no way to tell! Both Rome and Venice are four moves from Paris and five turns from Moscow, while Moscow is three turns from both of them!"

I too was dismayed by these facts. "Surely the Sultan would not have posed a problem with two solutions, Holmes?"

"No indeed, Watson, our clever friend the Sultan would not. We now we must consider the clue that Watson omitted from his list of last night." He indicated my notes with a wave of his pipe. "The clue that Watson failed to detect lies in the Sultan's statement to his ministers, and reads '...I cannot reach either of your offices more quickly than you can come here....' Since the convoy route through the Mid-Atlantic, Western Mediterranean, and Tyrrhenian Sea could deliver the Sultan to Rome in three turns, faster than either minister could possibly reach Paris, the second minister must be located in Venice!"

"Brilliant, Holmes, brilliant," Lord Fortescue proclaimed with a sigh of relief. "If only my mind worked as fast as yours. I will be very pleased to make my report this afternoon to the Prime Minster that our lease is safe once again." He shook Holmes' hand heartily, and nodded in my direction. "I am afraid I must bid you good-morning, gentlemen, for I now find myself anxious to immerse myself in my more mundane duties, that I might obtain some much-needed rest." And with that he was quickly gone from the apartment.

"Well Holmes, you have saved the British government from embarrassment once again," I said. "But tell me, how did you arrive at this conclusion so quickly? It is obvious once you state the logic, but your speed astounds me!"

Holmes smiled and replied, "Watson, a good Diplomacy player studies very carefully the inter-relationships between the provinces. Many a fatal stab has arisen because an unwise fellow has underestimated the movement capabilities of his supposed ally. Once you have carried out a few crushing stabs with a lengthy and unexpected convoy (as no doubt our friend the Sultan has done!), you will begin to see the benefits of an intimate knowledge of this mode of transportation."

I doubted whether I would be able to carry out such a stab in any Diplomacy game, certainly not one in which my friend was involved. Just the same, I sat down at the board to step through his logic once more, while Holmes flopped into his reading chair. He sat there the rest of the day, puffing on his pipe and gazing into the distance, seemingly content to await the next conundrum that would come to us at 221B Baker Street.

-- John H. Watson
via Eric Wagoner ([email protected])
and Derek McLachlin ([email protected])

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