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 meet...at 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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