Alone in the library of the Calhamer Club, I spent eighty minutes lost in
thought, devoted to my vain attempt to resolve what I had come to think of
as The Curious Case of the False Start.
On the surface, it seemed a simple enough problem, set for the British Crown
by the deviously clever Sultan of Suwat. To obtain an extension on an
important mineral lease at the Port of Suwat, my friend Sherlock Holmes
had been asked to supply the solution to the following puzzle:
We are given a Diplomacy board set up with the pieces in their starting positions,
but we are told that despite this fact, the game has already begun. Further, we
are told that although no neutral supply centres have been captured and all of the
players' home centres are in their own possession, it is also true that no piece on
the board started the game in its current position. In short, each power has managed
to depart from — and then return to — its original board position, without having
any of its original units sitting on the spot where it started the game.
We are given a Diplomacy board set up with the pieces in their starting positions, but we are told that despite this fact, the game has already begun. Further, we are told that although no neutral supply centres have been captured and all of the players' home centres are in their own possession, it is also true that no piece on the board started the game in its current position. In short, each power has managed to depart from — and then return to — its original board position, without having any of its original units sitting on the spot where it started the game.
The problem is to determine how quickly this rearrangement of pieces can take place. Not only that, but the Sultan required that the number of dislodgements be maximized, and that to ensure that the mystery was properly solved, the number of dislodgements should be brought by royal messenger to him. As if this was not enough, the Sultan's location (where he was waiting to receive this dispatch) was also left unspecified, except to say that the Sultan awaits the royal courier at the location into which the single unit that retreated in the puzzle withdrew after having been dislodged.
Before Holmes stepped away to dine with an acquaintance, leaving me in my puzzled state staring at the Diplomacy board, I had discussed the problem with my friend, exhausting all of the "trick question" possibilities. I was told that (far from being a variant game in which pieces do not begin the game in their usual location) the game is standard in every way. I also learned that the resolution to the problem did not revolve around the fact that the pieces had been accidentally or intentionally removed from the board, and simply replaced with different pieces of wood.
When Holmes returned to the game room, I was quite pleased with my efforts. "Holmes," said I, "I am happy to report to you that I have made significant progress toward solving the Sultan's puzzle."
"Indeed?" questioned Holmes. "Perhaps my continual urging to use your not inconsiderable talents for deduction have not been wasted! Let us see what you have determined."
With some measure of pride, I began. "I am fairly certain that the feat can be accomplished in a mere four movement phases. That is, a game could in fact be beginning 1903 with the positional and centre ownership situations identical to those as at the start of the game, but with the pieces in each position different from those that began the game there."
"Excellent," cried Holmes. "You are precisely correct. I admit, Watson, that I was slightly concerned for you after your initial ruminations had you sailing the Russian fleets all the way from one port to the other."
I flushed a bit at the recollection. My first thoughts had indeed concerned the lengthy trip that the two Russian fleets would need to make to dock themselves each in the other's port. With a twinkle in his eye, Holmes had left to take his sustenance while I was busily counting the length of time that would be required for a Sevastopol to St. Petersburg sail.
"Yes, Holmes," I responded. "I quickly realized that the task of rearranging the pieces could be done much quicker by cooperative action with foreign units. I reasoned that by dislodging the two fleets and rebuilding them with new fleets, the problem's central requirement (that the pieces on the board would be different from those that started the game there) would be fairly met."
"Quite so, Watson, and indeed dislodgements are the key to every facet of the solution to this fascinating puzzle."
"So I realized, Holmes, and I admit that it was with some measure of self-satisfaction that I have spent the past hour or more trying my level best to quickly return the board to its starting position — but with different pieces at each location — without taking possession of any neutral centres and while dislodging as many pieces as possible. It was while doing this that I realized with surprise that the task can be completed in a mere two game-years."
Holmes picked up from here. "Surprise, Watson? Surely it is more than obvious that it can be accomplished for any single power in two game-years."
"True, Holmes, but I admit I was astonished to find that all of the powers can do so in the same timeframe simultaneously."
"Indeed they can, Watson. In fact, as is easily seen, some powers can do it in half that time, finishing the task in only one game-year, although it is not a stipulation of the puzzle that any of the powers in fact do so."
"Yes, Holmes. France, for example, can arrange to have its fleet dislodged by an English attack into the Channel in Fall of 1901 and rebuild the lost fleet in the coming Winter adjustments phase, while having moved his Marseilles army to Paris and vice-versa."
"Correct, Watson. Like France, Austria, Russia, and Turkey can (with assistance) also accomplish the feat in a single year, by simply swapping the positions of their two armies while suffering the Falltime dislodgement of their fleet or fleets. As with France, the key, of course, is that the dislodged fleet is removed and then rebuilt, rather than retreated.
"Since the English army and the German and Italian fleets cannot possibly be dislodged and disbanded (and then rebuilt) in the first game-year, these three powers require two game-years to accomplish a rearrangement."
The great detective then concluded his lucid restatement of the task. "The ability of those four powers to accomplish the task in a single year is, of course, interesting, though not of consequence to us, as our task remains simply to accomplish it for all powers within the minimum time, which as you have found is two game-years, while maximizing the number of dislodgements."
"Yes, Holmes, and to this end, I have devoted the past hour to rather gleefully dislodging as many units as I could while recreating the game's starting position within the span of two years."
"Excellent, Watson, and to what conclusion have you come as to the number of dislodgements and the Sultan's location?"
"I am afraid, Holmes, that I remain a trifle confounded regarding those details. I have determined that a total of nine dislodgements can be made, but as to which single unit must have been retreated rather than disbanded, and the location into which the retreat must have been made, I am in a quandary. I fear that my nine dislodgements must simply be incorrect, for I am unable to determine from them where the Sultan awaits."
"Nine dislodgements is a fair attempt, Watson, and you are to be commended. You are correct, however, that a greater number is possible and until you discover those that you have thus far missed, the Sultan's location shall remain hidden to you."
"I regret, then, Holmes, that I feel I must request your assistance. I have been dislodging units willy-nilly throughout your luncheon, and despite being fairly sure in the knowledge (that you have just confirmed for me) that it must be possible — as I have begun to say — to 'squeeze' more dislodgements into the two game-years, I have been unable to better my total of nine."
"Right, then, Watson," said my friend, checking his pocket-watch. "Although my potted ferns will require their care before long, I believe that we have time for an illustration of the solution." With that, Holmes sat with me at the Diplomacy board, restored it to the starting position, and addressed me.
"Let us begin, of course, with the Spring 1901 moves. As you may have realized, the key moves of the solution will take place at the crossroads of Europe." At this point he pulled the Russian fleet on St. Petersburg's south coast to Livonia, the German fleet in Kiel into the Baltic Sea, and the Berlin army to Prussia, putting beyond doubt the short-lived fate of Russia's Northern fleet. He then pushed the Budapest army into Galicia and the Warsaw army into Silesia, where it found itself surrounded by red and black armies, when (rather to my surprise) he advanced the Munich army northeast into Berlin.
I had indeed determined that maximizing the number of dislodgements involved rushing the armies on the board toward each other. It seemed to me, and now it stood confirmed by Holmes, that the proximity of the German, Austrian, and Russian armies must be put to good use. However, while I was sure in the knowledge that the Bavarian army, being so centrally located at the start, would play a significant role, I wondered why Holmes would have chosen to concentrate the action where he did, putting the Munich army into Berlin, when he could just as well have let units congregate on Warsaw, Galicia, or, in cooperation with Vienna or Venice, on that very magnet of the board where it begins the game, Munich. I chose to stay silent at this time, and not reveal that this was where my thoughts had been attracted to for most of the evening.
"We have but two game-years to replace the Turkish fleet and Russia's southern fleets by dislodging and rebuilding them. Obviously, then, only one of the Sevastopol and Ankara fleets can be dislodged in 1901." Holmes moved the Turkish fleet to Armenia, the Sevastopol fleet into the Black Sea, the Moscow army to Sevastopol (both in preparation for a Falltime dislodgement of the Turkish fleet), and moved the Turkish armies into a position to allow them to end the first game-year in each other's original location, sending Constantinople to the vacated Ankara, and Smyrna into Constantinople. Turning his attention to the west, Holmes moved the French fleet from Brest to the English Channel, and to prepare a Falltime dislodgement of that fleet, Holmes then moved the English fleet from Edinburgh into the North Sea for cooperative action with London.
As if I should know his reasons, Holmes quickly moved the Italian units from Venice to Rome, from Rome to Tuscany, fleet Naples to Apulia, and the Austrian army from Vienna to Tyrolia and the Austrian fleet into Venice. My friend proceeded to finish with the French turn by pushing Paris to Burgundy and, to my surprise, Marseilles to Piedmont.
Taking all this in, I felt obliged to contribute something at this point. "I find your opening curious, Holmes. Most strange to me is your moving the Marseilles army to Piedmont. I had not considered such a thing and while I see that it masses a great many units together, the easy road to Paris through Gascony, to achieve the exchange of places of the two French armies, seemed far too obvious to avoid. Are you quite sure of your footing here? As you had mentioned to me, one of the powers meets the conditions a year early, and what you are proposing with Piedmont seemingly makes it impossible for France to do so....unless — I see! you intend to dislodge that army and rebuild it in Paris, with Burgundy moving to Marseilles to complete the French swap in the single year with one more dislodgement than I had divined! Brilliant!"
Holmes smiled. "Some but not all of that is indeed my intention, Watson. And yes, I believe we have found what may be the first of the dislodgements you overlooked."
"We have now created the game position going into Fall of 1901, Watson. Obviously, as with every standard game, no dislodgements have yet occurred. They will, however, begin to come quickly as we progress toward the solution."
"I can see that, Holmes. Here in the first Fall phase, you have managed to set up many units for destruction. The French fleet in the English Channel, the Turkish fleet in Armenia, the French army in Piedmont, the Russian army in Silesia, the Austrian fleet in Venice and the Russian fleet in Livonia are all vulnerable to attacks having the strength to dislodge."
"Well done, Watson. Indeed, each of these dislodgements does transpire. Attend while we step through the Fall turn. As you recounted, the French army in Piedmont is dislodged by an Austrian attack from Tyrolia, supported by the Italian army in Tuscany."
"Could it have been otherwise, Holmes? That is, could Italy have made the attack supported by Austria instead?"
"Perhaps so, Watson, but taking account of the geography in this part of the board, seeing as how only armies are involved here, and because Piedmont is not a supply centre, it matters not in this case. In other situations, however, the manner in which the units are dislodged is quite important to the solution. If the wrong unit is moved, the number of possible dislodgements is reduced."
Taking this as his lead, he then dislodged the French fleet by moving the English fleet from the North Sea, indicating that it was given support from the fleet in London.
"An amusing opening, Holmes. The normal move to develop the English position would be for North Sea to support London into the Channel. Why would you choose such a hideously compact form?"
"Foresight, my dear Watson. Knowing how the English army will be replaced, the choice should be obvious, no?"
I did not admit to the great detective that the question of the lone English army had stumped me before. It could not be swapped, so either would have to be dislodged or disbanded in Winter if a home centre were taken, something that was unlikely to happen now. But on the where and how, I was completely in the dark.
Holmes appeared to take no notice of my discomfort, as he pushed the Italian fleet from Apulia into Venice, indicating that it was supported in this move by the army in Rome, and in this way dislodging the Austrian fleet. This I could understand, as it put the Italian fleet immediately in peril from an attack by the Austrian army in Piedmont and a rebuilt Austrian fleet in Trieste. Furthermore the Italian armies would have no difficulty swapping positions, as Venice had already moved to Rome and the army in Tuscany was next to Venice.
Encouraged by this, I held out my hand to stop Holmes from picking up the next piece. "Allow me." The Turkish fleet in Armenia could either be dislodged by the Russian fleet in the Black Sea or the Russian army in Sevastopol. As in the previous case I reasoned that the fleet would need to be set up for a future dislodgement, but the army, which had started in Moscow, could end in Warsaw only if it stayed in Sevastopol this turn. Unhesitatingly I therefore attacked the Turkish fleet with the Russian fleet from the Black Sea, making the army in Sevastopol support the attack.
Before Holmes could comment I briskly tapped the army in Silesia, gesturing for a moment of thought. Either Galicia or Berlin could dislodge it. But if the attack came from Berlin, that army could just as well have held in Munich, as both spaces border Silesia. Knowing my friend's penchant for avoiding meaningless moves I concluded that Berlin was there to support, so I dislodged with the Austrian unit from Galicia instead. Observing this, Holmes seemed to relax as he took out his pipe and started to relight it.
For the Russian fleet in Livonia I had again a choice between the German fleet in the Baltic Sea and army Prussia. Applying the same logic as for Armenia I moved the fleet to Livonia supported by the army, as Prussia was only two moves away from Munich. While taking the Russian fleet off the board, I considered its options. The Prussian army could move either back through Berlin or through Silesia. Both spaces were occupied, Berlin by the second German army, Silesia by the Austrian army having just dislodged the Russian army there. Berlin would be able to support Prussia to Silesia, and again to Munich if an army would be waiting there. But what army? Tyrolia had just moved away to Piedmont and the French army in Burgundy was heading to Marseilles in order to complete the replacement in one year.
As I thoughtfully executed the move to Marseilles, I looked again at Silesia and the pieces taken off the board. And then it struck me! Barely hiding my excitement I picked up the discarded Russian army, placing it squarely in the centre of the board while triumphantly proclaiming: "It appears to me, Holmes, that you have informed the government to send a courier to Munich, where the Sultan will be waiting for him in anticipation!"
"Not unless I wanted the mission to fail," Holmes said shaking his head. "Did I not tell you that no centre would change hands at any time?"
"But that was not a criterion of the puzzle," I protested, before continuing in good stride: "Munich will be recaptured by Germany when Prussia moves through Silesia towards Munich, dislodging armies on every step and aided in this by the army in Berlin. As Prussia started in Berlin and Berlin in Munich, the end situation will have them swapped. Furthermore the temporary capture of Munich allows the disband of the German fleet, which can be rebuilt a year later when Munich is recaptured. Surely this much progress cannot be improved upon?"
Almost immediately I started to regret what I had just said. The great detective, with a bemused expression on his face, continued his attempts to relight his pipe before commenting: "Watson, you did an excellent job lining out the path that the German army in Prussia will be taking to arrive at the Bavarian heimat. However in your haste to get there, you missed a few opportunities to increase the number of dislodgements. Take this German fleet. Instead of disbanding, it would be more profitable if it could be dislodged, no?"
"Perhaps so, but it's stranded in the middle of nowhere with no enemy units in sight."
"They can be built. Tell me how many Russian units were dislodged."
"Well, Livonia of course, and Silesia. Ah, and no home centres lost. So Russia is entitled to two builds. The fleet in St. Petersburg and either Moscow or Warsaw, all border Livonia. So that's how you plan to dislodge the German fleet!"
"We have thus proven that there's no need to capture a home centre, that it even would prove to be counterproductive to do so." And with that Holmes picked up the Russian army in Munich and tossed it back in the box.
"But then there will be no unit in Munich to dislodge when the army from Prussia arrives there?"
"All in its own time. That army will only arrive in the Falltime, giving us one more season to station a victim there."
To my astonishment he then pushed the French army back to Burgundy.
"Holmes, wait, you said that one power would accomplish the feat in one year. If not France, then who?"
"I believe you overlooked this move," said Holmes, as he pushed the Turkish army in Ankara to Smyrna. "We have thus swapped the Turkish armies and are able to rebuild the dislodged fleet in its original home centre, all in the first year. Speaking of which, that Russian fleet could better stay in the Black Sea."
He moved the Russian fleet back and put the army from Sevastopol in its place.
"Really?" I exclaimed. "But then the army can't move back to Warsaw? You would need to dislodge them both!"
Holmes mysteriously pointed to the open space in Ankara. "See here the lion's den."
Smiling, he added: "We have now established the complete Fall 1901 season, and can move on to the Winter adjustments."
"Well, if there are no Fall retreats, ..." I said, leaving a small pause, but to my disappointment Holmes didn't interrupt. Vowing to do better in the second year I continued: "... the adjustments will hold no surprises. All the dislodged fleets are undoubtedly rebuilt in their starting centres."
I took up the discarded fleets and placed them back on the board, the Austrian in Trieste, the French in Brest, the Turkish in Ankara and the Russian on the south coast of St. Petersburg. "As for the armies, if you would do the honors?"
That was a smart move. With the French army holding in Burgundy France had two open build locations for its army, while Russia even had three. Of course in order to dislodge the German fleet Sevastopol would be of little use, but Warsaw and Moscow were equally eligible. It might not matter, but I had a feeling that Holmes still had something up his sleeve.
Holmes complied in good humor and unhesitatingly placed an army in Paris. Of course, the English army! To get rid of it Holmes was planning to convoy it to Picardy where it would be surrounded by the complete French military might!
Imitating the way he had referred to Ankara as the lion's den I pointed to the empty space of Picardy and said: "The pirate's cove." I laughed heartily at my own joke, but Holmes only frowned. "Picardy, Pirate's Cove. Pirate's Cove, Picardy. It's a wordplay!" I cried.
"I see, " Holmes replied and nodded. Then turning his attention to Russia he picked up the fleet I had placed in St. Petersburg, studied it for a moment, then said: "Undoubtedly this will come to you as a surprise, but it shouldn't. One should always take all possibilities into consideration. In this case the fleet is of limited help."
Putting the fleet aside he took two armies and placed them in Warsaw and Moscow.
"But then Russia will be lacking a fleet?"
"There's no condition that the same combination of units should exist throughout the game. To satisfy the condition of returning to the initial situation, it is sufficient to build the fleet at the end of the final year, which we are now ready to engage."
Holmes started off by moving the Austrian army from Piedmont to Venice indicating support from the fleet in Trieste to dislodge the Italian fleet stationed there. As if to further illustrate his previous point he remarked: "Observe how even this fleet could have been an army. In Fall, as the Italian army in Tuscany mounts an attack on Venice with Roman support, it relocates to Vienna or Budapest, allowing the dislodged army from Venice to be rebuilt as a fleet. But as this variation does not alter the result we will let it stand as is."
Turning his attention to the West he asked: "I believe you have discovered the reason for moving North Sea to the Channel?"
Eagerly I replied: "Indeed I have. The fleet in the Channel is needed to convoy the English army to the continent in Spring. In Fall it has to move to London, as that is the only English home centre bordering the Channel. Meanwhile London must move to Edinburgh, either through Yorkshire or the North Sea, to complete the fleet swap. If London would have moved to the Channel first, it wouldn't have been able to reach Edinburgh in time."
I moved the fleet from London to Yorkshire, then picked up the army in Wales and dropped it in Picardy, saying: "As to the destination of the convoy, it must be here, the pirate's cove!"
"Very well. Except that the true pirate's cove is not Picardy," said Holmes dryly, making the army trade places with the French fleet in Brest, "... but Brest. The reason is elementary, my dear Watson. If the English army were to be sent to Picardy, France would need to dislodge it there and would end up with a unit outside its home centres at the end of 1902. It is therefore that any unit dislodged in the final movement turn must be in a home centre belonging to the attacking power. The only exception is if that power is allowed to disband units in Winter after losing centres, something which is not the case here as no power has gained or lost any centres."
I had to admit that what he said made sense. "I can see that now. And Ankara fits the requirement, that's why you called it the lion's den. After the Turkish fleet moves out to dislodge the Russian army in Armenia with support from Smyrna, the Russian fleet in Black Sea can move in from behind. Turkey then simply needs to reverse its action to dislodge the Russian fleet in Ankara. It's as if it had unwittingly entered the lion's den only to be devoured when the animals return."
Executing the Turkish-Russian moves, Holmes smiled. "You're becoming poetic, Watson. To finish the puzzle only one part is left."
"But we have still not discovered the whereabouts of the Sultan. Now, Holmes, I have given this some thought. You might accuse me for being incorrigible, but if it's not the Russian army last turn, it must be the Austrian army this turn. Dislodged by Prussia with support from Berlin, it will retreat forward to Munich only to be dislodged once again by the same two armies. The Sultan must be in Munich after all!"
"As close as we have come to solving the riddle, it pains me nonetheless, Watson, to see that you have not yet understood the significance of the French army in Burgundy and the second Russian army build. A forward retreat, that cherished tactic in the arsenal of the offensive player, seeks out the best place to retreat only after all moves have been executed. As Silesia gets dislodged in the manner described, let Burgundy move to Munich and Warsaw dislodge the German fleet in Livonia supported by Moscow."
Executing the moves, Holmes held up the dislodged Austrian army, then firmly planted it in the vacated Russian home centre, saying: "Now here is where the Sultan is located. Instead of a single dislodgement, we have two, Munich and Warsaw. Had we built a fleet, we would have been unable to attack or support an attack on Warsaw, as Warsaw has no coastline. Quite elementary, Watson."
I sat stunned at the unassailable logic and beauty of my friend's reasoning. Holmes looked quizzically at me, then back at the board. Suddenly he stood up and paced to the lobby, where he had a short conversion with the desk clerk.
Clearly pleased with himself, he returned to his chair and sat down. Perplexed I asked: "Will you tell me what the sudden rush was?"
"With pleasure. I asked the clerk to transmit a message from me to the Foreign Office. It's an invitation to the Sultan of Suwat to visit London with excursions to Wales and the Channel islands. If they hurry, they can still wire the courier and let him deliver it to the Sultan along with the answer."
"For what purpose?"
"An innocent quid pro quo. Now, to return to the board, ..." said Holmes, indicating he considered the subject closed. "At this point, Watson, it should have become clear how many dislodgements transpired."
Obediently I started to count. "There were six in the first Fall season. In Spring I count four: Venice, Armenia, Silesia and Livonia. As for the second Fall season, let me see."
One after one I executed the final dislodgements: Picardy to Brest with support from Paris, dislodging the English army; Tuscany to Venice with support from Rome, dislodging the Austrian army; Armenia to Ankara with support from Smyrna, dislodging the Russian fleet; Silesia to Munich with support from Berlin, dislodging the French army; and Livonia to Warsaw with support from Moscow, dislodging the second Austrian army after it had retreated there. I also moved the English fleets from the Channel to London and from Yorkshire to Edinburgh.
Having done that, I announced: "And five in Fall 1902 for a grand total of fifteen. That's just fabulous, Holmes! Nearly two thirds of the starting units have been dislodged in a mere two years, one even twice, without ever increasing the unit tally as no neutral centres were captured. That's an average of five dislodgements per season!"
Holmes tapped his pipe to empty it. "I will agree with you that it is quite a marvel. Did you notice that the eight units that were never disbanded belong to a mere four powers? These four powers only had to replace the unit of which they only held one, while the other three ended up replacing every one of their units."
"You mean it is possible to divide the powers in two camps, those that lost all their starting units and those that kept their identical units by swapping them around? That's astonishing."
"In fact, there are many more remarkable facts about this solution. But let us first complete the circle by rebuilding all missing units."
And with that he placed one French, one English and two Austrian armies, and furthermore one German, one Italian and two Russian fleets in their respective home centres, restoring completely the starting position.
"How remarkable indeed," I marveled. "An equal number of armies and fleets respectively built by an equal number of powers. What a perfect equilibrium."
"Here are a couple of other curious facts in the solution as it was reconstructed here:
"In every season where there were dislodgements,
"Trivialities, Holmes. The more important question is: How can we know for sure that this is the optimal solution? Perhaps more dislodgements can be found in any one season? Take the first Fall season. Six dislodgements is awesome, but couldn't there be seven or eight?"
"You are correct that we have no ultimate proof, but we can gain confidence by exploring the alternatives. As a single dislodgement requires at a minimum three units involved, being the dislodged party, the dislodging party and the supporting party, when starting out with twenty-two units the maximum number of dislodgements is logically seven."
"Ah, right. And the limitations of the board setup will prevent us from reaching this maximum so soon."
"On the contrary, Watson, it is quite possible to get to seven dislodgements in Fall 1901, without even changing too much to the orders in our solution. The problem is rather that it is impossible from there to arrive at a solution."
Consulting his pocket-watch, Holmes rose up saying, "But before going into that,
Watson, we simply must return post-haste to the apartment in Baker Street, where my potted
ferns will be wanting my full attention."
Can you confirm for yourself the statements that Holmes made
in the time it takes Holmes and Watson to call a hansom cab and get back to
Baker Street? And what caused Holmes to invite the Sultan to London?
Was it something he saw on the Diplomacy board?
To be continued...
Can you confirm for yourself the statements that Holmes made in the time it takes Holmes and Watson to call a hansom cab and get back to Baker Street? And what caused Holmes to invite the Sultan to London? Was it something he saw on the Diplomacy board? To be continued...
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