"Incredible!" came the cry from Colonel Sykes, as he heard Sherlock Holmes proclaim that he had solved The Missing Master's Mystery. "I find it unbelievable that you could accomplish this!"
"Quite believable, my dear Colonel. A fascinating exercise in retrograde analysis, I must say, but quite soluble. If you would like, I would be happy to explain to you how the conundrum was resolved."
"Please, Holmes!" came the reply. And with that, our little group returned to the library, where Holmes had laid a Diplomacy board on the large reading table, along with the notes on the game that Colonel Sykes had provided.
"The problem, gentlemen, was simply to deduce, from the spare information left by the esteemed Captain Meriwether -- who was hastily called into Her Majesty's service and away from the game -- the position of the pieces currently on the board, so that Colonel Sykes could take over as Game Master of a particular ongoing Diplomacy game.
"This exercise is necessary, and player information cannot be relied upon, because the game in question is 'blind,' meaning that each player is only informed of the existence of any unit that comes adjacent to one of his own home supply centres or one of his own units."
"Yes, yes; we know all this, Holmes," I protested, eager to see if my friend had indeed been able to accomplish what seemed so impossible to me.
"A necessary preamble, Watson. Colonel, do you agree that this is the task at hand, and that my description of the variant in use is accurate?"
"I do," said the Colonel matter-of-factly. I could tell that Colonel Sykes, too, was anxious for Holmes to begin his explanation.
"That is well. I wished merely to double-check that I did not labor under a misapprehension of the manner in which this 'blind' game was conducted. Sometimes players are informed of any unit that comes adjacent to any territory within their border, not simply of units that come adjacent to their home supply centres."
"Yes, Holmes, I am aware of that mode of play. But your description of this game's parameters is accurate. A player cannot 'see' another player's unit unless it is adjacent to one of the player's own home supply centres or to one of the player's own units."
"This is important, Colonel, so forgive my continued questioning, but I must be sure that I interpreted the rules of the variant correctly."
Colonel Sykes was growing impatient. "Ask away, Holmes, but I know you are well aware of the standard mode of 'blind' play here at the Club. This game adheres to those rules, to the letter."
"I am sure it does, Colonel, but for our friend Watson's benefit, then, allow me to make two further enquiries, after which we shall then begin to bring some light to the cloudy situation. First, am I to understand that at the beginning of the game, the German player cannot 'see' the Russian army that begins the game in Warsaw?"
"Yes, Holmes. The army in Warsaw is not adjacent to any German home centre nor is it adjacent to a German unit. Therefore, the German cannot 'see' the Russian army in Warsaw."
"The fact that this army is adjacent to territories that lie within the German border (Silesia and Prussia) is irrelevant, then?"
"Correct, Holmes. But of course if that unit were to move across the border -- successfully entering either Silesia or Prussia -- this would bring the unit adjacent to Berlin and perhaps Munich, and therefore the German player would become aware of that unit and its location."
"Excellent," said Holmes. "My final question, Colonel, involves the handling of neutral (non-home) supply centres. Do they provide to their owners any information concerning nearby units?"
"None at all, Holmes."
This last question had me confused, so I stepped in to request an example. Holmes kindly explained as follows: "Consider, Watson, an Englishman who has taken ownership of Norway but who has, on a subsequent turn, left Norway vacant by removing the unit westward. This English player would no longer be informed of any unit that may be in St. Petersburg. Not unless and until an English unit is otherwise within proximity of St. Petersburg, of course (for example, a fleet in the Barents)."
"I see, Holmes. Neutral centres, when left vacant, do not provide their owners with any information on units coming adjacent to them."
"Correct, Watson. And with those clarifications out of the way, we can now assist Colonel Sykes in setting up the board position."
"Perhaps you can do so, Holmes, but not I! Your use of the word 'we' is quite improper!"
"Tut, tut, Watson, you give yourself too little credit. You too, Colonel Sykes. The position is not so obfuscated as it may seem. By simple consideration of the testimony that Captain Meriwether provided to us, it soon comes clear. I am confident that you will surprise yourselves. Let us begin."
Of course, I did not believe my friend, but Colonel Sykes and I did hope that some of Holmes's great powers of deduction would prove fathomable by our lesser minds, and so together we re-examined the list of facts that Captain Meriwether had entrusted to the Colonel. We saw that Holmes had prefaced the list by specifying the game's current phase, and that he had written numbers next to each of the ten facts, as a referencing aid:
"Now then," said Holmes. "Which fact strikes you gentlemen as the most interesting place to begin?" I read over the facts again, but each seemed more baffling than the last.
I was about to confess to Holmes that I had no idea where to begin, when the Colonel spoke. "Fact 3, Holmes," he declared.
"Well chosen, Colonel! Yes, fact 3 contains a treasure trove of information. It informs us that for the first three moves of the game -- Spring 1901, Fall 1901, and Spring 1902 -- every single one of the units on the board has been ordered in such a way that no player other than its owner has been able to 'see' the unit at all! We must bear this fact foremost in mind in all of of our efforts. Never in the first three turns of the game did any unit come adjacent to a foreign unit or a foreign home supply centre!
"This, of course, excepts the Italian army in Venice and the Austrian fleet in Trieste, which are each visible to the opponent to begin the game. Obviously, we immediately know from fact 3 that these two particular units did indeed HOLD on every one of the first three turns. This is the only way that the Austrian and Italian players could still be of the belief that every foreign unit may have been holding on every turn." Holmes, at this point, had busied himself with placing an Italian army onto Venice and an Austrian fleet into Trieste.
I was delighted with this. "Colonel! Holmes has now determined the position of two of the units -- at least their position before the final move that was made! There was an Italian army in Venice, and an Austrian fleet in Trieste, holding since the beginning of the game!" It seemed obvious now, and I immediately noticed something else. "Fact 5, Holmes! It tells us that only three units have ever issued a HOLD order. You have already identified two of them!"
"Quite so, Watson. And determining which is the third such unit is equally elementary." As he spoke, he took a Russian fleet from the box and placed it onto the board. "You see, the fleet that began the game in Sevastopol could not possibly have moved on its first three turns. If it had moved, another player -- either Austria or Turkey -- would have become aware of it. This is because the only spots to which the fleet can move (Rumania, the Black Sea, and Armenia) are adjacent to a foreign power's home centre."
"Excellent!" I exclaimed. "The third of the three units that ever issued a HOLD order!"
"Indeed," Holmes replied, calmly. "But of course there is much more that Fact 3 has to teach us, before we move on to another fact. It would be but a trivial undertaking to create a list of all of the spaces that could not be entered at all before the Fall of 1902, given the conditions that fact 3 imposes upon us."
I was a bit taken aback by his assertion, and felt forced to interrupt. "How might we recognize these spaces, Holmes?"
Holmes patiently provided this example: "Take the case of the English Channel, my dear Watson. Could it be entered without detection by any power?"
I thought about his question carefully. "Well Holmes, I suppose that nobody could enter the English Channel undetected. An English fleet would be seen by the French by virtue of the Channel's proximity to Brest, and a French fleet could likewise be seen by the Englishman in London. No other power's fleet could sail into the Channel, as it would obviously be seen by both England and France."
"Good show, Watson!" said Holmes. "Now perhaps with the Colonel's assistance, we might make a list of the rest of the spaces to which one of these restrictions apply."
The colonel and I proceeded to call out spaces which could not be entered by any power at all until the Fall of 1902, and, with Holmes' gentle suggestions, we eventually arrived at this list:
Personally, I felt rather pleased by this accomplishment, but Holmes rushed us forward. "There is a second class of space which will become even more crucial to us, as you shall soon see. In addition to the spaces that cannot be entered at all, by any power in the game, there are many spaces on the board that can be entered by one and only one of the powers without placing himself at risk of detection. Perhaps you could once again provide us with an example, Doctor?"
I thought for a long time about Holmes' question. Suddenly, I remembered our earlier conversation about Norway. "Norway!" I cried. "The Russian might travel there undetected, but any other power's presence would inform the Russian, because of Norway's proximity to St. Petersburg!"
"Quite right, Watson!" Holmes replied. "Now let us construct the list of the qualifying spaces, and I suggest that we organize it by power. We need not list those spaces within a power's own national boundaries (Clyde, for example), of course, as such spaces qualify de facto -- only that single power could conceivably move into such a space in the first three moves of the game." A quick glance confirmed my friend's claim, and after some more work, we arrived at the following list:
Again, the Colonel and I were satisfied by the novel view we were getting of the various spaces on the board Diplomacy from these studies, but Colonel Sykes expressed a shared frustration when he exclaimed, "This is all well and good, Holmes, but what good do these lists really do us?"
"Why, all the good in the world, my dear Colonel!" came Holmes' reply. "You see, we have taken the first and most important step toward determining to which units the good Captain referred when he wrote down fact number 4, which is the statement that one dozen spaces within the national boundaries were, as he put it, 'invaded' by foreign units in Fall of 1902."
I was quite stunned. "How is that possible, Holmes?" I asked.
"There are 42 spaces which lie inside the various powers' national boundaries on the Diplomacy board," he continued. We must simply determine which twelve of those forty-two spaces could have been entered that season, given that not one of the invading units was 'seen' beforehand by the invaded power. This is not quite the tall order it seems. As you yourself pointed out, Watson, almost no home supply centres could be among the twelve invaded spaces, as their owners would have received advance warning of the invader's advance. The only two exceptions to this are Venice and Trieste, which both, as we have already established, contain potential invaders that are already seen. I shall return to them momentarily. The remaining twenty home supply centres can safely be stricken from our list, however, leaving twenty-two possible spaces that can be entered. We know from Captain Meriwether's notes that twelve of these were indeed 'invaded,' and the other ten were not. Not quite as daunting a challenge now, I would submit."
I nodded, eager for Holmes to continue. He obliged. "Of those remaining spaces, many are surrounded entirely by either home centres or spaces from of one of the two lists we have just drawn up, and thus, these spaces could no more easily be invaded than could a home centre! For example, take the case of Gascony. It is bordered by Spain, the Mid-Atlantic Ocean, Burgundy, Marseilles, Paris, and Brest. Obviously, the invasion cannot have come from one of France's home supply centres. Could it have come from one of the other three spaces? Certainly not, because a foreign unit could not have been in any of them without France's advance knowledge! Therefore, Gascony can be safely eliminated as a target of foreign aggression. We can similarly rule out Ukraine, Syria, Apulia, and all of the spaces within England's national borders. We are left with a very comfortable list of 15 spaces. With that, Holmes reached into his pocket and produced a sheet of paper with the following written on it:
"Brilliant, Holmes!" I shouted excitedly. "However, we still must narrow this list of fifteen down to the necessary twelve invaders, and I am at a loss to see how to do so."
I was surprised at Holmes' answer. "That we cannot do without some small amount of difficulty," he said. "We must first recognize that there is some mutual exclusivity involved in the above invasions. First, Trieste and Venice obviously cannot each be invaded by the unit standing in the other, as the two would simply stand off. So one or the other of those spaces can certainly be eliminated from our list."
"Can we know which?" I asked excitedly.
A bit hesitantly, but with a perceptible sense of excitement, the Colonel spoke up to provide my answer. "I would suggest, Holmes, that it was Trieste that invaded Venice, for this reason: from your list, we see that Tyrolia (if it is one of the twelve) was invaded by either Italians from Venice or Germans from Munich. If we claim that it was the Germans, then no power could have invaded Bohemia, for Munich is the only unit available for that invasion. Therefore, our choices become having all three of Tyrolia, Bohemia, and Venice invaded, or merely Trieste and Tyrolia."
Holmes broke in. "Very good, Colonel, your logic is quite flawless! If Venice were to invade Trieste (rather than Trieste invading Venice), then only one of Tyrolia or Bohemia could have been invaded. Therefore, in resolving this problem of mutual exclusivity that Venice and Trieste pose for us, we would be striking two locations from our list (Venice and one of Tyrolia or Bohemia) rather than one. As there are still two other spots on the board that create conundra of mutual exclusivity (meaning, of course, that we shall be forced to remove one space from each of these two pairs from our list when we resolve these conundra), then to assume that Trieste were invaded would necessarily bring our list of fifteen spaces down to eleven, which is of course one space too few." Pointing at the board, where he had set up the three pieces whose positions before Fall 1902 were at this point known to us, Holmes continued. "Thus, we know that the Austrian fleet from Trieste invaded Venice, the Italian army from Venice invaded Tyrolia, and a German army from Munich invaded Bohemia. We can remove Trieste from our list of invaded locations." Holmes struck Trieste from the list and wrote check marks on the list next to Tyrolia, Venice, and Bohemia as being three of the twelve invaded locations. He also placed a German army on the board in Bohemia, and moved the Italian army from Venice to Tyrolia, and the Austrian fleet from Trieste into Venice, to update the positions of those units to their positions after Fall 1902.
"Next," Holmes went on, "we must decide which nine of the remaining eleven invasions occurred. As I have hinted, we will do so by resolving the two remaining problems of mutual exclusivity. As we shall see, there are two other sets of sites on the list that cannot possibly all be invaded."
Before the Colonel and I had time to look at the list to determine where these two situations that Holmes spoke about were located, my friend continued. "Let us move first to the interesting case of Picardy, Ruhr, and Burgundy. It is patently obvious that these three spaces cannot all be invaded, without the invading units being 'seen' by their intended victims. Since we have already committed the Munich army to an invasion of Bohemia, we know that if Burgundy is invaded, the attack must come from Ruhr. This, of course, would rule out any invasion of Ruhr, since no non-German unit could be positioned to move on Ruhr and yet remain 'unseen' if Ruhr was occupied by a German army scheduled to invade Burgundy.
"In fact, a simple look at the board will convince you that the only way that two of the three spaces Ruhr, Burgundy, and Picardy could be invaded is if German units advanced into Burgundy and Picardy from Ruhr and Belgium, respectively. There is simply no other way to rule out only one of these three spaces from our list, and I have noted that we can indeed only afford to rule one of them out, as we have one final pair of invasions that cannot possibly both take place."
Holmes placed a German army on Burgundy, but stopped short of placing one on Picardy. Noting this, I asked, "Holmes, did you not say that the German player invaded Picardy from Belgium as well?"
"Yes, we can be sure that he has, Watson, but we do not yet know whether he did so using an army or a fleet." Holmes struck Ruhr from his list of invaded locations, and added check-marks next to Picardy and Burgundy.
"We are now down to determining which seven of the remaining eight spaces were 'invaded.' A simple concentration on one final area of the board will eliminate the final candidate. Consider the situation around Silesia, Livonia, and Prussia. Livonia, we see, could be invaded by a German fleet leaving the Baltic Sea, but only if Livonia were empty in the Spring (as otherwise the German fleet would have been detected). Similarly, Prussia could be invaded by a Russian unit moving from either Livonia or Warsaw (but, of course, the attack could not come from Livonia if there was a German fleet in the Baltic Sea). The only way both Prussia and Livonia could be 'invaded' would be if a German fleet advanced from the Baltic Sea to Livonia and a Russian army left Warsaw for Prussia. However, an insurmountable problem arises if the Warsaw unit were committed to invade Prussia.
"Notice that Silesia can only be invaded from Warsaw, and we have already determined what must have happened on Germany's western and southern fronts -- he advances from Belgium to Picardy, from Ruhr to Burgundy, and from Munich to Bohemia. Obviously, German territory is not being invaded along those fronts, and no English unit could possibly invade German territory from the north or west without being foreseen. Therefore, if we assume that Warsaw is moving to Prussia, leaving Silesia off our list, this means that Germany is invaded by only that single Russian army. This is something that fact 4 tells us did not happen -- the Captain wrote that 'no power was invaded only by a single army.'
"We know, then, that Silesia cannot be the final location stricken from our list. This means that it was most definitely 'invaded,' and that invasion came from Warsaw." Absent-mindedly, Holmes took a Russian army and placed it on Silesia, and wrote a check-mark alongside Silesia on his list. "To deduce which of the other two locations -- Prussia or Livonia -- is (and which is not) invaded, we can simply rely on the same piece of intelligence we just used. Since the Captain told us that Germany could not have been 'invaded' only by the single Russian army that we have just established moved from Warsaw into Silesia, we know that it is Prussia, and not Livonia, that is the other location in this trio that is 'invaded.'"
"Amazing, Holmes! You have deduced the twelve invaded locations!" I cried.
"Elementary, Watson," said Holmes as he placed a Russian army in Prussia. Consulting his list, checking off as he went, Holmes also moved the Russian fleet on the board from Sevastopol to Armenia, and placed a Russian army in Galicia.
"Notice, gentlemen, that the fact that no power is invaded only by a single army also assists us in this way." Making something of a demonstration of it, Holmes placed a German fleet onto Finland. "Note that the only invasion of Russian territory that is among the twelve is an invasion of Finland, which could only have been accomplished by a German, and which -- per the note the good Captain left us -- must have been accomplished by a fleet rather than an army."
Holmes re-lit his pipe and puffed it for a moment before speaking again. "There you have it, Colonel. The twelve spaces invaded and their invaders. Germans entered Picardy, Burgundy, Finland, and Bohemia. French units entered Tuscany and the Piedmont. Russians entered Armenia, Galicia, Prussia, and Silesia, the Italian entered Tyrolia, and the Austrian entered Venice."
The Colonel and I slapped Holmes on the back, saying, "Good show!"
Holmes was enjoying the attention, and his pipe. After a brief pause, he said, "without proceeding further, we cannot yet say definitively whether armies or fleets entered Picardy, Tuscany, or Piedmont. In order to determine the answers to these questions, as well as the rest of the board position, we must take evidence from more of the facts that the Captain left us."
Holmes nodded seriously, and asked, "Shall we proceed?"
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