Solution to The Cultural Exchange Puzzle

When Holmes returned from ensuring Lord Fortescue was safely in his hansom cab, I was still staring at the ceiling trying to puzzle whether one of the seven Diplomacy powers could engineer a capture of all 22 home centres while abiding both by the in-force treaty protecting the Confederation of Neutral States and (unilaterally if need be) by the pending treaty requiring all units to occupy home supply centres at the end of each year.

Holmes, seeing that I was quite tired and yet unable to turn in before having decided the question, came to my aid.

"The Foreign Secretary asked me to relay to you his apologies for the swiftness of his departure, but you see, Watson, extraordinary circumstances called him away."

"Circumstances attending to the problem you have set me?" I asked.

"Indeed so, Watson, indeed so." Holmes moved to the bar and poured himself a half-snifter of cognac before continuing. "I had only just finished explaining that the feat under discussion could indeed be accomplished by one and only one of the seven Great Powers, when a message arrived by page for Lord Fortescue. The page had only just left us when you returned from your outing and joined our discussion. While you mused, lost in thought and in apparent study of the ceiling-paper, our friend took the opportunity to open the message. It was sent from one of his fellows, Samuel Meyer, at the Foreign Office, and the message included confirmation that the very power I had mentioned to Fortescue was quite possibly contemplating the nefarious deed under discussion. You certainly understand now why he was quick to his hat and coat."

"That I understand, Holmes, but you must tell me which power it is that may be considering the grand plan of attack! I have gone through each of the powers in my mind and each seems to have some weakness at the game's beginning which makes it impossible to achieve the feat."

Holmes smiled but did not answer, so I continued. "As I was saying, Holmes, for a power to reach a size of 22 centres by the end of 1903, the power must double its size the first year, and by the terms of the treaty, all new growth must come at the expense of another player -- no neutrals may be taken. Turkey can capture only Sevastopol, and England only Brest, so those powers are ruled out. Similarly, the Kiel fleet cannot reach a foreign home centre in 1901, nor yet the Neopolitan fleet or the St. Petersburg fleet. Only Austria and France may take possession of a foreign centre with every one of their initial units, but I cannot for the life of me see how either could achieve twenty-two centres by the end of 1903. Enlighten me, if you don't mind, Holmes," I pleaded, "for I do not see what mistakes I could have made."

"Certainly, Watson. One mistake you made was in claiming that England can take only one home centre from another power in 1901. This is patently untrue, as I'm sure you can see. In addition to Brest, England is also able to capture Kiel in 1901." Holmes smiled then, and added: "This, of course, has no bearing whatsoever on the problem at hand, though. Your more pertinent error lies elsewhere."

"Pray continue, Holmes, I am tired as a dog and must know my fallacy." Holmes thereupon took pity on me and allowed my tortured thinking to end.

"Quite right, then, Watson, I realize that your labours of the day have taken much from you, since simple mathematics is all that is necessary to determine which of the seven powers can realistically consider the achievement. Note that all the powers with three home centres can take only six units into 1902, and (although they may own as many as 12 centres) only nine units into 1903. Which means, Watson, that even if all that power's units were successful in gaining a new centre, the power would end 1903 with only twenty-one centres, one short of the requirement. Therefore, if any power can do it, that power must be Russia."

"But Holmes, am I mistaken in observing that the St. Petersburg fleet cannot gain a new centre in 1901?"

"You are not, Watson, which brings us to the one flaw in your reasoning. Specifically, you are mistaken that the power must double its size in the first year. Russia may gain three centres in 1901, to grow to a size of seven, which, coupled with the ability to build four units, puts the total of 22 centres by 1903 in reach."

"I follow you, Holmes, but I still do not think the feat possible. Yes, Russia begins with four, and if his three southern units gain new centres in 1901, he will reach a size of seven units at the end of the first year. But when he repeats the feat in 1902, he is yet only able to build three more units (owing to the fact that St. Petersburg is ever to be covered by its original unit), leaving him with ten to take into 1903. Even if all of these take a new centre, that is only twenty, not twenty-two."

"Watson, you jump to entirely too many conclusions. And in your fatigue, you have supply centre count and unit count confused. Firstly, there is nothing which forbids the Russian from building an army in St. Petersburg at the end of 1902."

"But surely, Holmes, there is! I have pointed out to you that the St. Petersburg fleet has no chance of ending a year in any home centre save St. Petersburg itself, and surely that precludes building in that centre in every year."

"Here Watson, come sit at the board, and I will show you how it is done. Do not feel badly, for apparently the puzzle has left the best men at the F.O. at a loss as well." Thereupon, with a great smile on his face, Holmes patiently arranged the units on the Diplomacy board and began to move them thusly.

Power	Unit	S01	F01	S02	F02	S03	F03
Austria	F Tri	-Alb	-ION	H	C Con-Nap	H	H
	A Vie	-Boh	H	H	H
	A Bud	H	H
England	F Edi	-NWG	-BAR	H	C StP-Lon	C StP-Edi	C StP-Edi
	F Lon	-NTH	-NWG	H	C StP-Lon	C StP-Edi	C StP-Edi
	A Lvp	H	H	H	H
France	F Bre	-ENG	-NTH	H	C StP-Lon	-ENG	C Lon-Bre
	A Par	-Pic	H	H	H	H	H
	A Mar	-Gas	H	H	H	H	H
Germany	F Kie	-BAL	C StP-Kie	H	S Pru-Lvn	H	H
	A Ber	-Pru	H	H	-Lvn
	A Mun	H	H
Italy	F Nap	-ION	-AEG	H	C Con-Nap	H	H
	A Ven	-Apu	H	H	H
	A Rom	H	H	H	H
Russia	F StP	H	H	-Lvn	H-OTB
	F Sev	-BLA	-Ank	H	-Con	-Smy	H
	A Mos	-Lvn	-Kie	-Mun	H	-Bur	-Par
	A War	-Gal	-Vie	-Tyr	-Ven	-Pie	-Mar
			A Sev	-Con	-Nap	-Rom	H
			A Mos	-StP	-Lon	H	-Bre
			A War	-Gal	-Bud	-Tri	H
					A StP	-Edi	-Lvp
					A Mos	-StP	-Edi
					A War	-Sil	-Ber
Turkey	F Ank	-Arm	-BLA	C Sev-Con	H	H
	A Smy	-Syr	H	H
	A Con	H	H

"Ah!" I exclaimed. "Of course! The useless fleet can be taken off the board by a successful attack made on it in 1902! Holmes, you are indeed astute!"

Still, I sat staring at the board. "Holmes, while I acknowledge that it can be done, yet am I astonished that it can be done with only nine units on the board. More puzzling to me is why Russia can do it with only nine units, but no other power could. I thought we had established that the greatest number of units a non-Russian power could bring into 1903 was nine, which is as many as your Russian uses. And yet we said that this very fact eliminated the other powers from eligibility. Can you explain this?"

"Elementary, my dear Watson. Note that a non-Russian power can bring ownership of twelve centres into 1903. With the maximum of nine units to take new centres, this means only twenty-one can be obtained by the end of 1903. But Russia, if playing in 1903 with nine units (as was effectively done in my illustration) can own as many as sixteen centres without the restrictions placed by the treaties, and as many as fourteen with the restrictions. Twenty-two is most definitely reachable, as I have shown."

I congratulated Holmes and headed straight for bed, silently hoping that the great detective would never again propose a problem to me while I am in such a great state of fatigue.

Needless to say, the rumour that Russia would attempt to control all home centres within a three year timespan proved to be unfounded. Undoubtedly it was started by a Russian nationalist who wished to exhibit his pride at being the only power that could possibly perform the feat. However, as in all cases of international affairs, the feat may only be accomplished with the cooperation of the community of nations, and the kind of suicidal cooperation required to make this particular feat a reality was certainly out of the question.