Given there's no spoiler tag here, I've sent my solution to you via PM TGM!

I got linked to these a few days ago. Really enjoyed doing them, thanks RoganJosh!

I made a puzzle of my own:

You're play testing a new map. You're playing as Blue, it's spring, and you're 1 SC away from winning. Elsewhere on the map you can guarantee to gain a centre, but Red is attacking your two coastal centres pictured here. How can you defend them both?

[url]https://imgur.com/kbUn5Uq[/url]

(Sx provinces are seas, Lx are land. A and F stand for fleets and armies, and are coloured by power)

I'm happy someone enjoyed them!

Thank you for these puzzles! 6 was really tricky and fun. 7 was easier, but the solution was something I'd never seen before.

Nice.

Now you've solved it with HEL moved to HOL, I prefer the puzzle to have the fleet in HOL instead of HEL.

(WARNING, MORE SPOILERS)



The first set is exactly as intended - well done!

Follow up is correct as well. And it seems to me that there is a solution even if HEL is moved to Hol (assuming bounce in StP):
After builds:
Nwy > StP supported from BAR ,
SKA>Swe, Hol>NTH, BAL > GoB

If R stays in StP there is no problem,
If R moves/retreats to Finland (after possibly bounce in GoB):
Sweden will attack Fin in the fall, cutting any possible support. The conclusion is that R/T can at most support the Russian unit back into StP, but they can't get a Turkish unit into StP.

Solution to Puzzle 6: (WARNING, SPOILERS)

Germany: F BAL S BER, A RUH & A BUR S MUN, A GAS-NWY, F HEL-DEN, F SWE S GAS-NWY.

England: F NWY-STP, F NTH, F ENG & F MAO C GAS-NWY, BRE-GAS.

The only interesting case is when STP-NWY is bounced by MOS, then NWY should retreat to BAR.

The hardest case is when this is a fall turn and Russia builds in STP (we assume Russia is hostile). If it was a spring turn, one can always wait a turn (ensuring T doesn't take STP of course) to reduce to this case.

Winter:
Germany builds in KIE
England removes F NTH

Stalemate part:
ENG S MAO, POR S SPA, GAS S MAR, BUR & RUH S MUN, KIE S BER.

Capture STP part:
Spring: A NWY-FIN, F SWE-NWY, F BAL-GOB, F DEN-BAL.

Fall: At least three of the four spaces adjacent to STP will be occupied by EG forces. Use a double supported attack on STP and BAL-LVN to guarantee that EG take STP.

From a puzzle composition point of view, I like how this solution relies on the German fleet being in HEL and doesn't work if it is in HOL (but I haven't checked that no solution exists in that case).

I think you can just post it here - it's been long enough since these were posted to first time. But maybe a homemade spoiler alert, in absence of a proper one.

I have a solution that works if Russia is hostile (and building).

What is the etiquette here with such things - do we post the spoiler, provide the solution in a private message or do you all trust my grandiose claims without proof?

Oh gees no - assume that Russia had abandoned the game and will not make any builds.

Or let it be Austria instead of Russia!

But nice catch.

[quote=RoganJosh post_id=140277 time=1586376951 user_id=171]
The intended solution works no matter if it's spring or autumn. But a shout out to CCR who found an alternative solution which works if it is autumn!
[/quote]
Thanks! I was also looking at a solution that works if it is Autumn... but actually I think it is foiled by a spiteful Russian build!

The intended solution works no matter if it's spring or autumn. But a shout out to CCR who found an alternative solution which works if it is autumn!

Is puzzle 6 in Spring or Autumn?

[url=https://www.youtube.com/watch?v=X_lSvr-wriA]Now in video form![/url]

This particular puzzle impressed me a hell of a lot so I asked RJ if I could put it up on my youtube channel. I'd strongly recommend you try both this one and his puzzle 6 here, both of them have really solid and interesting answers!

[b]Two classical puzzles![/b]

No game theory. No probabilities. Just two "classical" (but newly constructed) diplomacy puzzles. The first one is a variation on an existing puzzle, but I won't say which one.

[b]Puzzle 6:[/b] A variation. [url]https://imgur.com/Z04T3YI[/url]
That f*ing Russian! Sure, you'd been fighting him the whole games, but come on! Your German fleets were in order, and Turkey's solo push was doomed. The stalemate line was secured! The only thing Russia had to do was to retreat to St Petersburg. But, no, of course not. The Russian player [i]disbanded[/i] his last units. And now he sent you a message congratulating you on the loss. England has given up - claiming there is no way the solo can be stopped. Your task? Prove them wrong. Find a set of orders, for the German and English units, which secures the stalemate line [i]no matter[/i] what orders Turkey submits.

[b]Puzzle 7:[/b] A simple double. [url]https://imgur.com/a/jz5CLV3[/url]
Such a weird game. It's now the Fall of 1913. France and Italy have been complicit in taking you to a position where the solo seems guaranteed. But, of course, winning fast is better than winning slow. So, is it possible? Can you find a set of orders which secures the Russian solo in one move, [i]no matter[/i] what orders France and Italy submits?

Happy puzzling, all you quarantined smart-asses.

Feel free to discuss or post solutions in the thread, or in a PM.

The remaining solutions.

[u]Puzzle 2:[/u]
MAO > Por
WMS > Spa supported by Gas
Mar H supported by Bur

[u]Puzzle 4:[/u]
The key moves are Sil > Ber, supported by Kie, and Vie > Boh. For the other orders there are some alternatives. My personal favorite is Nwy > StP > Mos.

[u]Puzzle 1:[/u]
This is opening strategy, and hence the solution is somewhat subjective.

a)
This position emphasizes the importance of Tyrolia and the fact that retreats are decided [i]after[/i] all regular moves have been adjudicated. Tyrolia is one of the most important territories in FvA, as it borders six territories, each which is of strategical importance. The spring turn is the only turn when Austria can be guaranteed to push France out of Tyrolia. If Austria waits, then France can backfill Piedmont and get a unit attacking Munich. That is, an Austrian attack on Tyrolia is to be expected. What should France do with Tyrolia? The answer is: nothing.
Tyr H
By holding, you know that you will be allowed to retreat, forcing Austria to block all possible retreat options. Indeed, the Austrian standard continuation is:
Tri>Tyr supported by Mun
Vie>Boh
Bud>Vie
Ven H
War>Mos (or possibly War>Sil)
Notice that to block all retreat options (except Piedmont, which cannot be blocked) Austria has to use five units just to handle the one army on Tyrolia.

Now, you might say that Tyr>Tri or Tyr>Vie also forces Austria to use five units to dislodge Tyr, why not consider one of those? Hand on heart, Tyr>Tri is ok, though you would expect it to give the same outcome as Tyr H. The order Tyr>Vie, however, should be avoided. It might look like Austria gets a very strong presence in Germany, and that is true. However, Austria will have big problems capturing centers. If the army in Budapest is bounced, then not only can it pick up Rumania for a crucial build in 02, it can also continue to Moscow or Constantinople, solving Austria's center picking problem and paving the road for an Austrian rush.

What should France do with the rest of his units? Lon>NTH and Bel>Ruh (possibly Hol) is clear. For the remaining two armies, let's move on to part b) of the puzzle.

b)
If Austria would try to be a smart-ass, and not push France out of Piedmont, then his biggest fear is a backfill Mar>Pie. With army builds in both Paris and Marseilles, it's hard not to expect Par>Bur and Mar>Pie. But notice, if Austria follows the standard continuation and France orders Mar>Pie, then Tyrolia will be disbanded. Jackpot for Austria! That is, the French double army builds effectively rules out any Austrian smartassery.

The conclusion is that France can not play Mar>Pie. So, what to do with Marseilles? There is not that much to do. Support Paris into Burgundy, and then pick up Spain in the Fall. It has "bad" written all over it, as armies in Spain are mostly in the way.

No, France should not have built two armies. It should have been one army and one fleet. The army should have been built in Marseilles - you still need to threaten to backfill Piedmont - and the fleet should have been built in Brest.

And as I've clarified an a number of occasions, the phrase "probability" should be understood in the sense of a natural density. Which is the canonical interpretation in probabilistic number theory. (Which is where you deal with statements of the form "What is the probability that two natural numbers are equal?" Hint: it's zero.)

It's funny because you're not a stranger to this interpretation, yet you fail to connect the dots. This is how you used the word "probability" yourself in the following computation:

[quote=jay65536 post_id=126823 time=1579796495 user_id=9431]
Your last example is of course right, but let's dumb it down even further for non-mathematicians who are reading this. If we want to ask a question like "What is the probability that a natural number is even?" then what we do is consider an arbitrarily large N and ask the question "What is the probability that a natural number less than or equal to N is even?" In this case, if N is even, the probability is exactly 1/2, and if N is odd, the probability is (N-1)/2N. As N approaches infinity, this last expression converges to 1/2, so the limit behaves nicely and we can say without fear of contradiction that the probability is 1/2.
[/quote]

It is true that there is no uniform measure on the natural numbers. Or, equivalently, the sequence of uniform measures on the sequence of exhausting compact subsets of [b]N[/b] given by [0,N]\cap [b]N[/b], does not converge [i]to a measure[/i].
The notion "probability," however, is still well-defined, in the sense of a natural density (and it agrees with what the measure theoretical notion of probability would have been [i]had[/i] there been a uniform measure on the natural numbers.)

But, at the end of the day, it's a fucking puzzle. If you don't like solving puzzles then ignore it. If it's not clear what format the solution should be in, ask for a clarification. But proclaiming it "wrong" just because you couldn't solve it...

[quote=RoganJosh post_id=129230 time=1581155697 user_id=171]France strategy in simplified form is then: [i]Choose a natural number k. Play the waiting move k times, and then play an attacking move.[/i][/quote]

As I tried to point out, due to some math that is probably obscure to most people on the thread, this isn't actually a solution unless you specify a probability distribution that you're using to pick k, and I'm also pretty sure that once you do that, the probability of a French win can't be 1. It can be arbitrarily close, as we covered ad nauseam before, but it can't actually be 1.

[b]Puzzle 3:[/b]

To begin: the French waiting move is:

France Waiting Move (FW)
Bur > Mun supported by Ruh Ber
Kie and BAL support Ber H

To make the situation simpler, France will begin by playing the waiting move, picking up the English SC's and placing armies in Sweden and Norway. We assume that Austria enters Moscow, and that Austria has an abundance of armies that can backfill any region he would leave empty. There are three regions for France to attack: Munich, Prussia, and Livonia. The attacking moves that France should consider are:

France Munich Attack (FM)
Bur > Mun supported by Ruh Ber Kie
BAL S Ber H

France Prussia Attack (FP)
Swe > Pru convoyed by BAL
Ber S Swe Pru
Kie S Ber H
Bur > Mun suported by Ruh

France Livonia Attack (FL)
Swe > Lvn convoyed by BAL and supported by StP and GoB
Kie S Ber H
Bur > Mun supported by Ruh Ber

If France captures one of these centers he will win. (Actually, if he takes Livonia you get a new waiting move and a similar situation.) Austria has two counterattacks to consider:

Austria Prussia + Livonia counter (APL)
Lvn H with one support
Pru > Ber supported by Sil
Mun > Kie
Boh > Mun supported by Try
War > Pru

Austria Munich + Livonia counter (AML)
Lvn hold with 1 support
War S Pru H
Mun > Ber supported by Sil Pru

where the important thing to note is that AML is the only counterattack to FM, and it looses to the French waiting move. In addition, Austria will consider two defensive moves:

Austria Defensive 1 (AD1)
Lvn H with 2 supports
Pru > Ber supported by Sil Mun
Mun supported by Tyr Boh

Austria Defensive 2 (AD2)
Lvn H with 1 support
War S Pru H
Sil > Ber supported by Pru Mun
Mun supported by Tyr Boh

(There is some variation to exactly which defensive moves Austria considers, but the result in the end is the same.) Also in this game, we have a very magnanimous French player, who will simply accept defeat if APL and/or AML is met with FP or FL. Then, the possible outcomes are:

------- AD1 ---- AD2 ---- APL ---- AML
FW - replay - replay - replay --- F
FM - replay - replay ---- F ------- A
FP ----- F ---- replay ---- A ------- A
FL - replay ----- F ------- A ------- A

From here it's more or less the same as in Puzzle 5. France will play the waiting move (k-1) times and then an attacking move on turn k. Due to AD1 and AD2, France is forced to play either FP or FL as the attacking move with some probability p (for each), and he will play FM as the attacking move with probability (1-2p). Then, the game will be decided wither on turn k or on the turn l when Austria plays AML for the first time. The probability of an Austrian win becomes

P(l=k) + 2p P(l > k)

and, as in the previous puzzle, since France gets to choose both p and [i]how to choose k[/i] (i.e., France gets to choose P(l = k)), he can make this probability arbitrarily small, by making smart choices.

[b]Puzzle 5[/b]

France will consider the following moves:

France Waiting Move (FW):
Gas Bur supports Mar H
MAO > Spa supported by Por and Mar
(NAO > MAO supported by IRI and ENG)

France's Gascony Attack (FG)
Gas > Spa supported by Por
MAO > WMS

France Marseilles Attack (FM)
Mar > Spa supported by Gas & Por
MAO > WMS
Bur > Mar

Italy has the following options:

Italy Defensive (ID)
Either support holds or support himself with two into Spain

Italy Spain to Marseilles (IS) [Gascony counterattack]
Spa > Mar supported by GoL and Pie

Italy Piedmont to Marseilles (IP) [Marseilles counterattack]
Pie > Mar supported by GoL

Outcomes:
-------- ID ------- IS ------- IP
FW - replay ----- F ----- replay
FG - replay --- draw ------ F
FM ---- F ----- replay --- draw

The counterattack to FG is IS, which looses to the waiting move FW. Notice that if France would not consider FM, then Italy could just play ID indefinitely. That is, France needs to consider both attacking options.

In order to simplify the computations, the magnanimous French player has decided that in case FM is met with IS, he will accept the draw. That is, the outcomes will be:

-------- ID ------- IS ------- IP
FW - replay ----- F ----- replay
FG - replay --- draw ------ F
FM ---- F ------ draw --- draw

Let's reverse engineer how France should choose between the attacking options.

France's tactics: Choose an integer k. The first (k-1) turns France will play the waiting move FW. The k'th turn, France will play FM with probability p and FG with probability (1-p).

Notice that as long as France might play FM with some probability, there is no reason for Italy to consider ID at all, as it can only result in a loss or a replay.

So Italy has two choices: IS and IP - the two counterattacking options. Notice that the game ends when either Italy plays IS or when France plays an attacking move, whichever happens first. That is, Italy's only choice is which turn l to play IS.

What is Italy's probability to obtain the draw? If l < k, then Italy looses.
If l = k, then Italy draws. If l > k, then Italy draws if France plays FM. That is,
Italy's probability to draw the game is

P(l = k) + p*P(l > k)

[where P(x) stands for the probability that x happens]

Since France chooses p, he can make the second term arbitrarily small. Since France chooses [i]how to choose k[/i], he can make the first term arbitrarily small.

In practice, one option for France is to choose k uniformly between 1 and N. Then,
P(l = k) = 1/N
P(l > k) = (1-1/N)/2
so France can choose p as for example 1/N. This gives a probability for a draw which tends to 0 as N tends to infinity, so that France can make the probability of a draw as small as he wants by choosing N sufficiently large.