1v1 Puzzles

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RoganJosh
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Re: 1v1 Puzzles

#61 Post by RoganJosh » Thu Jan 23, 2020 5:17 pm

You are confusing probabilities and strategies again.

The probability distribution which describes the process of choosing an integer uniformly at random on the interval [0,N] converges as a function (to the function which is identically zero), but it does not converge as a probability distribution since the total mass of the limit is 0. Hence, that limiting function can not be used to define a probability.

Gees - since you did not get my hint, let me spell out the solution to you.

This is the 2x2 game.

France only choice is which turn to play the attacking move. When he plays the attacking move, the game will end (if it did not already end). Italy's only choice is which turn to play the counterattack. When he plays the counterattack, the game ends (if it did not already end).

That is, France choses a natural number k. Italy chodes a natural number j. The probability that Italy wins is the ”probabolity” that j=k. The probability that France wins is the "probability" that j != k.

These probabilities should be understood just like the probability you computed for that a number is even. The answer is that France's probability to win is 1.

Also in game theory, the notion ”strategy” is wider than just mixed strategies for games in normal form. This game is not in any of the standard forms ( that I know of). The reason is that the "payoff" matrix is not actually a payoff matrix - it includes the option ”play another round”. I might be wrong, but I would be very surprised if games like these have Nash equilibria.

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Re: 1v1 Puzzles

#62 Post by mhsmith0 » Thu Jan 23, 2020 5:19 pm

more on that concept (lol forum time limits on editing posts):

Austria has two distinct "assassin" move sets, and France two distinct "emperor" move sets. But ultimately, either Austria hits a French "emperor" with his "assassin", or France (eventually) wins.

Also in this concept of it, you can consider things like:
- Austrian "assassin" q3 against French "emperor" p2 may not actually be a French win (could be Austrian win, or draw, or "no result/change", perhaps?)
- Austrian "assassin" q2 against French "emperor" p3 may not actually be a French win (could be Austrian win, or draw, or "no result/change", perhaps?)
- 1/N probably isn't the exact optimal frequency of France's "emperor" move set where he attacks Berlin but not Livonia, and therefore Austrian optimal moves change too

But all of that is just dancing around the x value inside "Austria win equity is x/N". What matters most is that it scales to N; whether x is 0.5 or 10.0 [either would surprise me] is interesting for discussion, but it's still divided by N. imo anyway.

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Re: 1v1 Puzzles

#63 Post by mhsmith0 » Thu Jan 23, 2020 5:43 pm

PPPS You can probably torture the analogy to death even further by allowing for A Sev to get into Moscow and French armies to get into Norway and Finland. This then (I think?) creates three different French "emperor" type strategies

1) Convoy Finland to Prussia directly with support from Berlin, and use F GOB to cut support in Livonia (probably also support GOB-Liv from STP, and A Nor supports STP hold)
2) Aggressively attack Livonia, either STP-Liv supported by GOB/Baltic or Finland convoys to Liv supported by Baltic/STP (I'm not sure off the cuff which is better, or whether some form of mixed strategy, like flipping a coin for attack pattern A or attack pattern B one you decide to attack Livnoia, somehow improves it)
3) Aggressively attack Munich (Kiel supports the attack instead of defending Berlin - possibly there's value in randing whether you try to convoy Fin to Liv or move STP to Liv, but I'd GUESS it doesn't much matter)

There's no Austrian move set that can defend against all three French "emperors", but French "emperor" is vulnerable to at least one Austrian "assassin" (some of these result in a draw at worst instead of outright loss though)

So maybe it expands further into three French "emperors" and three Austrian "assassins", each of varying quality (maybe assassin A can kill emperor 2; maybe emperor 3 still requires Austria to guess right while playing assassin C on the move details; etc), but the analogy still holds conceptually even if the math and specifics become ever more complicated.

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Re: 1v1 Puzzles

#64 Post by RoganJosh » Thu Jan 23, 2020 5:59 pm

If you impose a turn limit N, then there are Nash equilibria. But the pure strategy space will have size 2^N (in the 2x2 game; it consists of all sequences of length N with entries 0 and 1), so I don't think it will make sense to talk about the limit of the equilibrium strategies.
mhsmith0 wrote:
Thu Jan 23, 2020 5:43 pm
PPPS You can probably torture the analogy to death even further by allowing for A Sev to get into Moscow and French armies to get into Norway and Finland. This then (I think?) creates three different French "emperor" type strategies

1) Convoy Finland to Prussia directly with support from Berlin, and use F GOB to cut support in Livonia (probably also support GOB-Liv from STP, and A Nor supports STP hold)
2) Aggressively attack Livonia, either STP-Liv supported by GOB/Baltic or Finland convoys to Liv supported by Baltic/STP (I'm not sure off the cuff which is better, or whether some form of mixed strategy, like flipping a coin for attack pattern A or attack pattern B one you decide to attack Livnoia, somehow improves it)
3) Aggressively attack Munich (Kiel supports the attack instead of defending Berlin - possibly there's value in randing whether you try to convoy Fin to Liv or move STP to Liv, but I'd GUESS it doesn't much matter)
This is what I meant with Puzzle 3 being the 4x4 version. France has one "waiting" order and three attacking orders.

In Puzzle 5, France has one "waiting" order and 2 attacking orders.

Yes, that Kaiji anime puzzle is more or less the same.

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Re: 1v1 Puzzles

#65 Post by Squigs44 » Thu Jan 23, 2020 6:03 pm

Okay I see what you are getting at. Basically you are saying that if you pick N to be infinity, the game never ends, and therefore France never wins nor loses, and the probability of him winning is undefined.

I guess what I meant was that AS N approaches infinity, the probability of a French win approaches 1. It never reaches 1, as the strategy of France waiting N turns cannot play out. But it gets closer and closer.

This is analagous to taking the limit of a piecewise function. If you approach a point from the negative direction, you can get closer and closer to a value. If you approach that same point from the positive direction, you can get closer and closer to a different value. Thus, we say that the limit of that function at that point does not exist.

Throwing aside mathematical rigor (did I mention I'm an Applied Mathematics major?), from a "how can we use what we have learned in real life" standpoint, everyone can agree that the best way to play France is to play the safe strategy a large number of turns, and then finally pounce with the risky strategy, leading to a *near perfect* win rate.

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Re: 1v1 Puzzles

#66 Post by jay65536 » Thu Jan 23, 2020 6:09 pm

RoganJosh wrote:
Thu Jan 23, 2020 5:17 pm
Gees - since you did not get my hint, let me spell out the solution to you.

This is the 2x2 game.

France only choice is which turn to play the attacking move. When he plays the attacking move, the game will end (if it did not already end). Italy's only choice is which turn to play the counterattack. When he plays the counterattack, the game ends (if it did not already end).

That is, France choses a natural number k. Italy chodes a natural number j. The probability that Italy wins is the ”probabolity” that j=k. The probability that France wins is the "probability" that j != k.
Aha! This is most decidedly *not* the same as what I was analyzing before. This is an entirely different can of worms, and I don't think I can say any more about it than what appears here: https://math.stackexchange.com/question ... finite-set

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Re: 1v1 Puzzles

#67 Post by RoganJosh » Sat Feb 08, 2020 9:54 am

Solutions

It's well about time to post solutions to these. Let's beign with the most interesting one - Puzzle 3 and its simpler versions. The 2x2 versions had one solution posted here:
RoganJosh wrote:
Thu Jan 23, 2020 5:17 pm
This is the 2x2 game.

France only choice is which turn to play the attacking move. When he plays the attacking move, the game will end (if it did not already end). Italy's only choice is which turn to play the counterattack. When he plays the counterattack, the game ends (if it did not already end).

That is, France choses a natural number k. Italy chodes a natural number j. The probability that Italy wins is the ”probabolity” that j=k. The probability that France wins is the "probability" that j != k.
Before looking at Puzzle 5 and Puzzle 3, let me explain the philosophy behind the strategies.

In each puzzle, France has a limited number of attacking options. That is, if the game is about F choosing (uniformly) an attacking option, and the defending power guessing which one, then there will always be a relatively large probability (33% in the 2x2 puzzle) that France does not win. Instead, France wants to make it into a game where the defending power has to guess which turn he will attack. Since there is a (countably) infinite number of turns, this greatly reduces the "probability" that the defending power makes the right guess.

There are a few things needed in order for France to change it into a game about when rather than where:
1) France needs a waiting move: a move that cannot lose,
2) France needs that the defending power has no waiting moves (for then the game would be stalemated), and
3) France needs an attacking move such that any counterattack looses to the French waiting move.

France strategy in simplified form is then: Choose a natural number k. Play the waiting move k times, and then play an attacking move.

In the 2x2 game, there is only one attacking move, so this describes the strategy completely for that puzzle. However, in Puzzle 3 there are two attacking moves to consider, and in Puzzle 5 there are three attacking moves to consider. For those two puzzles, we need to specify how France should choose between the attacking moves on the turn he decides to attack. The interesting (and not necessarily intuitive) thing is, for this tactic to work, France only needs that one of the attacking moves only has counterattacks that looses to the waiting move. The other French attacking moves can have counterattacks that either draws the game or even results in a French loss - that's ok. France just needs to be a little clever with how he chooses between his attacking options.

Notice that since the defending power has no waiting moves, he needs to play a counterattacking move, for that is the only way he can get out of the situation. If the defending power guesses k right, but does not counterattack, the the game will restart; France picks a new integer k, and the defending power will have to guess it correctly again.

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Re: 1v1 Puzzles

#68 Post by RoganJosh » Sat Feb 08, 2020 10:33 am

Puzzle 5

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 how to choose k, 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.

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Re: 1v1 Puzzles

#69 Post by RoganJosh » Sat Feb 08, 2020 11:39 am

Puzzle 3:

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 how to choose k (i.e., France gets to choose P(l = k)), he can make this probability arbitrarily small, by making smart choices.

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Re: 1v1 Puzzles

#70 Post by jay65536 » Mon Feb 10, 2020 4:20 pm

RoganJosh wrote:
Sat Feb 08, 2020 9:54 am
France strategy in simplified form is then: Choose a natural number k. Play the waiting move k times, and then play an attacking move.
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.

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Re: 1v1 Puzzles

#71 Post by RoganJosh » Mon Feb 10, 2020 11:06 pm

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:
jay65536 wrote:
Thu Jan 23, 2020 4:21 pm
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.
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 N given by [0,N]\cap N, does not converge to a measure.
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 had 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...

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Re: 1v1 Puzzles

#72 Post by RoganJosh » Mon Mar 16, 2020 6:53 pm

The remaining solutions.

Puzzle 2:
MAO > Por
WMS > Spa supported by Gas
Mar H supported by Bur

Puzzle 4:
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.

Puzzle 1:
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 after 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.

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Re: 1v1 Puzzles

#73 Post by RoganJosh » Mon Mar 16, 2020 6:54 pm

Two classical puzzles!

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.

Puzzle 6: A variation. https://imgur.com/Z04T3YI
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 disbanded 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 no matter what orders Turkey submits.

Puzzle 7: A simple double. https://imgur.com/a/jz5CLV3
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, no matter 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.
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Re: 1v1 Puzzles

#74 Post by captainmeme » Sat Apr 04, 2020 3:35 pm

Now in video form!

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!
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Re: 1v1 Puzzles

#75 Post by teccles » Wed Apr 08, 2020 7:48 pm

Is puzzle 6 in Spring or Autumn?

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Re: 1v1 Puzzles

#76 Post by RoganJosh » Wed Apr 08, 2020 8:15 pm

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!

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Re: 1v1 Puzzles

#77 Post by teccles » Wed Apr 08, 2020 8:23 pm

RoganJosh wrote:
Wed Apr 08, 2020 8:15 pm
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!
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!

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Re: 1v1 Puzzles

#78 Post by RoganJosh » Wed Apr 08, 2020 8:53 pm

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.
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Re: 1v1 Puzzles

#79 Post by bartogian » Thu Apr 09, 2020 12:51 am

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?

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Re: 1v1 Puzzles

#80 Post by RoganJosh » Thu Apr 09, 2020 6:39 am

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.

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