1v1 Puzzles

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RoganJosh
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1v1 Puzzles

#1 Post by RoganJosh » Thu Jan 16, 2020 10:52 pm

I'm was gonna see if there is any interest for some puzzles here.

I thought I'd market these puzzles as 1v1 practice. As a starter, I'll put four puzzles. One will be 1v1 opening strategy, one will be an 1v1 Austrian end-game puzzle, and two will be general stalemate line puzzles. The latter pair is of course relevant also for people who don't care about 1v1.

At this point, I would like to know if people enjoy these, and if the level is appropriate.

I should mention that there are already some diplomacy puzzles out there:
http://uk.diplom.org/pouch/Zine/W1995A/ ... oblems.htm
https://www.youtube.com/channel/UC_5f6rJS2cMPE986QDRf5p
..and I've seen at least one more collection that I can't find at the moment..

Oh, and maybe I should mention that these are not really puzzles that I "constructed" - these are taken from actual games here and on vDip.

Ok. Time for content.

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Re: 1v1 Puzzles

#2 Post by RoganJosh » Thu Jan 16, 2020 10:54 pm

Puzzle 1

https://imgur.com/wZ8gL5T

A 1v1 opening strategy puzzle in two parts!

a) Give the best orders for France and for Austria.
b) If you could, which build order would you change?

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Re: 1v1 Puzzles

#3 Post by RoganJosh » Thu Jan 16, 2020 10:55 pm

Puzzle 2

https://imgur.com/Gl3sr5V

A simple stalemate line puzzle:

Germany to win in one move.

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Re: 1v1 Puzzles

#4 Post by RoganJosh » Thu Jan 16, 2020 10:57 pm

Puzzle 3

https://imgur.com/0DOyI8T

A more challenging stalemate line puzzle!

France to win with probability 1.

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Re: 1v1 Puzzles

#5 Post by RoganJosh » Thu Jan 16, 2020 10:58 pm

Puzzle 4

https://imgur.com/yKg7WcA

And, finally, and Austrian end-game puzzle.

Austria to win in two moves.

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Re: 1v1 Puzzles

#6 Post by RoganJosh » Thu Jan 16, 2020 11:03 pm

If that was not clear, then "Power A to win" means to give a set of orders (with possible continuations) such that Power A wins no matter the opponents orders.

Feel free to post solutions and discuss in this threat. I'll check any proposed solutions.

There might be more than one solution.

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Re: 1v1 Puzzles

#7 Post by mhsmith0 » Fri Jan 17, 2020 5:46 am

Guesses sent in. I am admittedly stumped on #3, it SEEMS like a stalemate position...

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Re: 1v1 Puzzles

#8 Post by Tugster » Fri Jan 17, 2020 9:58 am

puzzle 3 is bullshit, there is no move set for France that will give a win with 100% certainty, none, but it is also not a stalemate, it is a series of guesses, and France might win, if he guesses well, but Austria might achieve a stalemate, if he guesses well.

Probably 1 means "In all cases" with perfect defense, therefore, think of a move set for France, then tell Austria the move set, then Austria gets to write his defense knowing in advance what France will do. With these parameters Austria can always defend, forever. I know that's not how the game works, but that's what it means to state "Probability 1" in that there is a move set that achieves victory for France, no matter what Austria does. This is false.

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Re: 1v1 Puzzles

#9 Post by Tugster » Fri Jan 17, 2020 10:16 am

and do this every turn, Each turn going forward forever, France decides on his moves first, then tell Austria, then Austria writes his moves. France will never get SC 18 in this scenario. If Austria guesses right forever, Austria will never lose. Yes, this is unlikely, but it is NOT 100% for France, it's not. It might be 95% or 98% but that's not 100%

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Re: 1v1 Puzzles

#10 Post by RoganJosh » Fri Jan 17, 2020 10:45 am

Tugster wrote:
Fri Jan 17, 2020 9:58 am
Probably 1 means "In all cases" with perfect defense, therefore, think of a move set for France, then tell Austria the move set, then Austria gets to write his defense knowing in advance what France will do. With these parameters Austria can always defend, forever. I know that's not how the game works, but that's what it means to state "Probability 1" in that there is a move set that achieves victory for France, no matter what Austria does. This is false.
Nope, probability 1 does not mean "in all cases."

For example, if you choose a real number uniformly at random on the interval [0,1], then by probability 1 it will be an irrational number. This does not mean that all real numbers on said interval are irrational.

But it is true (and maybe to mention as a hint for other players), that if you impose a turn limit, then there is a positive probability that Austria will get a draw or a win.

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Re: 1v1 Puzzles

#11 Post by bartogian » Fri Jan 17, 2020 11:01 am

A further puzzle that is a continuation of #3.

If you are GMing a game, or playing a game where a probability 1 win appears, what do you do?

P.S. For those confused, a probability 1 win means that for all p<1, France can find a strategy that wins with probability at least p.

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Re: 1v1 Puzzles

#12 Post by mhsmith0 » Fri Jan 17, 2020 4:42 pm

I guess I'll just lay out puzzle 3 thoughts a bit more. From where I see it...

1) If France can force an army into Prussia or Piedmont, the puzzle is simple (Piedmont taps Tyrolia, then Munich can be forced or at least 50-50'd; Prussia taps Silesia, Munich can be 100% forced without risk of unit swap)

2) The potential for France to force an army in
Piedmont: Even if France can force GOL (which probably takes risks to get a fleet into Marseilles), Ven/Tus support Pie H still works, since TYS definitely cannot be forced (Rom/Nap/ION support TYS H cannot be broken). I THINK it's ok to discount this path?

Prussia:
Presumably, first step is to force Livonia, next step is to force Prussia from Livonia

Livonia:
NOTES: I'm assuming Bur-Mun supported by Ruhr every turn barring other assumptions, mainly to cut support but with a non-zero chance of a win if Austria heavily screws up. Kiel either supports Ber H or Bur-Mun.
On Austrian end: Prussia supports Liv H unless explicitly noted otherwise. Tyr/Boh support Munich H, OR we run Tyr-Mun supported by Boh, and Mun-Kiel (to cut support from Kiel for Berlin H)

STP-Liv supported by GOB/Baltic
Austrian perspective: Sev-Mos succeeds (so if Liv isn't forced THAT TURN, it becomes much harder on France for I think obvious reasons), Pru/War support Liv H succeeds
While I'm at it, I think it's appropriate to assume A Tri moves towards Ukraine (Tri-Bud-Gal-War), arrives there in THREE moves, in basically all scenarios

STP-Liv supported by Gob; Baltic-Prussia supported by Berlin
Austrian perspective: If you take "try and hold Liv/Pru" approach, then Sev-Mos succeeds, either F Bal destroys Prussia or A STP destroys Livonia (warsaw can't support both to hold; probably A Liv is destroyed in this scenario)

NOTE THE RISK HERE: If Austria is being aggressive, he can take a shot at the win via a simple coin flip against the A Kiel order set (this is now open because F Baltic is doing something other than supporting A Berlin, making that spot temporarily vulnerable)
Option 1)
Mun-Kiel, Pru-Ber supported by Silesia. This wins outright against A Kiel support Ber H (Kiel gets cut, Baltic isn't defending Berlin, Tyr/Boh bounce against Bur/Ruhr at Munich)
Option 2)
Pru-SIl supported by Silesia, Tyr/Boh support Mun H. This wins outright against A Kiel support Bur-Mun (because 3 on 3 fails), with Berlin falling since it has no support anywhere

Even if it's the spring move set, I don't think there's any way for France to recover Berlin or Munich in the fall, giving Austria 18.

If there's a 100% way for France to take Munich without risking an outright loss, I don't see it. I don't think Livonia can be forced without using the Baltic Sea unit, and if Livonia can't be forced, I don't see how Prussia can be forced. And even if Livonia CAN be forced, I suspect there'd be some additional danger at work.

Very curious to see how that is wrong.

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Re: 1v1 Puzzles

#13 Post by Tugster » Fri Jan 17, 2020 6:15 pm

It's pretty clear from this lengthy analysis by MHSMith that it is not probability 1, it's not 100%. Because one of those ifs and guesses goes the other way, the French game crumbles, with distinct non zero probabilities of an Austrian win. And when you go into all the iterations, it is totally crystal clear that a French win is NOT certain. I am a mathematician, probability 1, does in fact mean 100% => "IN ALL CASES"

I quote the definition of probability 1 and 0:

The probability of an event is a number describing the chance that the event will happen. An event that is certain to happen has a probability of 1. An event that cannot possibly happen has a probability of zero. If there is a chance that an event will happen, then its probability is between zero and 1.

Note the words "CERTAIN TO HAPPEN"

A french win is clearly NOT certain to happen. One bad guess with Berlin trying to cut support on Prussia, and it's even possible, with a decent percentage chance, that Austria will win.

So I do think the guess are in France's favor, but for sure, it is not 100%, it's not.

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Re: 1v1 Puzzles

#14 Post by Tugster » Fri Jan 17, 2020 6:17 pm

and the example of random numbers between 0 and 1 being rational is really not applicable here, because in that example, we are talking about infinitesimal small chances of hitting a rational number, in the limit zero.

In this game, the chances of Austrian getting his guess right and taking Berlin are VASTLY highly than that, perhaps as high as 25% or more. It's not at all the same thing.

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Re: 1v1 Puzzles

#15 Post by Tugster » Fri Jan 17, 2020 6:28 pm

and as soon as you say this:

But it is true (and maybe to mention as a hint for other players), that if you impose a turn limit, then there is a positive probability that Austria will get a draw or a win.

you admit that it's not probability 1

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Re: 1v1 Puzzles

#16 Post by RoganJosh » Fri Jan 17, 2020 6:49 pm

Here's a link with some nice reading and references on the difference between "impossible" and "probability 0".

https://math.stackexchange.com/question ... ossibility

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Re: 1v1 Puzzles

#17 Post by mhsmith0 » Fri Jan 17, 2020 6:52 pm

I think I'd probably revise my earlier note, since France has three rather than two options in the event that F Baltic is busy doing something other than supporting or tapping Berlin...

France move sets
A) Ber-Mun supported by Bur/Ruhr/Kiel
B) Bur-Mun supported by Ruhr/Ber, Kiel supports Ber H
C) Bur-Mun supported by Ruhr/Kiel/Ber

Austrian move sets
1) Pru-Ber supported by Mun, Tyr/Boh/SIl support Mun H
2) Pru-Ber supported by Mun/Sil, Tyr/Boh support Mun H
3) Mun-Ber supported by Sil/Pru, Tyr-Mun supported by Boh
4) Mun-Ber supported by Pru, Tyr-Mun supported by Boh/Sil
5) Mun-Kiel, Tyr-Mun supported by Boh, Pru-Ber supported by Sil

France move set A:
v Aus1: Austria takes Berlin, Munich holds, Prussia empty space, Austria gets 18
v Aus2: Berlin/Munich control flipped, A Mun destroyed, Prussia is empty space
v Aus3: France takes Munich and doesn't lose Berlin, hits 18
v Aus4: France takes Munich and doesn't lose Berlin, hits 18
v Aus5: Berlin/Munich control flipped, A Mun destroyed, Prussia is empty space

France move set B:
v Aus1: no change in unit positions
v Aus2: no change in unit positions
v Aus3: Austria takes Berlin, Munich empty space, Austria gets 18
v Aus4: no change in unit positions
v Aus5: Austria takes Berlin, Munich empty space, Austria gets 18

France move set C
v Aus1: no change in unit positions
v Aus2: Berlin/Munich control flipped, A Mun destroyed, Prussia is empty space (also A Ber destroyed, but A Par presumably backfills Bur in the considered order set)
v Aus3: Berlin/Munich control flipped, A Ber destroyed
v Aus4: Austria takes Berlin, Munich empty space, Austria gets 18
v Aus5: Berlin/Munich control flipped, A Mun destroyed, Prussia is empty space (also A Ber destroyed, but A Par presumably backfills Bur in the considered order set)

From Austrian perspective:
Move set 1: 1/3 chance of winning, 2/3 chance of 0 net change
Move set 2: 3/3 chance of 0 net change (therefore sub-optimal, wouldn’t take it)
Move set 3: 1/3 chance of winning, 1/3 chance of losing, 1/3 chance of 0 net change
Move set 4: 1/3 chance of winning, 1/3 chance of losing, 1/3 chance of 0 net change
Move set 5: 1/3 chance of winning, 2/3 chance of 0 net change

Move set 2 does nothing, is sub-optimal, you can chuck it
Move set 5 is strictly better than move set 3 (against A, it draws instead of loses; against B, it’s same outcome; against C, it’s basically the same outcome)

So you can narrow it down to France move sets A/B/C against Austrian move sets 1/4/5
Each has a 33% chance of Austrian outright victory, where if France chooses move set A, he gets a shot at an outright win himself.

So it's not at the 50-50 level of "Austria has an outright coin flip to win the game if he predicts accurately when France will try for Livonia" but it's in the ballpark of 33% (adjusted for chance of outright loss that turn).

That said, there could certainly be some other tactical option I'm missing here. But I THINK that in order to take Livonia and then Prussia, France is required to accept the risk of an outright loss (also, if you get a status quo turn, but Sev gets into Moscow, you then add in Mos-STP and Mos support Liv-STP as possible moves, which makes it harder on France and probably creates the risk of an Austrian army in STP which also wins the game for Austria)

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Re: 1v1 Puzzles

#18 Post by bozotheclown » Fri Jan 17, 2020 9:41 pm

mhsmith0 wrote:
Fri Jan 17, 2020 4:42 pm
I guess I'll just lay out puzzle 3 thoughts a bit more. From where I see it...

1) If France can force an army into Prussia or Piedmont, the puzzle is simple (Piedmont taps Tyrolia, then Munich can be forced or at least 50-50'd; Prussia taps Silesia, Munich can be 100% forced without risk of unit swap)

2) The potential for France to force an army in
Piedmont: Even if France can force GOL (which probably takes risks to get a fleet into Marseilles), Ven/Tus support Pie H still works, since TYS definitely cannot be forced (Rom/Nap/ION support TYS H cannot be broken). I THINK it's ok to discount this path?

Prussia:
Presumably, first step is to force Livonia, next step is to force Prussia from Livonia

Livonia:
NOTES: I'm assuming Bur-Mun supported by Ruhr every turn barring other assumptions, mainly to cut support but with a non-zero chance of a win if Austria heavily screws up. Kiel either supports Ber H or Bur-Mun.
On Austrian end: Prussia supports Liv H unless explicitly noted otherwise. Tyr/Boh support Munich H, OR we run Tyr-Mun supported by Boh, and Mun-Kiel (to cut support from Kiel for Berlin H)

STP-Liv supported by GOB/Baltic
Austrian perspective: Sev-Mos succeeds (so if Liv isn't forced THAT TURN, it becomes much harder on France for I think obvious reasons), Pru/War support Liv H succeeds
While I'm at it, I think it's appropriate to assume A Tri moves towards Ukraine (Tri-Bud-Gal-War), arrives there in THREE moves, in basically all scenarios

STP-Liv supported by Gob; Baltic-Prussia supported by Berlin
Austrian perspective: If you take "try and hold Liv/Pru" approach, then Sev-Mos succeeds, either F Bal destroys Prussia or A STP destroys Livonia (warsaw can't support both to hold; probably A Liv is destroyed in this scenario)

NOTE THE RISK HERE: If Austria is being aggressive, he can take a shot at the win via a simple coin flip against the A Kiel order set (this is now open because F Baltic is doing something other than supporting A Berlin, making that spot temporarily vulnerable)
Option 1)
Mun-Kiel, Pru-Ber supported by Silesia. This wins outright against A Kiel support Ber H (Kiel gets cut, Baltic isn't defending Berlin, Tyr/Boh bounce against Bur/Ruhr at Munich)
Option 2)
Pru-SIl supported by Silesia, Tyr/Boh support Mun H. This wins outright against A Kiel support Bur-Mun (because 3 on 3 fails), with Berlin falling since it has no support anywhere

Even if it's the spring move set, I don't think there's any way for France to recover Berlin or Munich in the fall, giving Austria 18.

If there's a 100% way for France to take Munich without risking an outright loss, I don't see it. I don't think Livonia can be forced without using the Baltic Sea unit, and if Livonia can't be forced, I don't see how Prussia can be forced. And even if Livonia CAN be forced, I suspect there'd be some additional danger at work.

Very curious to see how that is wrong.
One thing I believe you are not considering is that France has the benefit of time. Austria must put Liv., Pru., or Mun. at risk every turn, while France can hold his position indefinitely. If you were playing Austria, and France entered the safe moves 1000 times in a row, what would you do as Austria?

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Re: 1v1 Puzzles

#19 Post by tr1285 » Fri Jan 17, 2020 9:52 pm

Would it be correct to interpret the solution for puzzle 3, "probability 1" as you are looking for some strategy by France that should allow him to win eventually? So Austria would have to continually guess right to perpetuity to stop it?

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Re: 1v1 Puzzles

#20 Post by Tugster » Fri Jan 17, 2020 11:55 pm

MH Smith goes over it well again. CLEARLY, there is no 100% certain way for France to win. He has to take risk to make a guess to go for the win, and if Austria counter guesses correctly at that moment, Austria can win. The guesses are in fact in France's favor, but not in a certain way. It's not probability 1, it's not, no way. We could debate forever about what the percentages are on the guesses for each player, but that of course isn't my point, my point is only that it is NOT 100%, it's not.

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