Reassessing optimal number of territories: Golden Numbers

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With respect to a map's territory count, what's more important to you as a CC player / cartographer?

Minimising neutrals
8
13%
Optimising fairness of the drop as detailed below
50
82%
Something else (please explain)
3
5%
 
Total votes : 61

Reassessing optimal number of territories: Golden Numbers

Postby benjikat on Thu Nov 22, 2007 10:24 am

I have been thinking recently (both from the perspective of maps to play and those I want to create) about the optimal number of territories for a "fair" map to have.

I know that the foundry likes to minimise the number of neutrals in any given game, with "nice" numbers like 60, 72 etc... but this (IMO) is an incorrect approach (especially for large maps). Hopefully I can explain why:

Territory count is a fundamental principle within the game of CC - i.e. the obtaining of 1 extra man per 3 territories controlled.

In all forms of the game, but especially those that amount to 1v1 (i.e 2 player, 4 player dubs, trips, and 4v4), knocking one or more players into a lower territory band is a fundamental tactic. When each player starts with 12 or 15 territories, then in say 4 player doubles, the team that happens to go first gets a huge advantage. Personally CC is a strategy game for me, and thus I like to minimise as many of the luck factors as possible - the dice are enough randomness surely :).

The setup (in terms of territory count alone) that is least advantageous to the player and/or team that happens to go first is for for each player to start with one less territory than the next band, i.e. 11(or less),14,17,20,23 etc.

I have analysed the numbers (within a spreadsheet you can download from here), and found that the following are the "golden numbers", which create a drop that in all (or most) forms of the game require a full 3 territories to be taken from a player before they are disadvantaged because of territory count and happening to not go first:

35 and less (the most you can start with is 11)
42,43,44 - It's no surprise that the many maps with these "classic" numbers of territories are so popular.
52,53 - the best sizes for slightly larger than standard maps
70,71 - the best sizes for large maps
80 - 5 & 6 player games require 2 conquers
88,89 - 4 player games require 2 conquers and 7 players only 1
104 - a great number (what a saddo I am :) ) - only 8 player games require less than 3 conquers - but still 2 - the best size for a very large map
141,142,143 - 5 player games require only 2 conquers
160 - 4 & 7 player games require only 2 conquers
190,191 - 6 player games require 2 conquers and 7 only 1

I have included 7 and 8 player games as they (well 8 anyway) are on Lack's to do list. I also think that 3,5 and 7 player games are significantly less important to optimise than 2,4,6,8, which of course also incorporate all the team games.

It should also be noted that for the very large maps a fair bit of this is irrelevant (for games with few players, 2 in particular): If you receive 9 or more armies going first then simply by deploying them in one place and attacking 3 3s, you are more than 60% (Coleman's recent retracted definition of a "broken first turn") likely to knock them into the band below. This occurs at 81 territories for 2,3 player games, 108 for 4, 135 for 5, 162 for 6, 189 for 7 and 216 for 8.

Moreover, with the very large maps gaining one less army is the least of your worries if your opponent(s) got great dice. In fact I think that this analysis is only really relevant for maps up to 104 territories (which I really hope Cairnswk chooses as the number for Waterloo :wink:)

Note: territory count in all the above is after enforced neutrals are discounted.

As a related aside, a practical example: I adore the D-Day map except for a couple of minor things (which I wish I had brought up at the appropriate time, but which have really only been solidified in my mind by actually playing some games - btw when will we get a playtest area?): a) with 72 territories (a very very poor number within the parameters of the above analysis), it clearly advantages those going first. b) whilst I totally agree with the strategic benefit of the paratroopers and thus their -1 "bonus", this strategic benefit almost never applies to the drop. In fact I have faced my first turn with only two armies to deploy where most others had four! I know that mibi and Coleman may well disagree with me (as well as the wider CC community for that matter), but for me the map would be greatly improved by making the 2 planes start neutral (maybe set at only 1 or 2). This would slightly increase the number of neutrals deployed by the game engine on the drop, but only significantly for 8 player games.

Up until this point it seems as though minimising the number of starting neutrals has been the main (sole?) consideration, I hope this can start a debate as to how many territories we want in our maps - although of course the final opinion will always be that of the individual cartographer...

(Also - I'm not infallible and would appreciate it if someone could double check my findings)
Last edited by benjikat on Fri Nov 23, 2007 12:07 pm, edited 2 times in total.
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Postby benjikat on Thu Nov 22, 2007 2:29 pm

Would those who have replied "other" please explain....
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Postby yeti_c on Thu Nov 22, 2007 2:40 pm

I've not checked the numbers - but an excellent analysis...

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Re: Reassessing optimal number of map terrtitories

Postby BaldAdonis on Fri Nov 23, 2007 1:56 am

benjikat wrote:88,89 - 3 player games require 2 conquers and 7 players only 1
104 - a great number (what a saddo I am Smile ) - only 8 player games require less than 3 conquers - but still 2 - the best size for a very large map
141,142,143 - 4 player games require only 2 conquers
160 - 3 & 7 player games require only 2 conquers
190,191 - 6 player games require 2 conquers and 7 only 1

Anyplace you have 3 there should be 4, and any 4 should be 5. I think that's the only mistake in the sheet, and it makes sense, because once you get that far down, it's easy to forget that 2&3 go together.

Like you said, by the time you get to these large maps, the bonuses for two and three players are enough to take down a string of three 3s, but who's playing two player games on a map this size?

benjikat wrote:the map would be greatly improved by making the 2 planes start neutral

Why not the paratroopers instead? As it is, there's no good reason to attack a paratrooper, so anyone holding one from the drop is disadvantaged, in that they can't take the outer ships without a punishment. If they're set at one neutral, the back door is still there, but anyone who takes that route will have to suffer the penalties in subsequent turns.
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Re: Reassessing optimal number of map terrtitories

Postby benjikat on Fri Nov 23, 2007 2:47 am

BaldAdonis wrote:
benjikat wrote:88,89 - 3 player games require 2 conquers and 7 players only 1
104 - a great number (what a saddo I am Smile ) - only 8 player games require less than 3 conquers - but still 2 - the best size for a very large map
141,142,143 - 4 player games require only 2 conquers
160 - 3 & 7 player games require only 2 conquers
190,191 - 6 player games require 2 conquers and 7 only 1

Anyplace you have 3 there should be 4, and any 4 should be 5. I think that's the only mistake in the sheet, and it makes sense, because once you get that far down, it's easy to forget that 2&3 go together.


oops - yeah I forgot it seems that I put 2&3 player games in the same column - have now edited the first post.

BaldAdonis wrote:
benjikat wrote:the map would be greatly improved by making the 2 planes start neutral

Why not the paratroopers instead? As it is, there's no good reason to attack a paratrooper, so anyone holding one from the drop is disadvantaged, in that they can't take the outer ships without a punishment. If they're set at one neutral, the back door is still there, but anyone who takes that route will have to suffer the penalties in subsequent turns.


In this particular instance there isn't any particular difference - the planes aren't part of the ship bonus either, so doesn't really make much difference.
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Postby waseemalim on Fri Nov 23, 2007 3:13 am

go benji!! you are much more estoric than I thought.
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Postby benjikat on Fri Nov 23, 2007 6:51 am

waseemalim wrote:go benji!! you are much more estoric than I thought.


The fact that I had to look up the proper definition of esoteric (which is what I presume you mean), suggests I'm not!


Back on topic: Mathematically the two opposing approaches I have presented are pretty mutually exclusive due to the fact that the very numbers that minimise neutrals are multiples of 3 and 6, the very multiples I would wish to avoid.
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Postby rebelman on Fri Nov 23, 2007 10:43 am

I like the way this debate is opening up , but before I make a detailed contribution I first need to find my thinking head ala wurzel gummidge.
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Postby oaktown on Fri Nov 23, 2007 2:06 pm

benjikat wrote:Back on topic: Mathematically the two opposing approaches I have presented are pretty mutually exclusive due to the fact that the very numbers that minimise neutrals are multiples of 3 and 6, the very multiples I would wish to avoid.

I like this discussion. To me the worst numbers of of territories to begin with are those numbers that gives a clear advantage to whomever scores the first turn: multiples of 12, 15, 18, 21. The biggest problem I have with World 2.0 is that the first player gets a big enough bonus to knock the following players' bonus down an army.

Starting neutrals are not inherently bad... pre-determined starting neutrals play an important role in in maps with special territories that might give an advantage to the player that gets it on the drop.

However, limiting the number of random neutrals is important because random neturals; 1) effectively take territories out of play for the entirity of some games, 2) force a player to waste armies attacking something other than his enemies if a neutral stands between him and a bonus, and 3) often serve as defensive shields for a player who is lucky enough to get a drop behind one or more neutral territories.

Both of the choices in this poll are equally important to me because both are all about limiting the role that luck plays in the first round of the game.

Starting a map with 36, 48, or 60 territories is attractive because it limits the number of neutrals; however, these numbers also lend an advantage to the player with the opening move as they are guaranteed a full initial placement. It would be nice to come up with a formula for "golden numbers" that somehow take both factors into account and mimimize the roll of luck.
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Postby yeti_c on Fri Nov 23, 2007 2:20 pm

Also - another point I just thought of...

If you start with a drop that means if you lose 3 territories then you lose 1 army from your count...

This also means that 1 territory capture means you gain 1 army...

Of course - in the larger maps - if you have enough to drop the opponent 3 territories - then you also gain 3... so that puts you 2 ahead of your opponent...

So whilst I see your point Benjikat - there is a flip side...

Perhaps the best numbers to go for then are the ones in the middle...

i.e. you need to lose or gain 2 territories to lose/gain 1 army?

Thoughts?

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Postby AndyDufresne on Fri Nov 23, 2007 3:01 pm

Hm, numbers have never been something I enjoyed. I'm more of a pleasant word man. But what I can say in words...is that I agree with Oaktown's statement of
Both of the choices in this poll are equally important to me because both are all about limiting the role that luck plays in the first round of the game

And as Oaktown also mentioned...if the golden numbers could be formulated...hm...that would be interesting.

I wish I could contribute more to this, but unfortunately I'll have to leave that up to those more mathematically inclined. Coleman might have some insight on all this when he gets back.


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Postby Aerial Attack on Fri Nov 23, 2007 3:42 pm

yeti_c wrote:Also - another point I just thought of...

If you start with a drop that means if you lose 3 territories then you lose 1 army from your count...

This also means that 1 territory capture means you gain 1 army...

Of course - in the larger maps - if you have enough to drop the opponent 3 territories - then you also gain 3... so that puts you 2 ahead of your opponent...

So whilst I see your point Benjikat - there is a flip side...

Perhaps the best numbers to go for then are the ones in the middle...

i.e. you need to lose or gain 2 territories to lose/gain 1 army?

Thoughts?

C.


Interesting counter argument yeti - alas I believe it is slightly flawed (I haven't done the math ...).

This all depends on the likely hood that someone will conquer three OR four territories on the initial turn.

For larger the maps the probability of taking TWO terrs is fairly great. Take ONE is almost a certainty.

Assuming same dice [and cards] for both parties - if only one or two terrs needs to be captured to lessen the opponent's reinforcement rate - the initial attacker is guaranteed to always have a higher reinforcement rate [and therefore eventually more armies to win with].

If you calculate out the probability of taking three terrs in benji's situation - then it is more difficult to capture three terrs back to knock the initial attacker down 1 reinforcement, but fairly easy to capture one terr to regain the lost reinforcement. Additionally, one would need to conquer SIX terrs to knock someone down 2 reinforcements (and gain a FOUR army per turn advantage). People will most likely stop at either 1 or 3 terrs taken

Conversely, in the way it currently exists - people are more likely to try for FOUR terrs to knock the opp down 2 reinforcements. Here, they would be down 3 armies per turn. People will most likely stop at either 2 or 4 terrs taken.
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Postby benjikat on Fri Nov 23, 2007 8:27 pm

oaktown wrote: Starting neutrals are not inherently bad... pre-determined starting neutrals play an important role in in maps with special territories that might give an advantage to the player that gets it on the drop.


Agreed, I think there is much scope for creative use of starting neutrals, as I hope DiM's new map will prove, and which allows for the fairest drop yet of any CC map.

oaktown wrote:
However, limiting the number of random neutrals is important because random neturals; 1) effectively take territories out of play for the entirity of some games, 2) force a player to waste armies attacking something other than his enemies if a neutral stands between him and a bonus, and 3) often serve as defensive shields for a player who is lucky enough to get a drop behind one or more neutral territories.


Agreed, although I think different forms of the game suffer more from neutrals. The larger the map the less of a problem they are, and the more "open" a map the less problematic too. An intricate map with lots of bottlenecks and small bonuses is clearly going to be affected more.

oaktown wrote: Both of the choices in this poll are equally important to me because both are all about limiting the role that luck plays in the first round of the game.


I agree with this too, but felt I had to take the approach I did in my opening post so as to most effectively challenge the status quo.

oaktown wrote: It would be nice to come up with a formula for "golden numbers" that somehow take both factors into account and mimimize the roll of luck.


Let's try to do just that.

In order to do so, we need to be able to mathematically describe how destabilising random neutrals are.

I suggest % of deployed territories is used, meaning that we really want to minimise neutrals on smaller maps?

yeti_c wrote:
If you start with a drop that means if you lose 3 territories then you lose 1 army from your count... This also means that 1 territory capture means you gain 1 army...

Of course - in the larger maps - if you have enough to drop the opponent 3 territories - then you also gain 3... so that puts you 2 ahead of your opponent...

So whilst I see your point Benjikat - there is a flip side...

Perhaps the best numbers to go for then are the ones in the middle...

i.e. you need to lose or gain 2 territories to lose/gain 1 army?

Thoughts?

C.



Aerial Attack wrote:Interesting counter argument yeti - alas I believe it is slightly flawed (I haven't done the math ...).


Yes I believe this to be slightly flawed too, as the potential for 1 extra army next turn if it isn't taken away from you, is far far less of an advantage, than actually taking away an army from your opponent for their very first turn.

If you start with 3x+2 territories, then you need to take 1 territitory to gain the potential for an extra 1 next turn for you, 3 to take away one from an opponent, and 4 to take 1 away and gain the potential for 2 extra.

If you start with 3x+1 territories, then taking 2 territories takes 1 away and gives you the potential for an extra one next turn.

If you start with 3x territories, then a single take will take an deployable army away, and 2 more would give you the potential for an extra one next turn.

I think that the advantages gained should be seen in relative rather then absolute terms, i.e % loss in deployment. So for example if each player has 12 starting territories in 4 player dubs (I know I keep using it as the example, but it is on my mind as it is the form of the game I am concentrating on learning most at the moment.), then each of the starting players taking 1 territory from their immediately following opponent means that team one gets, 4 and 4 to deploy, but team 2 gets just 3 and 3. This is a disadvantage of a full 25%. If however each player starts with 17 territories, then team 1 gets 5 and 5, and team 2 gets 5 and 4 (if team 1 successfully take 3 territories between them). This is only a disadvantage of 10%.

During the course of typing all this I think I have come up with a potential formula for determining our "golden numbers".... but I'll put it in the next post....
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Postby benjikat on Sat Nov 24, 2007 3:42 am

ummm - quite a few hours later (I fell asleep :oops: ).


So a potential framework for calculation optimum territory numbers - optimal in the sense of minimising the effect of happening to go first in particular.

For each game type (2,3,4,2v2,5,6,2v2v2,3v3,7,8,4v4, 2v2v2v2), and each territory size, calculate the following:

(for the ones in italics we have to decide how important they are to include as their "current popularity" is non-existent)

a) The probability of taking away a deployable army from the next player / team.

Some discretion needs to used here as the the best tactics to use to do this. i.e in 4 player dubs, could involve each player targetting the next, or both targeting the fourth, depending on likelihood (the threshold of this strategic switch could be a global variable)

Multiply this probability with the % disadvantage losing that army gives, as defined in the previous post.

b) the probability of gaining the potential of an extra army (or 2) for next turn.

Multiply this probability by the % advantage this potentially gives, multiplied by a coefficient that represents the difference between actual loss and potential gain (IMO it should be as low as 0.2)

c) The % of random neutrals created with respect to the total number of deployable territories

Add a) and b) together, and then add to c) in some proportion (which also needs to be a global variable).

Then for each game mode, weight these according to their popularity. Do we need to take the scoreboard into account for our "fairness index". i.e do we need to also weight by potential pts at stake, or potential points winnable?? I think we need to do something to not have the figures swamped by their suitability to 1v1 games (which currentyl account for a third , and growing, of all games)....

This then would provide a "fairness index" for each map size, which along with our 2or3 global variables, could help tease out the "golden numbers", which I suspect will look very similar to those I initially posted, unless you greatly favour minimising neutrals, especially for the larger maps.

My guess is that this would show that the small and classic size maps (less than 36, 42,43,44) are significantly "fairer" than any others (btw do you think that analysis like this was done for the original board game? - I hope so because I think they were spot on in the sweet spot). It will also be a very jagged plot, which I hope would revael our "golden numbers" when plotted.

What do you all think?

I would need some help to work all this out (I don't think my clunky(even on a Mac Book Pro) Open Office in X11 would cope) and the only way I have of calculating the dice probabilities is one at a time with an online calculator ( which I will do if I have to, be think is a waste of my time really - and others would always wonder where I made my mistakes - there would surely be a few due to human error).

The other thing to be agreed is the standard tactics used to attack, I think sticking them all on territory and attacking is sufficient (other, better btw, methods would make the calculations way more complicated).

1v1 - easy - attack opponent
3 - first 2 attack 3rd
4 - what would be a realistic scenario?
2v2 - either attack next person, or both target 4th
5 - ???
6 - ???
2v2v2 - ???
3v3 - team 1 each target the next player, or all target the 6th
7 - ???
8 - ???
2v2v2v2 - ???
4v4 - team 1 each target the next player, 1 and 3 target 4 with 5 and 7 hitting 8 or all target the 8th

Enough for now - even my head is started to hurt...
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Postby benjikat on Sat Nov 24, 2007 3:00 pm

I really don;t mean to keep bumping my our thread, but I genuinely have some further thoughts and want to make sure that the handful of people who are actually reading all this stuff see it.....


Some other ways of mathematically describing the disadvantage of neutrals:


1) straight up number of random neutrals
- doesn't take map size into account at all.

2) % of random neutrals compared to total deployable territories
- is there a direct inverse correlation between map size and disadvantage created by random neutrals?

3) % of random neutrals compared to each players deployable territories
- so for a 4v4 game on a 47 territory map, each player gets 5 territories, but there are 7 random neutrals

4) some combination of the above?

It is important to also note that an calculation of relative fairness, can include a global slider that shifts emphasis between our 2 opposing objectives. This allows for maps that have many bottlenecks to be more concerned with neutrals than normal....


I really hope that somebody can help me reach some firm conclusions on all this - and that it is done in a way that can actually be communicated to the foundry in a simple way...
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Postby Coleman on Sun Nov 25, 2007 5:52 pm

In Response To The Topic In General
I'm not sure what all people were hoping I'd have to say about this.

I agree with most of the analysis, numbers don't lie.

I also agree that random neutrals are bad and most of these numbers result in a lot of them in most player counts. Turning this disadvantage into math is difficult and I have no idea how to go about it. Instead I'd want to focus on avoiding them entirely within the numbers provided.

Something to add...

Code: Select all
Players Pages
6       2193
5       582
4       2179
3       1835
2       3667


So the numbers that give 5 players a high number of neutrals don't concern me as much because 5 players is played least often. Solving 6 and 4 is probably best.

So with 6 and 4 players in mind. I believe 42 and only 42 is truly optimally fair for 6 players.

For 4 players 44, 52, 80, 88, 104, 116, and 160 are best.

Just for completeness if we wanted to start designing maps with 3 or 5 players in mind...

For 3 Players 42, 141

For 5 Players 35, 70, 80, 160, 190

Finding the absolute ideal for 2 players merits some extra discussion as 1/3 of the numbers are going to be neutral anyway so I'm not sure a random 1 or 2 on top of that matters a lot in the long run. My opinion right now is anything between 18 and 35 isn't too bad and anything above will favor the first player.

It is important to note that no one number is optimally fair for all the player counts while avoiding starting neutrals. :(

If you are wondering why I ignored 7 and 8... They are not a reality yet so they are not really on my mind.

In Response To D-Day
Starting both planes neutral 3 (my ideal solution) drops the total number into your desired 70, but that is only optimal for 5 players in avoiding starting neutrals, which is the least played (but still played enough) player count. I don't have a huge problem with this myself but mibi might, as it changes the focus from avoiding neutrals for 2, 3, 4, and 6 players to a game optimally designed for 5. The xml change would be negligible.

My largest concern with it is that the majority of players used to no neutrals would suddenly be confused by the high count of them when playing the map and may send false bug reports or unjustly dislike it as nothing says "D-Day is a map optimally designed for 5 players" and there is no sane way to notify everyone who plays it of the change.
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Postby Aspect on Mon Nov 26, 2007 11:34 pm

I dislike the d-day maps planes too... just weighing in on that point

But anyway, one thing you are all not mentioning is the player to go second, while he may have one less army to drop, has the advantage of attacking ones instead of threes. Now I'm pretty sure the odds are better 6v1 than 7v3.

As for neutral territories, I think that they are being unduly criticized. They provide an additional element of strategy to the game. Sure it adds a bit of luck, but so does a random drop in general. It's possible to start out with all of Australia, but it doesn't happen very often. The same goes for the random neutrals. They can impact the game, but what can't? Cards, initial drops, ROLLS, they are all luck based. Random neutrals should be a minimal consideration.
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Postby Heimdall on Tue Nov 27, 2007 12:26 am

I don't mind games with many neutrals, however i have noticed some cases like Battle for Australia map you'll eventually end-up with Neutral territories that are totally ignored as it's better to attack another player than waste your armies on a Neutral player. Not sure what solution can be brought forward to counter this.

Keeping neutral territories at only 1 army might be a solution as this doesn't require too many resources to conquer and this lessens the luck factor.
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Postby Coleman on Tue Nov 27, 2007 12:33 am

If all the neutrals are 1 then the players landing next to them actually gain an advantage instead of a disadvantage, so the imbalance would be switched.

Unless they were counting on neutrals to block for them... Either way, an increase in randomness usually results in a decrease in strategy, so I still feel avoiding random neutrals is a priority, but these sweet spot numbers are worth knowing.
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umm

Postby WL_southerner on Tue Nov 27, 2007 7:32 am

neutrals are a good thing and a bad thing, if you ever played the world knock out risk, then you know that all matches are played with neutrals they are prefix, and you need to throw a 7 on the dice to kill one army, there are 3 armys on neutrals at start of game,could it be possible to do the same have neutrals prefix like one in each continent and maybe 2 in the larger continent
oh we going to need more pixels for maps if lack up the number of players
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Re: Reassessing optimal number of map territories

Postby yeti_c on Tue Jul 22, 2008 3:15 am

I'm just going to Necrobump this thread - as it is really an interesting piece of analysis that should possibly be looked at by all budding Map Makers.

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Re: Reassessing optimal number of map territories

Postby mibi on Tue Jul 22, 2008 2:43 pm

i've kind of given up on trying to fit in the right number of territorities in there. neutrals are either good or bad depending on your view point, so i'll just let the digits fall where they may.
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Re: Reassessing optimal number of map territories

Postby MrBenn on Tue Jul 22, 2008 4:28 pm

Coleman wrote:If you are wondering why I ignored 7 and 8... They are not a reality yet so they are not really on my mind.

Perhaps someone is able to re-run the figures with these included?
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Re: Reassessing optimal number of map territories

Postby MrBenn on Mon Nov 16, 2009 5:36 am

Current active games:
    2 players - 3123

    3 players - 1290

    4 players - 3797
      (4p doubles - 1080)
    5 players - 2307

    6 players - 3579
      (6p singles - 2342)
      (6p doubles - 381)
      (6p triples - 855)
    7 players - 654

    8 players - 4261
      (8p singles - 3134)
      (8p doubles - 482)
      (8p quads - 645)

(more analysis to follow...)
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Re: Reassessing optimal number of map territories

Postby MrBenn on Mon Nov 16, 2009 6:51 pm

benjikat wrote:35 and less (the most you can start with is 11)
42,43,44 - It's no surprise that the many maps with these "classic" numbers of territories are so popular.
52,53 - the best sizes for slightly larger than standard maps
70,71 - the best sizes for large maps
80 - 5 & 6 player games require 2 conquers
88,89 - 4 player games require 2 conquers and 7 players only 1
104 - a great number (what a saddo I am :) ) - only 8 player games require less than 3 conquers - but still 2 - the best size for a very large map
141,142,143 - 5 player games require only 2 conquers
160 - 4 & 7 player games require only 2 conquers
190,191 - 6 player games require 2 conquers and 7 only 1

I have done some further analysis - the following territory values are optimal for at least 2/3/4/6/8 player games - ensuring that nobody starts with a multiple of 3 on the first turn (therefore requiring a single-territory capture to knock down their opponents deployment). Figures in bold are completely optimal (for ALL games). This list varies slightly from benjikat's original list, as I've used fractionally different criteria to define "optimal" :
35 and less
42,43,44
66,67,68,69
70, 71, 80
88,89
104, 114, 115, 116
138,139
140,141,142,143,160,161
176,177,178,179
186,187,188

215
224
232,233
248,249,250,251
259
260, 283, 284
285,286,287,296,304
305
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