edit1: added note about construction tech.
What is it
This is a build designed to be used in conjunction with liberty opening tree on any difficulty including deity. With this build you quickly get the National College up and running and plonk down an additional 3 cities shortly after it finishes. This way you have both the National College and 4 cities up around turn 60-70, giving you an excellent platform from which to launch any strategy you like.
There are two variations on this strategy. The standard version, and the Great Library variation. The standard version can be used on any difficulty but is better used on immortal and deity since it allows you to get out a lot of early fighting units out before expanding, and also because the AI will likely grab the GL quickly anyway. The standard version I've used on two of the deity challenges as well as other immortal and deity maps to great effect. The GL version I have used very successfully on immortal and emperor, but it can be risky on immortal as there's a chance the AI will get the GL before you. Don't even attempt the GL on deity.
Standard version
Tech Order
Pottery -> Writing -> Mining -> AH (optional) -> Calendar -> Philosophy -> Archery -> Other luxuries -> Wheel -> research construction until 1 turn to go and then tech towards civil service.
Build order
Scout -> Monument -> Granary -> Library -> Shrine -> Additional workers/warriors as needed -> NC as soon as philosophy finishes -> Settler
Buy order
Tiles around capital as needed -> Settler -> Save for archers/archer upgrades
policy order
Liberty -> Citizenship -> Republic -> Collective rule -> representation -> meritocracy
So you basically want to use this build if you've got space to expand and feel like liberty would be a good choice on the map. I would not recommend this if you're surrounded by AI or if you have a jungle start, but it should be okay for most starts. Often you'll get your first rush bought settler and/or free settler from collective rule before you finish the national college, take each settler with a warrior to where you want to settle and wait.
When you get your first worker from citizenship, start improving farms first. On this version of the build you're not getting the GL, so you need to get your population up as high as possible to help you tech philosophy as fast as you can. I would suggest getting 2-3 farms and then improving a luxury. If you don't have a mining or Calendar luxury near the capital you're out of luck, but it's not so bad, your rush bought settler will just be delayed (from not being able to sell the luxuries), so you'll get one city a bit later. Once you start the national college, chop down a forest or two (if you can) to make it build faster.
A note on religion: I never have a problem getting a pantheon, even with such a late shrine, however if you want to delay the library and/or granary a little bit to get the shrine earlier this is also perfectly acceptable, and will not delay your NC by more than 2-3 turns.
Examples
I tried this out on a few random deity and immortal games. On a good start you'll get national college and 2 additional cities up around turn 55-60, with the 4th city around turn 65. On a poor start you'll get the national college and first few expansions around 65-70 and your 4th a bit later.
Here is a very good deity start with Greece. National college finished at turn 56 with two cities ready to settle, by turn 64 I have 4 cities with luxuries improving and some good troops, I've also used the warriors to kill barbs, getting favour with a city state:
Here's an average immortal start with Siam. National college finished at turn 60, but due to slow culture have not got free settler yet, nevertheless I was still able to get my 4 cities up and running by turn 69 with a few warriors, workers and good defence on the way.
Here's an example of a bad start with Polynesia (deity); no river, few hills and lots of tundra nearby, NC doesn't finish until turn 66, but will still have 3 cities soon. In this case I'm not building my 4th settler, as there aren't enough good spots. Instead I would likely churn out archers and attempt to take the Swedish satellite city above for control of the natural wonder, cows and luxuries.
Great Library variation
Tech Order
Pottery -> Writing -> Mining -> Calendar -> (get philosophy with GL when it finishes) -> Archery -> Other luxuries -> Wheel -> tech towards civil service picking up construction when appropriate.
Build order
Scout -> Monument -> Granary -> The Great Library -> NC -> units (situational) -> Settler
Buy order
Tiles around capital as needed -> Worker -> Settler -> Archers
policy order
Liberty -> Citizenship -> Republic -> Collective rule -> representation -> meritocracy
If you think you can get the great library, then this version will give you a bigger tech advantage than the standard build, however it will leave you with a bit less defences, since your cap will be building the wonder instead of producing units. Also you won't be able to produce a second worker so you pretty much have to rush buy one, if you steal one as well this will give you 3 workers, which should be enough.
This build will get the GL usually around turn 35-40, which is pretty much when the AI usually gets it on immortal. The NC will come around turn 50-55.
If you have a good start with a few forests and an easy mining lux that you can sell off quick to buy an additional worker, then I would recommend giving it a go on immortal, because with chopping you should be able to get the GL before 35 which should be quick enough (although occasionally the AI gets it super early on immortal if it pops a tech rune with writing)
Example
Here is a game with the Celts on immortal. The starting location is okay but not brilliant; no wheat or deer, and jungle/marsh covering most luxuries. GL finishes at turn 36, a couple of turns later than I'd like but still got it. NC finishes turn 52, after the NC I produce one Pictish warrior for a bit of extra defence before cranking out a settler. By turn 66 I have my 4 cities up and running. I'm producing monuments in all 4 cities but will likely switch to archers and harass Korea with composite bows. I've rush bought a couple of archers already.
What is it
This is a build designed to be used in conjunction with liberty opening tree on any difficulty including deity. With this build you quickly get the National College up and running and plonk down an additional 3 cities shortly after it finishes. This way you have both the National College and 4 cities up around turn 60-70, giving you an excellent platform from which to launch any strategy you like.
There are two variations on this strategy. The standard version, and the Great Library variation. The standard version can be used on any difficulty but is better used on immortal and deity since it allows you to get out a lot of early fighting units out before expanding, and also because the AI will likely grab the GL quickly anyway. The standard version I've used on two of the deity challenges as well as other immortal and deity maps to great effect. The GL version I have used very successfully on immortal and emperor, but it can be risky on immortal as there's a chance the AI will get the GL before you. Don't even attempt the GL on deity.
Standard version
Tech Order
Pottery -> Writing -> Mining -> AH (optional) -> Calendar -> Philosophy -> Archery -> Other luxuries -> Wheel -> research construction until 1 turn to go and then tech towards civil service.
Build order
Scout -> Monument -> Granary -> Library -> Shrine -> Additional workers/warriors as needed -> NC as soon as philosophy finishes -> Settler
Buy order
Tiles around capital as needed -> Settler -> Save for archers/archer upgrades
policy order
Liberty -> Citizenship -> Republic -> Collective rule -> representation -> meritocracy
So you basically want to use this build if you've got space to expand and feel like liberty would be a good choice on the map. I would not recommend this if you're surrounded by AI or if you have a jungle start, but it should be okay for most starts. Often you'll get your first rush bought settler and/or free settler from collective rule before you finish the national college, take each settler with a warrior to where you want to settle and wait.
When you get your first worker from citizenship, start improving farms first. On this version of the build you're not getting the GL, so you need to get your population up as high as possible to help you tech philosophy as fast as you can. I would suggest getting 2-3 farms and then improving a luxury. If you don't have a mining or Calendar luxury near the capital you're out of luck, but it's not so bad, your rush bought settler will just be delayed (from not being able to sell the luxuries), so you'll get one city a bit later. Once you start the national college, chop down a forest or two (if you can) to make it build faster.
A note on religion: I never have a problem getting a pantheon, even with such a late shrine, however if you want to delay the library and/or granary a little bit to get the shrine earlier this is also perfectly acceptable, and will not delay your NC by more than 2-3 turns.
Examples
I tried this out on a few random deity and immortal games. On a good start you'll get national college and 2 additional cities up around turn 55-60, with the 4th city around turn 65. On a poor start you'll get the national college and first few expansions around 65-70 and your 4th a bit later.
Here is a very good deity start with Greece. National college finished at turn 56 with two cities ready to settle, by turn 64 I have 4 cities with luxuries improving and some good troops, I've also used the warriors to kill barbs, getting favour with a city state:
Spoiler :
Here's an average immortal start with Siam. National college finished at turn 60, but due to slow culture have not got free settler yet, nevertheless I was still able to get my 4 cities up and running by turn 69 with a few warriors, workers and good defence on the way.
Spoiler :
Here's an example of a bad start with Polynesia (deity); no river, few hills and lots of tundra nearby, NC doesn't finish until turn 66, but will still have 3 cities soon. In this case I'm not building my 4th settler, as there aren't enough good spots. Instead I would likely churn out archers and attempt to take the Swedish satellite city above for control of the natural wonder, cows and luxuries.
Spoiler :
Great Library variation
Tech Order
Pottery -> Writing -> Mining -> Calendar -> (get philosophy with GL when it finishes) -> Archery -> Other luxuries -> Wheel -> tech towards civil service picking up construction when appropriate.
Build order
Scout -> Monument -> Granary -> The Great Library -> NC -> units (situational) -> Settler
Buy order
Tiles around capital as needed -> Worker -> Settler -> Archers
policy order
Liberty -> Citizenship -> Republic -> Collective rule -> representation -> meritocracy
If you think you can get the great library, then this version will give you a bigger tech advantage than the standard build, however it will leave you with a bit less defences, since your cap will be building the wonder instead of producing units. Also you won't be able to produce a second worker so you pretty much have to rush buy one, if you steal one as well this will give you 3 workers, which should be enough.
This build will get the GL usually around turn 35-40, which is pretty much when the AI usually gets it on immortal. The NC will come around turn 50-55.
If you have a good start with a few forests and an easy mining lux that you can sell off quick to buy an additional worker, then I would recommend giving it a go on immortal, because with chopping you should be able to get the GL before 35 which should be quick enough (although occasionally the AI gets it super early on immortal if it pops a tech rune with writing)
Example
Here is a game with the Celts on immortal. The starting location is okay but not brilliant; no wheat or deer, and jungle/marsh covering most luxuries. GL finishes at turn 36, a couple of turns later than I'd like but still got it. NC finishes turn 52, after the NC I produce one Pictish warrior for a bit of extra defence before cranking out a settler. By turn 66 I have my 4 cities up and running. I'm producing monuments in all 4 cities but will likely switch to archers and harass Korea with composite bows. I've rush bought a couple of archers already.
Spoiler :