Here's my report for the T-hawk 20k challenge game.
As noted in T-hawk's post, the key to speed up a 20k victory are great leaders to rush wonders and cultural improvements. To get many great leaders, war has to be waged, and since wonders and cultural buildings will have to be built nearly the whole game, near constant war is required.
This being an emperor game on a tiny map with only three opponents, there's a catch, though: You have to be careful not to cripple or eliminate all your opponents. Generating leaders requires your elite units killing enemy units, so you have to make sure that there are enough enemy units to kill until the end of the game.
So the plan was to expand only slowly and to do oscillating wars, mainly fishing for leaders against attacking units and only to capture cities with key resources or luxuries. So here we go...
I founded my capital one tile SW from the starting spot as T-hawk suggested, and indeed it was an amazing site! Two cows, a game tile, three bonus grass, fresh water and coastal access...wow.
The first thing I built was a temple, to get culture going as fast as possible, and helped it by chopping the game forest after irrigating the cow. With this high amount of food, I decided not to build an early granary but instead built two warriors next, then a settler, a second worker, another warrior and then started to build the colossus in 2670BC. I somehow thought it would give the same amount of culture the oracle gives, don't know why...
Anyway, I met France during the same year. I had started to research warrior code, but now traded away the wheel for warrior code and some chump change because I wanted to get to horseback riding as fast as possible to start my first war. Osaka was founded then, too - I wasn't satisfied with its location, but failed to come up with a better spot (see next screenshot).
In 2630BC, India found me, and England was met in 2510BC. Lots of trading occured, and I finally got pottery in 1870BC - I don't know if I ever got it later than that.
Osaka had been building a warrior and a worker, then barracks and was starting to build my first horseman in 1830BC. Then, Kyoto finished the colossus in 1600BC.
It immediately started to build the pyramids next, but since London was building them already, it was more a prebuild for anything else I could get. I was researching literature at max rate now. Osaka built a settler, and Tokyo was founded north of Kyoto in 1350BC to get the gems and a higher unit support.
In 1275BC, literature came in and I started the republic at minimum science. Although the AIs had literature now as well and the cascade wasn't broken yet, I managed to get the great library in 850BC! One turn before, I decided I could really need a leader now for my next project, and declared war on India although I only had two horsemen and one spear. Another motive was to pillage their only iron:
From that iron hill, I was able to kill several warriors and got my first elite units. Before I got any leaders though, England finished the Pyramids, France the oracle and India the great wall.
Unfortunately I was so occupied with the war that I totally forgot to build a library! I remembered it as late as 270BC and lost valuable culture there. Anyway, in 750BC republic popped out of the library and I immediately revolted. I killed 8 archers, two warriors and some horsemen and a second horse promoted to elite, but I was already in the middle ages before I scored my first great leader in 370BC!
Most conveniently the colosseum was finished next turn in Kyoto, and the GL rushed Sun Tzu's. Golden age! After finally building a library and then a cathedral, I started to build the hanging gardens in 210BC. I also decided to capture Lahore for its luxury, and founded Edo on the iron near Delhi in 10AD, intending to raze the Indian capital next.
Unfortunately, France decided to declare war on India too, which I didn't like at all - the Indians shouldn't be crippled too much! But I couldn't stop now and had to raze Delhi to protect Edo. I killed five spears in 70AD, which wasn't enough, but then razed the city in 90AD. I hadn't scored another leader yet, though.
France knew education already, so I started max research on chivalry. In 170AD, I scored a second leader which I kept for some turns. Kyoto was swapped from hanging gardens to Sistine's, finished it in 230AD, then the leader rushed the hanging gardens. I could have also had Leo's since invention had come out of the library, but the gardens were worth more culture.
During the same turn the leader had appeared, England had threatened me and I was at war with them now. That was good, because India was gassed and I needed new units for fishing. It took some time before the first English units appeared, though.
Education came in 270AD, and the turn thereafter I had enough money to rush a university in Kyoto. Then I started Leo's, which was still available, as a prebuild or to rush it in case another leader appeared. In 400AD, I decided to give India some rest and made peace for gunpowder and 16gpt. See you again in 20 turns, Gandhi! And since England's offense left a lot to be desired, I decided to declare war on France next.
I had no intentions of attacking Joan's cities, though. Unfortunately, England completed Leo's in 460AD before I got another leader, so I made peace with Lizzy for a discount on music theory and switched Kyoto to Bach's. But the leader draught continued, and England completed Bach's too in 550AD, so I made peace with France for astronomy and some money and completed Copernicus' the next turn.
Hm, so now no wonders were available and all the warring had slowed down the tech pace, so I started to research on my own again. I also was stuck at peace for several turns...or so I thought. But during the next interturn, Lizzy threatened me and declared war already. Thanks!
And in 600AD, the peace deal with Gandhi expired and war broke out again, too. I decided that India was only making tech buys more expensive and I could kill one civ, and so started to raze Gandhi's cities, but I still failed to get another leader. France actually managed to get Magellan's, damn!
Then, finally, in 750AD I got my third leader and made peace with England to get free artistry and rushed Shakespear's. I also declared war on France again, but was careful not to let a musketeer win any fight - I always redlined them with my catapults before attacking them. And the reward for this second war against France was another leader in 850AD, who rushed Smith's.
But in 900AD, the first French cavalry units appeared and slaughtered some of my poor samurai, so it was time for peace again, yielding me physics and metallurgy. I made peace with Gandhi too, who was still alive, for military tradition and navigation. Unfortunately, France started to build Newton's only a few turns later and was the only one with the required tech! I switched Kyoto to the most expensive building available (a courthouse) and slowed down production, giving me two turns time to get ToG. Next turn, England had it too, and most conveniently, our peace deal was about to expire...so I renegotiate and got the tech for 19gpt.
I switched Kyoto to Newton's, and it would be finished in 16 turns. Hm...I investigated Paris, and...it needed 16 turns as well! Oy, that was close. Ties go to the player, right? I didn't want to rely on that and mined an irrigated plains tile for more production at a food deficit, but that didn't help, so all I could do was wait and hope...
But before that was decided, England offered me magnetism for 530 gold, which I accepted, and then sneak attacked me on the same turn. Uh...yeah. They attacked with three cav and one knight, killing a cav of mine. That war caught me at a very bad time - my capital was building a wonder, and my other cities were in infra mode, building banks - and with cavs against muskets, losses are very high. England had also an MPP with France, so I had to be very careful here. But after my cities had finished banks and could produce units again, and by using the terrain to my advantage, the war went okay - but without getting me any leaders.
In 1060AD, France declared war on poor India. And four turns later, I finished Newton's - yes, I made it!!! On the same turn, France extends peace, which I had overlooked but which was okay, and India left the game. Steam power came in in 1150AD, but I had no coal - or so I thought. Later I saw that coal was in one of the two jungle tiles not yet improved south of Lahore, which I had overlooked completely. (Stupid me!) So I traded away my only gems and 100 gold to France for their wines and coal, which was a rather odd deal...
In 1250AD, I finally scored another leader, who rushed Universal Suffrage which France had been building with a headstart for some time already. Maybe I should have expanded a bit more, because I was behind in tech now. 1295AD got me another leader, which I saved for now, and peace was made again with England for some money.
Then, in 1350AD, waiting for scientific methods in six (and having skipped nationalism), this:
(Note the coal one tile south of Lahore, which I still hadn't noticed...)
Again, that caught me unprepared since I wanted to build hospitals. I had 10 cannons, 7 cav, and several MDIs, some of them still regular (ex-MP-warriors from the ancient age). I dialed up France and demanded their units should leave, and Joan of course declared war. I managed to kill 6 cavs and wound one, but lost 2 cavs and 4 MDI myself. In the counterattack, I lost an elite musket and killed another cav.
Next turn, France bought England into the war as well! I had heavy losses during the next turns, and when my leader rushed ToE in 1395AD, I decided to take atomic theory and then replacable parts instead of the usual electronics. I couldn't build Hoover in Kyoto anyway and needed better defenders than muskets!
Well, I had no rubber. At least I found my source of coal now while looking for it.
So research was set on nationalism next, for better defenders and because I wanted to build the CIA. But in 1420AD the first French infantry appeared, so I decided to make peace now. I gave electricity inthe peace deal to England to get nationalism, and from France I managed to get espionage and 200 gold. I was grateful this was civ3 and not GalCiv, otherwise France would have realized that they were much stronger than me and I would have had to pay them heavily...
Kyoto finished the CIA in 1450AD and was generating 105 culture per turn now. Battlefield medicine was added next. France actually managed to beat me to the Hoover Dam despite me prebuilding it, probably with a great leader (Joan had declared on England earlier), but it didn't matter for the 20k outcome anyway. In 1560AD wall street saw the light of day, and in 1595AD I discovered motorized transportation. France didn't have oil, so I sold them motorized for flight, rubber and lots of cash, and built several tanks, planning to attack them soon. That war should serve two purposes, getting me my own source of rubber and generating leaders for SETI and the UN.
Bangalore had wines and spices, and I squinted and somehow believed the rubber tile would be in reach, too - I don't know what was wrong with my eyes this game! I had declared war the turn before and now captured the city, giving me happiness but no rubber. And since the defending infantry had caused heavy casualties on my side, I needed some turns to defend against counterattacks and regroup.
While doing this, at least my second objective was met: I scored another leader to rush SETI! And in 1680AD I captured Rouen, generating another leader who finally formed an army so that Kyoto could build the heroic epic and the military academy.
If I remember correctly, Kyoto was generating 122cpt now. In 1762AD, I noticed I would win next turn and so wanted to have a last turn of fun: I threw all I had against France and razed Paris. Then:
Kyoto would have finished the UN two turns later.
So that's my try for the fastest 20k victory: 1764AD. Due to self-restricting myself in the beginning (delibarete slow expansion; fighting only defensively to keep my enemies strong), it had been a surprisingly hard game in the end. Okaý, not really hard - I was never really in danger. But I had lived through some interesting times when France attacked me with lots of cavalry, while I only had muskets and was in full infrastructure mode.
I think my biggest mistake this game was not to do more research of my own, which could have given me the techs needed for more cultural buildings earlier. That, and forgetting to build a library for so long.