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Zerg Basics
Zerg are a very good and versatile race. Playing with them requires more attention unlike Terrans and Protoss. Probably one of their biggest disadvantages is inability to queue units. This forces you to either be a master of macromanagment or to build many hatcheries, spending both time and resources. Their other bigger weakness is that overlords, both control, detector and transporter unit are weak and can be easily killed - two scourges, one irradiate, few shots from any flying unit is all it can take. Nevertheless, with good build orders a considerably bad player can win a game against other medium/good player by simply building a overwhelmingly large amount of units. Remember one thing - a good Zerg player is the one that doesn't rush for victory.

Zerg Build-UP
Zergs have millions of build orders, most of them used for powerful rushes, others for fast expansions. Most of the players on Battle.NET became famous just because of their good rushes. This is kind of a economical build order.

1. Build up your workers until you reach 8 control, build overlord and 9th drone,
2. Build extractor and 10th drone, cancel extractor (a.k.a. extractor trick),
3. Keep cranking drones until you reach 12 control,
4. With 11th drone Build spawning pool,
5. 12th (not 12 control) builds the extractor,
6. Keep pumping drones as you get larva,
7. When the extractor is finished, pull 3 drones from the minerals on it,
8. Build drones until you get 300 minerals for the 2nd hatchery (expand if you like),
9. Upgrade 1st hatchery to lair. Build 4 more drones and 4 sunken colonies,
10. When the 2nd hatchery pops up, build an overlord.

Now you can choose what to do next. A good way to continue would be to morph a hydralisk den and few hydralisks to strengthen your defences. If you wish to go for mutalisks, build an expansion as soon as you can as you will need much gas.

The 2 Minute Zergling Rush
A bit overpowered (it can be defended by properly using workers), the 4pool rush is the fastest rush in the game. Use it when you know exactly where your opponent is or if he's close to you.

1. Put all of your drones on minerals (don't build 5th drone as yet),
2. As soon as you gather 150 minerals, build a spawning pool,
3. Build a drone (!),
4. After the pool is finished use all three larvae to morph zerglings,
5. Rally zerglings to your enemy's base and hope for a win.

6pool Zergling Rush
Somewhat slower, and considerably more economic, the 6pool rush is still very good if used against non-zerg player.

1. Send all of your drones to minerals,
2. Build 5th and 6th drone and send them to minerals,
3. After 6th drone has gathered minerals once, use it to build spawning pool,
4. Build 7th drone, overlord and 8th drone,
5. Use 8th drone to scout out for your opponent, or put it on minerals if you know where he is,
6 When the spawning pool is finished, build 6 zerglings and rally them to your opponent base.

Keep producing zerglings until you have build 12 of them, build an expansion.

3 Hatcheries Strat
One of the best ways to start out a game would be with 3 hatchs. But, if you think that you'll be rushed early in game (about 4-5 minutes) then this may not be very good start.

1. Keep building drones until 9 control,
2. Build overlord, extractor, 10th drone and cancel extractor,
3. With 10th drone build 2nd hatchery (if you want, you can expand right away),
4. Keep cranking drones until 14 control, and then build the 3rd hatchery,
5. Build a drone and a spawning pool.

From this point you can mass zerglings (even as much as needed for defending the zealot rush) or get extractor and develop.