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Vault City is an anomoly in the wastes. It's one of the few places to seemingly have it made, but requires citizenship or a day pass to even enter. However, you can enter the courtyard area where the "plebs" stay at no cost. It's in the courtyard where your first quests will be.
When you're ready to enter the city proper, speak with Wallace at the customs office in the north end of the courtyard, and you'll find there are several ways to get in. The main way is with a Day Pass. You can talk your way into one, wear just your vault suit and speak with him, wait until night and steal one from the locker, and several other ways. You can also get false citizenship papers from Skeev nearby, but be careful flaunting them around the city... however the amenities office merchant won't mind, which is a good thing. The only true way to be at ease is to become a citizen, which if you follow the quests below, will be no problem.
You can pick up the quite good NPC Cassidy from the bar, and you can fix up Charlie who's sick in a tent with some Rad-Away for a bit of xp.
Did you keep Vic's Water Flask like I recommended in Arroyo? It can come in handy here with Ed (in a tent on the west of the main path in the courtyard) for some xp and new locations on your world map.
In the council area, and insice the council building on the north end, you'll find Gregory, McClure and Lynette. Gregory can give you a citizenship test but you'll need Perception, Intelligence and Luck to be at 9... which for most people will disqualify them. You'll get xp if you pass. See McClure for almost everything else (except when the quests below call for it), including after you learn of the Jet problem in Redding. When you find a cure for Jet, you can go back to him for 2,500xp.
In the council area, and when you're a citizen, you'll have access to the Vault.
On level 1 of the vault, if you're male, you'll have xp opportunities to court Phyllis. If your Doctor skill is at 75% or higher, you can talk with the doctor in the room behind Phyllis and receive a couple of perks, one right away and another if you're treated for radiation or poisoning. As a bonus for such a high Doctor skill, you can also search the archives in the nearby computer using Science for information on combat implants, which you can then go back to the courtyard and ask Dr. Andrew about performing the surgery. Remember to fix the auto-doc in the back room where Andrew is first or the implants won't work. You may simply wish to skip them as 1, they're really expensive and 2, can cost a ton of money and lower your charisma at the same time, but do give you the benefit of damage resistance. It may be worth talking to him at least if you're curious.
On level 2 of the vault, if your Perception is fairly good, you'll notice a rattling vent on the northeast end of the level, which you can then fix for some energy ammo and xp. You'll find a Computer Voice Module which you can use at a later time as well on this level. To get in the locked doors you'll need to increase your strength to 8 or better. Use buffout if you need to, as behind the doors you'll find some nice ammo and a Red Memory Module which you can later use to surgically enhance your Strength by 1.
On level 3 of the vault, use Science on the southernmost computer in the west room for xp. You can also access the central computer in the southeast end which will give you the location to Vault 15. If your Perception is high (7 or greater), you'll notice a slot you can plug your Pipboy into. Do so and you'll add Broken Hills, Redding, New Reno, New California and Vault 15 to your map, if you do not yet have them.
Quests & Opportunities
Here is a list of some of the quests and other opportunities in Vault City, each with a brief synopsis containing at least one way to solve it. The remaining ways, and possibly other minor quests I will leave up to you.
1. Deliver Moore's briefcase to Mr. Bishop in New Reno:
In the council area, look for a citizen preaching by the name of Moore. If you agree with him for a bit while in dialogue, he'll give you a briefcase to deliver to Mr. Bishop, who can be found in the Shark Club in New Reno. Do so to get on Bishop's good list and for a bit of xp and cash.
2. Get a plow for Mr. Smith:
In the courtyard area, you'll find Mr. Smith outside his tent. He needs a plow for his family. You can pick one up at the shop nearby on the west end of the area, but it'll cost you $800 or slightly cheaper. No matter, it's for a good cause, xp and a Desert Eagle .44 with ammo.
3. Solve the Gecko powerplant problem:
In the council building, talk with first citizen Lynette, and be polite with her, at least until you're a citizen. You'll understand why I said that when you meet her. Talk to her about citizenship until she mentions the special favor having to do with the Gecko powerplant. When you're ready to leave Vault City, refer to Gecko quest #1 to finish this up. You'll probably not want to return to Lynette again about the powerplant problem and instead pay McClure a visit after you solve the problem, as he'll make you a citizen.
4. Rescue Amanda's husband, Joshua:
In the courtyard, locate Amanda in one of the tents. She'll mention her husband, who's being held captive in the Servant Allocation Center in Vault City. Head back up inside the city and speak with Barkus inside the center. Alternatives for completing this quest include talking him into believing Joshua has a contagious disease with a successful Speech check, or with a bribe.
5. Deliver beer and booze (10 each) to Lydia:
Lydia runs the local bar just inside the city, downtown area. Once you have your citizenship (as they'll check you at the gate), deliver 10 of any real booze or beer for payment.
6. Deliver pliers (tools) and a wrench to Valerie:
Valerie is in the maintenance center near the Tap House downtown (just inside the city). She's Vic's daughter and can upgrade your weapons as well. First you'll need to bring her a set of pliers and a wrench. In the courtyard, you'll find Curtis the little boy who's usually just south of the bar running around. He's looking for his Mr. Nixon doll. Head to the north of the bar and you'll find his doll against the wall. Bring it back to Curtis and listen to what he says about the wrench being near some rocks. Just to the east of the bar is a pile of rocks. Search there for a wrench, and take that and a tool (the normal kind used for repair) to Val. She'll give you the Super Repair Kit, needed for a quest in Gecko (so keep it), and offer to upgrade some of your weapons.
7. Scout the eight sectors around Gecko and return to Stark:
In the Corrections Center within the city, find Stark who will give you this quest. Simply run around the eight grid squares around Gecko to the northeast, and return for your reward.
8. Enter NCR and return to Stark:
After quest #7, Stark will want you to scout out New California Republic and return, and he'll even add it to your map. Return when you've done just that for some cash, xp and a motion sensor.
9. Deliver Lynette's holodisk to Westin in NCR:
You may not complete this one for awhile. First you'll need to defeat the Raiders, an optional area, and also take Mr. Bishop's holodisk from his New Reno safe (see Raiders and New Reno for details). After these two things have been accomplished, come back and speak with Lynette and you'll receive another quest. Now it's simple. Find Westin in NCR and deliver the holodisk for decent xp.
10. Deliver a sample of jet to Dr. Troy:
On the vault's first level, speak with Dr. Troy with at least 75% Doctor skill. If you're not up to the task, you'll need to come back later (or never). Take a tour of the facilities with him, convincing him that you're also a doctor, and as he ends the conversation he'll ask you to smuggle some jet to him. Do so for some cash. This is simply an optional part of the more major quest to solve the jet problem, so you'll want to refer to quest #14 in New Reno for that.