A Meeting of the Minds

 is a Fallout 3 quest and achievement that is added to the game with the Point Lookout add-on.

Ending the feud
After telling Desmond that you are ready to go, you will both make your way to the Lighthouse. Once you make your way to the base of the Lighthouse, enter it. When you enter, a section of the Lighthouse floor slides away and reveals a door to what is called the “Underground Lab”. In the lab, you will make your way to a room with a closed metal door. Destroy or disable the turrets that begin to attack you and make your way through the hallway behind the closed door. As you make your way through the hallways, look in the rooms and get supplies that are in the ammo boxes.

Who to kill?
When you find the brain you must talk to him and decide what you’re going to do. If you decide to kill Desmond, it is an easy fight for a high-level character. Make your way through the complex and talk to Professor Calvert, who will then reward you... with death. Fight him by shooting his glass tank, and eventually it will shatter. The Protectrons he sends after you should be a minor hindrance at best. Only a few of the Protectrons in the room will attack you. If you choose to kill Calvert, then the battle will unfold in the same way.

Once the fighting is finished, go to the door in the information room, and inside you will find a workbench covered with lots of Energy ammo, a Microwave Emitter, two ammo boxes, and a safe with some money and other random stuff in it.

Notes

If you choose to kill the Brain but forgot to shoot the Brain after breaking the glass a message will appear stating that "The Brain is unconscious" (Confirmed on both Xbox 360 and PC)

Killing Desmond
If you decide to kill Desmond, the Professor will reward you with the best thing any human could ever hope for... death. Or at least he'll try, at which point he'll send the robots after you. Destroy them and loot Desmond's body to find his unique glasses (+5 to small guns and explosives).

Killing Calvert
If you don't want to kill Desmond, shoot at the brain until the glass shatters and the water completely drains, then shoot the brain itself. After the brain is dead talk to Desmond, he will give you a key which unlocks the nearby door which requires a key. Inside this door is a cache of energy weapon ammo and weapon components, as well as the Microwave Emitter. And if you feel like being dirty and don't really care about Desmond, you can kill him afterwards and claim his Eyeglasses. He also carries a Combat Shotgun and a Sniper Rifle. Killing Calvert does not make any of the tribals at the Cathedral hostile to you, but if you chose to side with Desmond in Thought Control the unnamed tribe members will be hostile already.

Note: you can also get Desmond's Eyeglasses by using the theft trick, just plant a helmet on him that can't go with glasses, (ie.power helmet) exit the mansion, save the game then load the game, re-enter the mansion, and Desmond should be wearing the helmet. then pickpocket it from him

Bugs

 * In some cases if you choose to kill Calvert the Protectrons don't attack. (Confirmed on xbox 360)
 * After killing Desmond or Prof. Calvert, in the room where you need the alpha security badge, the game thinks that the door where you came in (from the outside) is closed and there is an invisible barrier that you cannot pass. (in XBOX 360). This is not a problem since you can leave the Lighthouse through a ladder in the reward room with the Microwave Emitter.
 * After siding with Calvert originally (crushing the Cogwave Jammer), and then killing Desmond under the lighthouse, instead of seeing the large glass tube containing "The Brain", there was no glass tube at all, but a Vault-tec bobblehead floating at the top of where the tube should have been. I moved to activate the bobblehead and it was named "The Brain"; upon activating, an empty dialog box opened for a moment, and then disappeared. Nothing further happened, and the reward vault door was standing open. (XBOX 360)
 * This may or may not be a glitch but after beating the main quest and siding with Desmond he gives you the vault key for the door but, when you walk up to the door it does not say "Requires Key" it simply says "Open Door" so you don't really need the key. Besides it becomes unlocked after you kill The Brain anyway so...