Science

Science is a Fallout, Fallout 2, Fallout 3, Fallout Tactics, Fallout: New Vegas, Van Buren, and J.E. Sawyer's Fallout Role-Playing Game skill.

Fallout, Fallout 2, and Fallout Tactics

 * $$\text{Initial level}\%=25+(2\times\text{Intelligence})$$

Example: A starting Intelligence of 5.
 * $$25+(2\times5)=35\%$$

The Fallout Tactics 1.27 patch notes readme file says:

suggesting that Science somehow affects critical chance or effectiveness against robots (at least in Tactics).

Fallout 3

 * $$\begin{matrix}\text{Initial}\\\text{level}\end{matrix}=2+(2\times\text{Intelligence})+\left\lceil\frac{\text{Luck}}{2}\right\rceil$$

Example: A starting Intelligence of 5 and Luck of 5.
 * $$2+(2\times5)+\left\lceil\frac{5}{2}\right\rceil=15$$

Science primarily allows you to hack terminals using a mini-game. Certain terminals have different difficulty levels, and different Science skill levels are required to hack them. The difficulty ratings and skill levels are: very easy (15), easy (25), average (50), hard (75), and very hard (100). Your Science skill relative to the difficulty rating affects how hard the minigame is (at Science 100, average terminals are a breeze, presenting only about 5 possible passwords). Harder terminals have more words, longer words, and fewer brackets.

Ways to increase Science

 * Permanent
 * Leveling up (10+INT (+3 with Educated perk))
 * Bobblehead - Science (+10)
 * Big Book of Science (+1 or +2 with Comprehension)
 * Daddy's Boy/Girl (+5 for each rank, max 3 ranks)
 * Tag! (+15)
 * Survival Expert (+2, 4 or 6)
 * Covert Ops (+3) (Operation: Anchorage)
 * Temporary
 * Mentats (+2-10 depending on Intelligence, a perfect Intelligence of 10 will see no benefits)
 * Enclave scientist outfit (+5)
 * Lesko's lab coat (+10)
 * Scientist outfit (+5)
 * The Surgeon's lab coat (+5)
 * Vault lab uniform (+5)
 * Lab coat (+5) (Broken Steel)

Interactions that require Science skill

 * 50 Science to hack the RobCo production facility mainframe during the quest Wasteland Survival Guide.
 * 50 Science to reactivate the robots during the quest Big Trouble in Big Town.
 * 60 Science to unlock certain dialog options with Doctor Lesko during the quest Those!. This allows you to learn he had been using a modified form of Forced Evolutionary Virus to induce pyrosis reactions in giant ants, creating fire ants.
 * 75 Science to modify healing archways to fully heal the player, though this reduces their number of uses to three. (Mothership Zeta)
 * 75 Science to hack the Mobile Base Crawlers' mainframe. (Broken Steel)
 * 80 Science to deactivate the repulsion fields aboard the Mobile Base Crawler. (Broken Steel)
 * With a high Science skill you can bypass the speech check required to convince John Henry Eden to self destruct during the quest The American Dream.
 * Although a high Science skill increases your chances of disarming a slave collar, the Explosives skill is also taken into account. Even when both skills are high, the slave could still blow up, so saving before attempting to disarm a collar is advised.

Fallout: New Vegas

 * $$\begin{matrix}\text{Initial}\\\text{level}\end{matrix}=2+(2\times\text{Intelligence})+\left\lceil\frac{\text{Luck}}{2}\right\rceil$$

Example: A starting Intelligence of 5 and Luck of 5.
 * $$2+(2\times5)+\left\lceil\frac{5}{2}\right\rceil=15$$

Like in Fallout 3, your Science skill relative to the difficulty rating affects how hard the mini game is. Notably, at Science 100 even Hard terminals will have very few words to choose from. This can possibly be as low as 5 possible passwords which, combined with the Computer Whiz perk, means an 80% chance of getting most terminals off the bat without even having to look at the words.

Unlike Fallout 3, your Science is used for more than a few random dialogue checks and hacking. Many crafting recipes (primarily at the Workbench, but also at Campfires) rely on Science, specifically ones involving Energy Weapons ammunition and chem crafting.

Ways to increase Science

 * Permanent
 * Leveling up (up to +15 or +17 with Educated)
 * Big Book of Science (+3 or +4 with Comprehension)
 * Logic Co-Processor (+2 depending on Intelligence, a perfect Intelligence of 10 will see no benefit.)
 * Good Natured (+5)
 * Skilled (+5)
 * Temporary
 * Programmer's Digest (+10 or +20 with Comprehension)
 * Night Person (+2-4 at night depending on Intelligence, a perfect Intelligence of 10 will see no benefit.)
 * Mentats (+2-4 depending on Intelligence, a perfect Intelligence of 10 will see no benefit.)
 * Party Time Mentats (+2-4 depending on Intelligence, a perfect Intelligence of 10 will see no benefit.)
 * All-purpose science suit (+5)
 * Scientist outfit (+5)
 * Followers lab coat (+10)

Interactions that require Science skill

 * 25 Science to use the chemistry set in Doc Mitchell's House.
 * 25 Science to discover that Victor has been overridden when you asked him to help fighting the Powder Gangers in Goodsprings.
 * 30 Science to reprogram Primm Slim to become the Sheriff of Primm.
 * 45 Science to obtain the rebreather without finding a pressure cooker for Jack in Volare!.
 * 50 Science to hack Daniel Contreras' terminal and download the weapons manifest to get the weapon This Machine.
 * 50 Science to figure out who hacked Primm Slim in Primm.
 * 50 Science to teach Jack how to make Party Time Mentats, Slasher, Rocket and Super stimpak in Aba Daba Honeymoon.
 * 50 Science to get Loyal to give you the sonic emitter in Ant Misbehavin'
 * 50 Science to suggest and alternative to Fixer for Jacob Hoff.
 * 55 Science to upgrade the rocket trajectory at the end of the quest Come Fly With Me.
 * 55 Science to run the bypass on ED-E, thus lowering the Repair requirement to 35 (instead of 65). However on the Xbox 360 (no patches) this is reversed, 55 Repair lowered to 35, requiring 65 Science.
 * 60 Science to repair Rhonda in Black Mountain.
 * 60 Science to reprogram Fisto for Wang Dang Atomic Tango without a program disc.
 * 60 Science to persuade Clanden to make a chlorine bomb in How Little We Know.
 * 60 Science to persuade Ralph to give you his chlorine for free in How Little We Know.
 * 70 Science to convince Keely that the player must return the data of Vault 22 to Thomas Hildern in the quest There Stands the Grass.
 * 80 Science to bypass the credit check to enter The Strip.
 * 80 Science to convince Scribe Ibsen to delete the Brotherhood of Steel's files on the computers at Hidden Valley.
 * 90 Science to convince Henry to test the Stealth Boys on nightstalker brains in Guess Who I Saw Today.

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