Fallout 3 skills

Background
As in the previous games, the player chooses three Tag Skills out of thirteen to be the character's specialties. Unlike previous games, tagging a skill only grants the player a 15 point bonus to that skill. Five skills have been cut from the game in comparison to Fallout and Fallout 2.

Formula
The initial value of each skill is based on a character's value in the relevant attribute plus a bonus determined by their Luck attribute, rounded up.
 * $$2 + (\mbox{Stat} \times 2) + \left\lceil\frac{\mbox{Luck}}{2}\right\rceil$$

Example: A starting Endurance of 5 and a starting Luck of 5 will give you an initial Unarmed skill of 15.
 * $$2 + (5 \times 2) + \left\lceil\frac{5}{2}\right\rceil = 15$$

Initial values also function as base value, meaning any later change to the SPECIAL stat would also have similar change to the respective skill.

At level up, the character will distribute 10 (13 with the perk Educated) skill points plus a number equal to their Intelligence. With a IN of 10, you can distribute up to 23 skill points when leveling up.

Temporarily improving skills

 * By wearing certain armor and clothing that offer skill modifiers.
 * By using chems that alter their base stats, thus increasing the related skills.
 * Carrying the Lucky 8-ball from Timebomb. The Luck increase raises all skills slightly, but is lost if the item is removed from inventory.

Permanently improving skills

 * Reading a skill book increases the relevant skill by one point, or two if the character has the Comprehension perk.
 * The Intense Training perk will raise the skills governed by the stat chosen (or all for Luck), as will the Ant Sight(Perception) or Ant Might(Strength) quest perks.
 * By increasing a base stat by picking up a stat Bobblehead.
 * Picking up a skill Bobblehead will increase the corresponding skill by ten points.
 * Receiving Yew's Bear Charm will give you a permanent ten point increase to your Speech skill.
 * Certain perks offer a skill increase.

Maximizing skills

 * In order to reach the maximum number of points available in all skills, you only need to train all skills to 40. This is because most skills have 25 skill books, which will give 50 points in all with the Comprehension perk, and then ten more points can be acquired through a bobblehead. Additionally, a bug with two skill books gives the player an opportunity to max the two related skills without ever putting points into it; see Glitches/Exploitable Bugs for details. One other caveat—since the Yew's Bear Charm acts like a second Speech bobblehead, one only needs to increase Speech to 30.
 * A brief explanation on how to raise all skills to level 100 can be found here, courtesy of Mr. Sark and G4TV.com (http://g4tv.com/videos/36608/Mr-Sark-Fallout-3-Leveling-Tips/).
 * It isn't possible to raise a skill above one hundred points, but the game keeps track of increases past 100 just in case a skill is lowered (By something such as radiation poisoning or power armor), so you may want to raise some skills to (theoretically) 102 or 104.
 * Leaving Vault 101 you can have a maximum of 308 skill points (including Medicine Bobblehead), with a S.P.E.C.I.A.L. of 1, 10, 1, 1, 10, 1, 9, with another 7 split between Agility and Endurance. It's not a practical loadout however, as you are forced to choose between action points and hit points, and you will have a ridiculosly small weight capacity. (You may want to keep a maximum of each stat at 9, so that the S.P.E.C.I.A.L. bobbleheads are not wasted)

Exploits
It is possible to never have to put points into certain skills to max them out. In particular, certain skill books are on respawning non-player characters - merely waiting 73 hours after killing will result in a new copy being created.
 * Big Guns - U.S. Army: 30 Handy Flamethrower Recipes: It is possible to obtain infinite copies of this book in Bethesda Offices East. A raider with a flamer normally on the top floor has this book in his inventory. When killed he'll respawn in 73 hours with another copy of the book in his inventory.
 * Science - Big Book of Science: It is possible to obtain infinite copies of this book in Shalebridge Tunnels. An Ant Researcher at this location has the book in his inventory. When killed he'll respawn in 73 hours with another copy of the book in his inventory. (This method is very glitchy and can end making you never able to enter Shalebridge Tunnels again.)

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