Fallout 3 skills

Background
As in the previous games, three Tag Skills out of thirteen can be chosen as the player character's specialties. Unlike previous games, tagging a skill only grants the character a 15-point bonus to that skill. Five skills are removed 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 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 stat of 10, up to 23 skill points can be distributed 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 the Speech skill.
 * Certain perks offer a skill increase.

Maximizing skills

 * In order to reach the maximum number of points available in all skills, one only needs 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 give 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.
 * 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 one may want to raise some skills to (theoretically) 102 or 104.
 * Leaving Vault 101 one 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 one is forced to choose between action points and hit points, and one will have a ridiciously small weight capacity. (One 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 will 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 will respawn in 73 hours with another copy of the book in his inventory.

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