Magic resistance

' (or ') is a stat that all units have, including minions, monsters, and buildings. Increasing reduces the magic damage the unit takes. Each champion begins with some which may increase with level. You can gain additional from abilities, items, masteries, and runes. stacks additively.

All champions begin with with the exception of  and. Currently no ranged champion gains per level, but most melee champions do except  and.

,, ,  and  do not gain  per level because they are not always melee champions due to their variable attack range.

All champions that do gain per level gain  per level, reaching  at level 18.

Damage reduction

 * Note: One can include the in all the following ideas by enumerating it with a due amount of corresponding negative .

reduces the damage of incoming by a percentage. This percentage is determined by the formula: Damage Reduction(100). For example, a champion with 150 points of would receive 60% reduced damage from. Incoming is multiplied by a factor based on the unit's  (same with ):

$$\pagecolor{Black}\color{White}{\rm Damage\ multiplier}=\begin{cases} {100 \over 100+{\it MR}}, & {\rm if\ }{\it MR} \geq 0\\ 2 - {100 \over 100 - {\it MR}}, & {\rm otherwise} \end{cases}$$

Examples:
 *  &rarr; (20% reduction).
 *  &rarr; (50% reduction).
 *  &rarr; (20% increase).

Stacking magic resistance
Every point of requires a unit to take 1% more of its maximum health in  to be killed. This is called "".


 * Example: A unit with has 160% of its  in its, so if the unit has , it will take  to kill it.

What this means: by definition,  does not have diminishing returns, because each point increases the unit's against  by 1% of its current  value whether the unit has  or.

For a more detailed explanation, see this video.


 * Unlike, increasing makes healing and shielding more effective because it requires more raw damage from your enemies to remove the  granted. This is called indirect scaling.

Optimal efficiency (theoretical)
''Note:, commonly referred to just as 'effective health', describes the amount of raw burst damage a champion can receive before dying in such a short time span that he remains unaffected by any form of (even if the actual considered damage is of sustained form). Unless champion's resists aren't reduced below zero, it will always be more than or equal to a champion's displayed in their  bar and it can be increased by buying items with extra,  and. In this article, will refer to the amount of raw  a champion can take.

In almost all circumstances, champions will have a lot more than  such that the following inequality will be true:  > 100.

If this inequality is true, a single point of will give more '' to that champion than a single point of.

If ( < 100), 1 point of will give more  than 1 of.

If (100), 1 point of will give exactly the same amount of  as 1 point of.

Because of this relationship, theoretically, the way to get the maximum amount of from a finite combination of  and  would be to ensure that you have exactly 100 more  than  (this is true regardless of how much  and  you actually already have).


 * Example: Given a theoretical situation where you start off with and  and are given an arbitrary sufficient number of stat points (x ≥ 100), each of which you can either use to increase your  or  by 1 point, the way to maximize your  is to add points to your  until your (100)(0100)100, and then split the remaining stat points in half, spend half on your  and half on your .

However, this is only theoretically true if we consider both and  to be equally obtainable resources with simplified mechanism of skill point investment. In reality a player buys these stats for instead. As of  (derived from cost of basic magic resistance item) is currently (as of season six)  times higher than  of  (derived from cost of basic health item), we theoretically can maximize  represented by product of  (100) with  as input variable by satisfying the following equation: (100). The graph and conclusions obtained by solving it are mentioned in the analogous section about armor.


 * Example: Given a theoretical situation where you start off with and  and are given an arbitrary sufficient amount of  (x ≥ ), which you can either use to increase your  or, the way to maximize your  is to buy  until your (100)(0100)675, and then split the remaining  in half, spend half on your  and half on your  (as former is  times cheaper than the latter, it would lead to buying  times more additional  than  and thus naturally reaching equality in the equation above).

Now we just formulated a simple rule of preserving equilibrium (or maximum ):

Once equilibrium state is reached, all we need to do to preserve it is to always distribute equally into all involved stats for the rest of the game.

... or in our case, always into  and  into.

Again this model is highly simplified and cannot be exactly applied in cases when we are buying any other item than, or  (for example if our decision-making process would involve  instead of , the above model would need to use equilibrium constant ). Even considering the purchase of different or  items with differing  (quite natural expectation under real circumstances) makes use of single constant utterly impossible. Going even further, the continuous model simplifies a discrete character of real shopping, as you cannot really buy for, so with that much  you opt to buy either a single  or a single , drastically changing the equilibrium constant to .

However, thankfully to almost linear item stats' a player can use weakened base equilibrium condition in a form:  ≈ (100) safely enough to speed up decision-making. The important thing to remember is that there is no reason to hold to it too strictly.

Note: In case of only the basic constant  is slightly changed to .

This information is strongly theoretical and due to game limitations from champions' base stats, innate abilities and non-linearity of of item stats ( of stats differs for different items or is even impossible to be objectively evaluated due to interference of unique item abilities), the real equilibrium function is too complicated to be any useful.

The complexity of this problem provides space for players' intuition to develop and demonstrate their itemization skills. If given sufficient amount of time, each player could perfectly analyze situation at any given moment when he exited the shop and tell what should he buy at that moment for available gold to maximize own. The sheer impossibility of doing such thing in real time creates opportunity to develop the skill. Not only that but often choosing to maximize current leads to suboptimal decision branches in the future. The summary on end game screen about type of fatal damage taken is a key part of this decision process as well.

Instead, broadly speaking, items which provide both and  give a very high amount of  against  compared to items which only provide  or only provide. These items should be purchased when a player is seeking efficient ways to reduce the they take by a large amount. Furthermore, these items are among all available items the best ones to distribute their equally among both  and, thus working perfectly for rule of preserving equilibrium.

Magic resistance as scaling
These use the champion's to increase the magnitude of the ability. It could involve total or bonus. By building items, you can receive more benefit and power from these abilities.

Champions

 * grants equal to  to himself and the target ally (in addition to a base amount) for 3 seconds.
 * grants him equal to.
 * grants equal to   for its duration.
 * passively grants to herself equal to, increased to  while below.
 * takes and gains an equal amount of . Half of this  is stolen on cast, and the next half is taken over 4 seconds. The  bonus/reduction lasts for another 4 seconds after the drain completes.

Champion abilities
Note: Only the buff effect of these abilities is shown here, to read more information on each of these abilities, follow the link on each of them.
 * allows her to enter an egg-state for up to 6 seconds upon taking lethal damage. While in this state, she will receive a modifier of.
 * increases by   to himself and the target ally for 3 seconds.
 * increases an allied champion's by  for 4 seconds.
 * increases his by (2Gnar's level) when transformed into.
 * permanently gains every time he kills an enemy, up to a maximum of 30.
 * increases his by  for 4 seconds, stacking up to 8 times, up to a maximum of.
 * increases his by   for 8 seconds.
 * increases his by.
 * increases his by  for 4 seconds.
 * increases her by   for 3 seconds, deals damage after that time to units around her, and retains the defensive buff for an additional 3 seconds if any enemy is struck by the blast.
 * increases his by  upon activation and an additional  upon activation and an additional   per second for 15 seconds up to a maximum of.
 * passively increases his by . He loses the passive while his ability is active.
 * increases an allied champion's by  as long as the ball is attached to it.
 * increases her by 15%, doubled to 30% while below.
 * increases his by  for 6 seconds.
 * lets out a battle roar, damaging enemies increases his by  for each enemy champion or large monster hit for 4 seconds.
 * passively increases her by . This bonus is doubled while she is in dragon form.
 * increases his by  for 25 seconds.
 * immediately steals, and a further 20% over 4 seconds. These bonuses last for another 4 seconds after the drain completes.
 * grants him for each visible nearby enemy champion.
 * increases his by  for each enemy champion hit for 8 seconds.

Ways to reduce magic resistance
See magic penetration. Note that and  are different.

List of champions' magic resistance

 * 66 champions have from level 1 to level 18. These champions are mostly ranged champions, with the exception of, and.


 * 1 champion starts with at level one, and at  at level 18. This champion is.
 * 1 champion starts with at level one, and at  at level 18. This champion is.


 * 66 champions start with at level one, and at  at level 18. These champion are all melee champions, without exception.

Trivia
(Last updated 13/11/2016 on patch 6.22)


 * One of the biggest amount of any champion can obtain is  (which reduces  by %), being a level 18.


 * Base stats: 
 * Runes:
 * 9
 * 9
 * 9
 * 3
 * Masteries:
 * 5 points in
 * 5 points in
 * Items:
 * 6
 * Buffs:
 * (this mastery can not stack)
 * (this mastery can not stack)
 * (this mastery can not stack)


 * Relevant mathematics:
 * Items
 * Runes
 * Mast. & buffs
 *  multiplier
 * (42015)
 * bonus25 =
 * bonus25 =


 * Base stats: 
 * Items
 * Runes
 * Mast. & buffs
 *  multiplier
 *  multiplier
 * ((420))
 * ((420))


 * Having an enemy with the same setup use  on  will yield a total of , which reduces  by %.


 * bonus =
 * bonus =


 * Base stats: 
 * Items
 * Runes
 * Mast. & buffs
 *  multiplier
 * (420)
 * (420)


 * Prior to Patch 3.10, a level 18 with 1, 5 , 3 points in , 3 points in , , , , , an allied  aura, a full page of  runes, and  active gave a total of approximately . Switching  for  and having an enemy  with the same setup use  on  and the allied  use  on the enemy , yielded a total of approximately . This is the highest possible amount of , which is  reduction.
 * If disconnected from the fountain and had the same setup, he will have . This is a  reduction.
 * NOTE: This calculation does not include the mastery.

Magic resistance Magieresistenz Resistencia mágica Résistance magique Odporność na magię Магическое сопротивление 魔法抗性