I made a post about a day ago about most efficient rune combinations and got alot of feedback. After reading through the comments I realized that DPS cannot be calculated without taking into account the base stats of each champion. So I wrote a program in Mathematica 9 that takes into account: base stats, starting item and offense masteries. I used constrained optimization of multiple variables with multiple constraints to find the combination of**attack speed** and **attack damage** **marks and quintescences** that maximizes DPS (damage per second) of fighters, assassins, tanks and marksmen at **level 1** – all in one click. However, I did not take into account champions’ abilities.

I’m deciding whether I should upload the program or not. Also, for someone to use the program they must have Mathematica 9. **Here’s a preview of how it looks:**

**here’s a graph**

If anyone wana help create an app or web application of this program, message me. .

**Fun Facts**

- Maokai has the highest DPS (73.8488) of all junglers, given that he starts with Hunter’s Machete, have all [21] offense masteries (as shown in the preview above), and a combination of
*9 AS marks*+*3 AS quints*. - Twitch has the highest DPS (60.6584) of all marksmen, given that he starts with Doran’s Blade, offense masteries, and a combination of
*7 AD marks*+*2 AS marks*+*3 AS quints*(I guess that’s why he’s getting nerfed in 4.10) - Shaco’s optimal rune combination for maximum DPS at level 1 (67.7924) is
*6 AS marks*+*3 AD marks*+*3 AS quints*(and yes its the same if you factor in his backstab passive – just a 20% increase in his DPS) - Scaling health per level seals are efficient at level 6 (breakeven point with flat health seals)
- AS quints OP
- Rule of Thumb: The higher base AD is, the more efficient AS runes are; The lower base AD is, the more efficient AD runes are.

**Now feel free to ask for optimal combinations of other champions.**

Also I can create other programs that would optimize various rune combinations such as armor/health/health regen seals/quints, magic pen/ability power marks/quints, magic resist/CDR/mana regen glyphs/quints/etc; But I just don’t think its worth the time. Maybe sometime in the future; I just did this one for fun.

EDIT: I’m not an expert, so don’t take my advice as written in stone – I’m only doing the math. Again, I did not factor in champions’ abilities so even though these runes maximizes basic attack DPS, ability scaling was not considered. That being said, *efficiency was calculated for only level 1 where only one ability is unlocked (level 1 damage comes mainly from basic attacks)*. However, it may be the best for champions that rely mostly on basic attacks.

EDIT 2: *The whole point of this is just to show how useful math can be to solve problems, not to make strong judgements on best rune combinations – the math in this program isn’t powerful enough yet to determine that as there are some variables left out and other runes to consider, but still may be used as a guide to better strategies and theory crafting*

EDIT 3: I’m tired calculating DPS lol XD if this post gets popular, I’ll consider uploading the program.

EDIT 4: Someone asked for a graph that shows the trade off between AS runes and AD runes, well here it is:

Note that the axes are labeled AS marks and AS quints but it already takes into account AD marks/quints (for example at the origin where AS mark = 0 and AS quint = 0, AD marks = 9 and AD quints = 3…AD marks = 9-AS, AD quints = 3-AS). Also this is holding base AD and base AS constant. All I did was maximize this function, subject to the constraints {0 <= ASmarks <= 9, 0 <= ASquints <= 3}, as base AD and AS varies to get optimal DPS for each champ.

I’ll update this post with more facts as time go by

**UPDATE: Here’s the program:**

After getting much feedback from my first post, I decided to create a program in Mathematica 9 that shows the various combinations of Attack Speed and Attack Damage **marks** and **quints**, that **maximizes DPS** (damage per second) from **basic attacks** for various AD champions.

The computations take into account:

- Starting items
- Masteries
- Base stats

Here is a link to **download**: http://speedy.sh/FcrE2/DPS-rune-efficiency.cdf

(You gotta click on the file name at the top of the page where is says “Download: DPS rune efficiency.cdf”, click on “DPS rune efficiency.cdf”)

**OR** http://www.4shared.com/get/WfvUADcoce/DPS_rune_efficiency.html

For those who **don’t have Mathematica 9**, you will need to download the CDF Player:http://demonstrations.wolfram.com/download-cdf-player.html

Feedback is appreciated.

Enjoy.