I will give a case study to show which runes are optimal for Tryndamere, armor penetration, flat attack damage, or attack damage per level, by comparing their qualities and the various ways in which they scale. Below are Tryndamere's innate statistics, from lolwiki

Hp 461 +98

Hp5 7.9 +0.9

Ad 56 +3.2

As 0.644 +2.9%

Arm 14.9 +3.1

Mr 30 +1.25

Ms 320

Rng 125

The left hand column are his level 0 values, and the right hand column are his increases for each level. Now consider two possible runepages, one emphasizing armor penetration, the other attack damage. The mastery pages will be identical.

__RUNE PAGES__
__Armor penetration rune page:__
+24 magic resist at level 18

+13 armor

+25 armor penetration

__Attack damage rune page:__
+24 magic resist at level 18

+13 armor

+15 attack damage

__MASTERY PAGE__
Summoner's wrath

Executioner

Demolitionist

Tough skin (two ranks)

+3 attack damage

+0.5 attack damage per level

+10% critical strike damage

+4% attack speed

+10% armor penetration

+6 armor pentration

+6 armor

+6 magic resist

The innate statistics for flat armor penetration Tryndamere (fapT), and flat attack damage Tryndamere (fadT) are below. I have excluded armor, magic resist, health, range, movespeed, hp5 and attack speed because they are identical. I have included flat armor penetration and percentage armor penetration as new base statistics.

*fapT*
Ad 59 +3.7

fap 31

pap 10%

*fadT*
Ad 74 +3.7

fap 6

pap 10%

The equation for damage dealt with a single attack against a target with more armor than you posses in armor penetration is below.

**fapT**
DAMAGE DEALT = ATTACK DAMAGE * 100 / (100 + ARMOR - 31)

**fadT**
DAMAGE DEALT = ATTACK DAMAGE * 100 / (100 + ARMOR)

At the bottom of the thread is a 3d graph of the following equation which you may magnify in size by right clicking it and opening it in a new tab.

z = x * 100 / (100 + y)

with the following boundaries, and their reasons:

x is greater than or equal to 59, the minimum attack damage for fapT

x is less than or equal to 108, the maximum attack damage for fadT, full bloodlust passive with an attack damage item.

y is greater than or equal to 31, the minimum armor fapT for which the equation is valid.

y is less than or equal to 100, arbitrary value that i chose.

z is greater than or equal to 29.5, computed.

z is less than or equal to 103, computed.

[ATTACH]attachmentid=494879&stc=1&d=1343250762[\ATTACH]

The best case scenario for fapT at level 1 is for when all of the opponents armor is penetrated, and has an attack damage item and full bloodlust passive is the point (89, 0, 89)

Therefore fapT will do no more than 89 damage on each normal auto attack at level 1.

The same scenario for fadT at level 1 against the same opponent, with 31 armor, yields the point (108, 22.5, 88.16). 31 armor was penetrated by 6 armor penetration, then of the remaining 25, 10% of that was penetrated to produce effectively 22.5 armor.

Therefore against the same opponent, fadT will do no more than 88.16 damage on each normal auto attack at level 1.

This shows that fadT does less damage than fapT can do in its ideal case, to an opponent with 31 armor at level 1. Consider the same case but without maximum passive from bloodlust.

fapT (74, 0, 74)

fadT (93, 22.5, 75.91)

This shows that a full health fadT does more damage than a full health fapT to an opponent with 31 armor at level 1.

The point at which fapT does equal amount of damage to the same opponent that fadT does, can be figured by solving the following equation, where B stands for the passive attack damage boost from blood fury passive:

100*(93 + B ) / (100 + 22.5) = 100*(74 + B) / (100)

.8163*(93+B) = 74 + B

75.92 + .8163*B = 74 + B

1.92 = .1837*B

B = 10.45

The passive on bloodlust will only grant integer amounts of attack damage, so 11 points of bonus attack damage must be gotten before fapT does more damage than fadT at level 1 against an opponent with 31 armor. At least 11 points of attack damage will be granted to tryndamere from his bloodlust passive if he has less than 26% of his health remaining.

We can make the following conclusion, for tryndamere hitting an opponent at level 1 who has 31 armor:

If tryndamere is above 26% of his max hp at level 1, fadT does more damage.

if tryndamere is below 26% of his max hp at level 1, fapT does more damage

the difference in damage never exceeds 2 points.

If the damage difference never exceeds 2 points, then why bother analyzing it? At levels above level 1, tryndamere will continue to earn attack damage from his items and from the flat bonus on bloodlust, but the passive scaling attack damage on bloodlust will be dwindled by those. This shows that even at level 1, fadT and fapT do comparable damage regardless of the bloodlust, and is a hint that we can ignore the scaling passive attack damage bonus, and simply account for the flat attack damage bonus in further analysis. Simply put, both fapT and fadT scale comparably with loss of health, so we can exclude that from further study.

But how do they scale with levels?
Recall the base statistics:

*fapT*
Ad 59 +3.7

fap 31

pap 10%

*fadT*
Ad 74 +3.7

fap 6

pap 10%

We can use the previous equation for comparing fapT and fadT over how they scale with loss of health, but adapt it for use with levels. What we are really interested in is how they scale with increase in attack damage, which is granted at each level in the same amount.

y = bonus attack damage for fadT granted by levels

x = bonus attack damage for fapT granted by levels

100*(74 + y) / (100 + 22.5) = 100*(59 + x) / (100)

.8163*(74 + y) = 59+x

60.41+.8163*y = 59+x

1.41 + .8163*y = x

Now for example, if fadT had y = 66.6 for being level 18, then fapT would only need x = 1.41 + .8163*y = 55.78 attack damage from levels, implying that fapT would only need to be level 55.78 / 3.7 ~ 15 to do the same damage.

Therefore without items or skilling bloodlust, a level 18 fadT does the same damage that a level 15 fapT does. So here we can see that fapT scales better with levels than fadT does.

But does fapT scale better with levels than any other choice of runes? Consider attack damage per level runes, for attack damage per level Tryndamere (adlT).

__Attack damage per level rune page:__
+24 magic resist at level 18

+13 armor

+35 attack damage at level 18 (+1.95 attack damage per level)

*adlT*
Ad 74 +5.65

fap 6

pap 10%

now we use the equation from earlier

1.41 + .8163*y = x

if fapT is level 18, then x = 66.6 = bonus damage from levels for fapT, and y = 79.86 = the required bonus damage from levels for adlT. Therefore adlT must be level 14 ~ 79.86 / 5.65 in order to do the same damage to a target with 31 armor that a level 18 fapT does.

But how do they scale against armor?
I will use the damage equations from earlier:

x = bonus damage for fadT and fapT

y = armor of opponent which is not to be less than 31

z = damage dealt

100*(74 + x ) / (100 + (y-6)*.9) = z

100*(59 + x ) / (100 + (y-31)*.9) = z

In other words, we want to know the armor values for which the damage is the same for both fadT and fapT. Any value below that and fapT will provide more damage, and any value above that and fadT will provide more damage. We will always assume that the opponent has 31 armor or more.

100*(74 + x ) / (100 + (y-6)*.9) = 100*(59 + x ) / (100 + (y-31)*.9)

(74 + x) * (100 + (y - 31) * .9 ) = (59 + x) * (100 + (y - 6) * .9 )

7400 + 100*x + (74+x)*(y-31)*.9 = 5900 + 100*x + (59+x)*(y-6)*.9

1500 = (59+x)*(y-6)*.9 - (74+x)*(y-31)*.9

1667 = 59*y + x*y - 354 - 6*x - 74*y - x*y + 2294 + 31*x

0 = 273 -15*y +25*x

**y = 18.2 +1.67*x**
So for example, if x = bonus damage for fadT and fapT, including that gained by levels, items, and the blood fury passive = 100, then the required armor value for which fadT and fapT will do identical damage is y = 18.2 +1.67*100 = 185.2.

In other words, fadT with 174 attack damage will do the same damage with his auto attacks to an opponent with 185 armor that fapT with 159 attack damage will do.

Performing the same analysis but including last whisper gives us the following equation and result, where i have included the attack damage bonus of last whisper. The .55 multiplier comes from the 40% armor penetration of last whisper, and the 10% penetration of the armor left after that, .6 *.9 = .55

100*(114 + x ) / (100 + (y-6)*.55) = 100*(99 + x ) / (100 + (y-31)*.55)

(114 + x) * (100 + (y - 31)*.55) = (99 + x) * (100 + (y - 6)*.55)

11400 + 100*x + (114+x)*(y-31)*.55 = 9900 + 100*x + (99+x)*(y-6)*.55

2727.3 = (99+x)*(y-6) - (114+x)*(y-31)

2727.3 = 99y - 594 -6x -114y + 31x + 3534

0 = 212.7 -15*y +25*x

**y = 14.18 + 1.67*x**
So for example, if the bonus damage besides last whisper for fadT and fapT, including blood fury passive, items, and whatever is gained by levels = 60, then the required armor in order for fadT and fapT to have identical damage is y = 14.18 + 1.67*60 = 114.38.

In other words, a fadT with last whisper and 60 ad from all sources except masteries and runes and the last whisper itself, will do the same amount of damage to an opponent with 114 armor that a fapT would do, given that fapT also has 60 ad from all sources except masteries and runes and the last whisper.

But look what happens when you consider a Tryndamere with a very large amount of attack damage, say 250 on top of that given by runes and masteries and the last whisper. How much armor would be required in order for fapT and fadT to be equal?

**14.18 + 1.67*250 = 431.68**
The opponent would need more than a staggering 432 armor in order for the flat attack damage runes to beat out the flat armor penetration runes, and thats because armor penetration scales with attack damage very well. The more attack damage you have, the more flat damage increase you would get by penetrating armor, so the armor required to make a measily 15 flat attack damage become better than 25 armor penetration, when you already have a large amount of attack damage, is staggering, even with the last whisper.

**In conclusion**
- adlT scales with levels best, fapT second best, and fadT least.

- fapT and fadT scale comparatively in increase in damage with loss of health at all levels.
- adlT scales harder with loss of health at early levels than either fapT or fadT and scales less than fapT and fadT at later levels.
- at level 1, fadT does more damage than fapT except when below 26% hp.
- at level 1, adlT does less damage than either fapT or fadT.
- fapT scales better with attack damage than fadT and adlT

adlT attack damage per level Tryndamere

fadT flat attack damage Tryndamere

fapT flat armor penetration Tryndamere