I've played around 500 games as Tryndamere, and I am an average player at around 1200 elo. I've played against skilled players climbing their way to the top, and against poor ones falling to the bottom. I've played him in every position and against nearly every champ. Over time, I've been able to identify the central issue that plague's Tryndamere, causing him to be a nightmare to balance, resulting in him being either extremely weak or extremely strong. Surprisingly it doesn't have anything to do with being a melee carry, a significant, although separate problem, which is also addressed in this thread.
His biggest issue is building bonus fury on critical strikes, and I'll explain why.
1. remove the bonus fury on critical strikes and vastly buff his fury generation from all sources.
2. vastly increase his fury decay and decrease the delay before fury begins to decay.
3. make his bloodlust heal scale with percentage health missing rather than fury, still consuming fury.
4. make undying rage last for as long as tryndamere has fury, make him only immune to death from damage dealt by champions and minions, and seriously reduce its cooldown.
Currently tryndamere with full fury can easily maintain his fury through last hit critting. A tryndamere with no fury cannot build or maintain fury anywhere nearly as easily because he has no chance to crit without fury. If riot balances with the assumption that tryndamere will have full fury throughout the laning phase, then a tryndamere without full fury is a pushover and is so weak he probably will not be able to build fury without being harassed to death and pushed out of lane. If the balance assumption is the opposite then tryndamere will be decent without fury and overbearing with it. This is essentially one of the biggest reasons tryndamere has a snowbally lane.
Tryndamere is held back (riot refuses to buff him) because when he is doing well enough to maintain fury, regardless how that came to happen, then he is almost overbearing. If Tryndamere is slightly ahead he can magnify that advantage through fury, and at the same time reduce the difficulty required to maintain that advantage. If Tryndamere is going to receive buffs that make his generally stronger even when not ahead then there needs to be counterplay to him when he is ahead.
This skill is supposed to help Tryndamere bounce back into lane when he is doing poorly, or getting zoned from minions. In reality it is most powerful when he is already ahead and doesn't need the advantage, making him even more overbearing when ahead. The skill is balanced around that, so the base value is pathetic and it doesn't do its intended purpose. Riot doesn't necessarily like that a well placed ignite will kill Tryndamere after Undying rage automatically even if Tryndamere heals regardless of whether Tryndamere managed to escape or not, but buffing the heal is out of the question because of how precarious Tryndamere's kit already is.
The delay before losing fury should be about 1 second, and it should take around 3 hits to fill it completely. It should take about 2 seconds after the delay to lose it all. After Tryndamere uses undying rage he will simply survive for as long as he can stick to a target. Being disabled or merely kited for 3 consecutive seconds will stop Tryndamere's undying rage. Tryndamere will die from monster damage so he can't solo baron or dragon with just this ultimate. Tryndamere will die from tower damage so this ultimate will not hold him back (riot won't use it as an excuse to not buff him) by merit of it being used to snowball him in situations where his opponent has no counterplay options in lane if Tryndamere is somehow ahead (how that happened or how difficult it is is obviously almost irrelevant in riots point of view) due to Tryndamere's kit already being precarious and an ultimate like this, pretty much married to the identity of Tryndamere, is designed to push him over the edge.
Currently so much holds Tryndamere back that he can't properly carry in his short 5 seconds of lifetime. Tryndamere used to have a 7 second ultimate and was a veritable carry. At the end of the day a carry simply needs to be able to do large amounts of damage safely, and that safety needs conditions that the enemy can work around to kill you if they use team work, and which can be reinforced by team work on your own team. Tryndamere lasts 5 seconds period, team work or not. Tryndamere ought to last for time that depends on team work between his team and the opponent team. This would give him something like a lesser late game Sion effect (who can legitimately carry as a melee champ in the traditional sense but was not designed to melee carry and who has no innate skills to help him play that way before amassing the tremendous gold cost of a carry build), alive while hitting something, dead after a few seconds of not hitting something.
I've given some particular number just to convey the idea. The actual numbers required to make it balanced can only be discovered with iterated testing.
What is an ad carry, and what are they doing for their team? Ezreal, Vayne, Kogmaw, they all require varying amounts of protection and do varying amounts of damage. If they get caught with CC they are pretty much dead or very near dead.
An ad carry simply does sustained damage safely. There are carries that can burst some champs from full, there are some ad carries that sport powerful disables, but the one thing that all ad carries share is that they do a relatively large amount of sustained damage and they do it (or want to do it) with relative safety (relative to most other champs).
Ranged carries can be protected by their teams with simple positioning and team work. Melee carries cannot and therefore need to either do much more damage or sport some survivability method. In practice neither of those are sufficient.
In order to make melee carries an actual competitive choice for literally carrying a game in the traditional ranged carry sense, in other words, in order to make the distinction "melee" and "ranged" for ad carries largely irrelevant, the melee carries cannot be intrinsically inferior to the ranged carries for doing the job that ad carries do, which is putting sustained damage out safely, given team work.
The average amount of difficulty required to isolate and kill a typical ranged carry in a typical team fight with typical teams should be about the same as isolating and killing Tryndamere on average in typical circumstances. The difference is purely tactics; you kill the two targets in two entirely different ways, and that is what makes a proper melee carry a unique experience.
Mikael's Crucible and Shard of True Ice are support items that can help to protect tryndamere in team fights by cleansing the cc on him and creating a field slow that will make it easier for him to stick to targets. There are a variety of means for stopping tryndamere.
I do not think tryndamere should be able to roll into a team fight and be immune to death for the duration of the fight regardless of how the enemy responds, and I do not think he should just die in 3 seconds regardless how he approaches the fight and from what angle. There are plenty of variable to tweak in order to find the balancing point at which handling Tryndamere is around as difficult as handling a ranged carry, and that is a model for all melee carries.
Right now the vast majority of people in your ELO bracket that play Tryndamere on a regular basis are the dedicated veterans, the people who have played him in 500 games and can whoop your arse in the laning phase just because they are playing with the skill and confidence of someone who belongs in a higher ELO bracket, and will be stuck there even if they play 500 more games as long as those are Tryndamere games simply because Tryndamere played skillfully is practically designed to stomp people who are less skilled. Bonus fury generation on critical strikes, causing him to be too precarious to be used reliably against people of the same skill level and too overbearing against people of lesser skill level is unacceptable. Why on earth does his average fury generation per hit need to be proportional to the fury that he has accumulated? That is an exponential function, and it causes him to have amplifying characteristics that are impossible to balance appropriately. His resource and theme execution should be rethought or even if the ideas in this thread aren't agreeable, change his fury generation from having amplifying characteristics to having decaying characteristics.
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