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How to Build Jayce?

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How much does Jayce rely on auto-attacks (which benefit from crit) vs how much does he rely on purely AD-scaling abilities - Q (cannon) Q (hammer) and E (hammer) - (which get no benefit from crit)? Given that, how much of a benefit are you really getting by buying a more expensive IE instead of a BT? And then how much are you delaying other items in your build by buying IE instead of BT? It's a question of gold spent vs benefit gained.

Just because I haven't had much time to play beyond a quick co-op first win of the day for the past few weeks doesn't mean I can't do simple math.

I wrote a Jayce guide for dominion a while ago that has some of the basic math theory as to why hypercharge is so strong with IE. (cut&paste into browser)

The basic theory is that maxed hypercharge acts as an attack resetter, AND has a 3.9 AD ratio (1.3 AD autoattacksx3).
So thats a 4.9 ratio that happens in about 1 second.

Copying a piece of it for illustration: (note I don't assume the attack reset for hypercharge, just a 3.9 ratio)

Assume 300 AD, no crit.

Q/E Nuke does 400 +200x1.68 = 750ish damage.
HyperCharge does 3.9x300 = 1150ish damage.
Q melee gapclsoer does 200+200 = 400 damage.
E hammer knockback does 160+200+20% = 750ish damage.

Imagine we can do 1 of 2 things for sake of comparison: Get 90AD or 75% crit (I am assuming the Inf Edge passive in effect which you obviously can't have with 0% crit but go with it for now). Both have similar costs (40g per AD, 45-50ish per Crit)

HyperCharge would do 3.9x390 = 1500ish damage with the +90 AD approach (up from 1150). Other skills would get about +90 dmg (or 140ish Q/E nuke)

With the crit approach, we are looking at 3.9 x 300 total AD +75% chance of adding 160% of 300 AD damage.
== 3.9 x (300 + .75x1.6x300) == 3.9 x 660 = 2574! (up from 1150)

Wait...whaaat? That's right. The crazy 260% multipler from crit turns HyperCharge into an amazing engine of DEATH, once a decent amount of AD is obtained.
Specifically, for the same gold-cost that crit increases HyperCharge damage by 120%, compared to the 25% or so gain from more AD (though the AD also boosts his other skills too).

THIS is why we focus on crit lategame because it increases Jayce's burst potential to ridiculous levels. Basically we don't try to turn him from a burst caster into a sustained output ADC, but instead tweak itemization to use HyperCharge to increase his burst exponentially during the endgame.

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Full build:

Ninja Tabi/Merc Treads
Infinity Edge
Trinity Force/Last Whisper
Randuins/Frozen Heart/Aegis
Force of Nature/Maw/Guardian Angel

Try this build out before saying it is bad.

Black Cleaver should be Jayces main item. Why? Because if the armor shredding, CDR and health, that item alone makes Jayce able to chase nearly anyone our of a lane. I've been playing Jayce as an ADC and I can tell you even in that roll INF edge is out classed by pretty much every other item for him because he doesn't rely on auto attacks enough for it to mater. Blade Of The Ruined King will allow you do deal more reliable damage than an INF edge. You can win while building an INF edge, but then again you can win playing AP pantheon but that still doesn't make it good.

He should be played as a Tanky AD bruiser if you spend all your gold on an unnecessary item that brings him no real advantage he cant fill that role, and if hes taking the ADCs role then you will end up with two useless champs on your team.