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You should probably include a void staff, once they start getting magic resist, you're going to be hitting for around 10% of their health instead of 20-50%. And as previously stated, building Rylai's is actually good because it allows you to chase with your team. Other than the triple Archangel's this guide is pretty decent.
The creator of the guide said 2nd archangels and third archangels or a void staff, so read the stuff inside the build (A voids staff was include but it was assumed that if they don't get much mr then you don't get a void staff... also your assessment is off on the amount of damage done. Also the more mr they get the less deadly they are to the rest of your team... unless your team is all ap for some reason.... plus the lich bane makes up for those who get mr... unless most of them get mr it's poitnless.
Also you are not maximising the cost/benfit health:mana:damage:speed ratio. By this I mean if your oppoent has boots of switness they can outrun you unles you ghost, this is where a rylai's would come in handy, so if the oppoent jumped you, you could have more survivability. I do not believe lich anes passivies stack nor do archangels thus the 2nd lich bane is just for ap and movement speed I assume and the archangels for mana and ap. In other words this build is ok, but I don't believe it does AP kog justice... although I wont downvote it for it is viable... although not my preference.
(The only time I ever run out of mana is when I use other abilities on top of my ult, and then jsust a quick 5 second break and I'm good to startup again if I have blue, if not a quick 10 second break and I have enough for a real teamfight and not artillery spam. If no blue and fully loaded on mana then I have enough for around 20-30 shots dependant on how fast I fire... for sometimes there's nothing to shoot at. Basically you ahve 4k aman... at 200 mana per ult that's enough for 20 shots... then your mana regen per 5... each ult taking a second so.. 20 seocnds times 105 this another 2 ults minimum without chalice or blue buff...
I can understand the summoner spells and bravo on the ranked play... for I strongly agree champs like nocturne and whatknought can be annoying (althouh since nocturne makes you blind you might as well jsut looka t yourself... for that's proably where he's going... then jsut ult on yourself even if he's not there.... and then if he shows nuke him down asap or run for your life, yet I disagree that Xerath is a worse version of kog... his ult has a way logner cooldown. The chalice is wise early game. The reason I don't do it is I get a rod of ages for survivability, mana, and ap, and those stacks are needed sooner than later.
To add to the annoying early game champs I would have to say orianna... you rarely see her, but with her ult she has enough range to fight you until your level 11... even if you are great at skillshots... (especially since she can pull you closer).
To add to late game hinderances... competnet twitches... for your usually behind the battlelines... thus I say kog needs a bodygaurd for if you don't see twitch coming... your screwed more than if it were any other champ... Of course this is assume twitch built properly... (I even find it hard when with my build which has more ap and health than this 1.... damn those lifesteal champs!). AKA don't let lifsteal champs near you... unless they're squishy... for your lich bane and all your abilities will nuke them... lee sin is annoying
When it comes down to it... I find no champ can fight ap kog unless they have some serious lifesteal and attack speed... haven't found a champ with spell vamp able to outdo ap kog.
Additionally you will be far away from any team fights anyway. So you don't need to flash out of enemies! You will stay in bushes or behind thick walls and give your team artillery support.
If an enemy get you there you should be able to cleanse his disables and kill him while you try to escape from his attack range.
In the most emergency cases you will find yourself running for you life to reach the next tower to beat the enemies up while they try to push it or you will chase enemies down that are far away from you. But normally you will just stand somewhere in the desert and press Living Artillery and sometimes Void Ooze. So how could Flash and Ignite be of any use to you?!
If you are selling you Chalice of Harmony you will lose the ability to shoot for as long as you want with your Living Artillery even when you have blue buff. You have to keep that in mind! But your mana should last for maybe 10-20 rapid fired shells
Ban Shaco and Nocturne. Those guys will be a really pain in YOUR ***.
If those guys get in melee range to attack you while you don't get any help, your escape chance are below 20%. I recommend a "Stand and fight" tactic in those situations, your chance to beat them up is 30/70 and with a Lich Bane it will be around 80/20.
Mid game:
Try to reach lvl 16 as fast as possible and then get the blue buff then. If you manage this you have for a short time a giant advantage. Most of the enemies will lose 20-50% of their hp when you hit them with your Living Artillery. And now you can spam it till the blue buff last. Use it to win the game or push as much towers as possible!
Teamfights:
Stay out of enemy range and spam Living Artillery
If you really want to help your team as much as possible shoot a Void Ooze through the enemy team and do as much aoe damage as possible with your Living Artillery.
if someone comes to close try to nuke him, if Void Ooze is on cd melt him with Bio-Arcane Barrage and Caustic Spittle.
If enemy tank is trying to initiate give him the proof that this was the worst decision he could made:
1st Void Ooze, 2nd Caustic Spittle, 3rd Bio-Arcane Barrage and finally some Living Artillery.
This will kill him or cost at least 50% of his hp.
Avoid teamfights far away from the lanes:
Team fights around the red buff and blue one are your absolute weak spots! You are an artillery that needs clear sight on the enemies and you don't want to be ambushed from the flanks.
Always try to bind the enemy team on the lane. If they're @ the lane and pushing your tower or guarding their tower you have a big advantage. Its like hunting sitting ducks!