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Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
Almost all items are very situational.
First: Regrowth Pendant -> Philosopher's Stone
You probably have no mana regen whatsoever. This item will give you that along with survivability via health regen, and will also pay for itself by the end of the game (usually).
Second: Boots of Speed -> Ninja Tabi OR Mercury's Treads
This is somewhat dependent on the other team. As long as they have at least two physical damage champions I get Ninja Tabi. One or less Physical Champions should mean Mercury's Treads. If other factors come into play (They have two magic but both seem to be playing very well and are getting fed) use your own jusgement.
Third: Force of Nature OR Sunfire Cape
Anything beyond this point is greatly dependent on the other team. Two physical three magical should probably mean Force of Nature here, since Ninja Tabi should hold off the physical champions long enough to get other items. Two or less magical probably means Sunfire Cape.
Fourth: Sunfire Cape OR Force of Nature OR Banshee's Veil
If you got Force of Nature last item and there's two or more physical on their team, go for Sunfire Cape. If there's one or less physical on their team, it may be better to get Banshee's Veil and just try to avoid the physical opponent or rely on your HP/Team. Mercury's Treads + Force of Nature + Banshee's Veil = Magic attacks do barely anything to you. Also, if they have 4+ Physical and you got a Sunfire last item, you may want to grab another here.
Fifth: Warmog's Armor
If you went Force of Nature + Banshee's Veil and their Physical champions are pounding on you, it may benefit you more to go Sunfire here. Otherwise, by now you should have a good amount of Armor and Magic Resistance to counter the enemy team and should work on some more HP. Whenever you get a chance, farm minions to stack the passive. Champion Kills work ten times better than minions(10 minions = 1 champion), but playing as Amumu should get you mostly assists and not many kills. When fully stacked this one item should give you over 1k health.
Sixth: Open Choice
Very subjective to what items you already have and how both teams are playing. Most games won't last long enough to get you another item here. If they're heavy physical and you went Sunfire + Sunfire + Warmog's, another Sunfire could help quite a bit with Health, Armor, and doing AOE damage. If they're heavy Magical and you went Force of Nature + Banshee's Veil + Warmog's, another Force of Nature would make you near invincible (The passives don't stack, though.). If they're pretty balanced and you went Ninja Tabi + Sunfire + Force of Nature + Warmog's, Banshee's Veil could help even out your Armor and Magic Resistance. If your team is roflstomping theirs and for some reason they just keep holding on, grab Rylai's Crystal Scepter for the extra punch on all your attacks.
More Item Stuff:
Don't bother with any potions at the start. Amumu isn't a good laner, especially early on, so you're going to have to play somewhat defensively and therefore probably won't run out of mana before going to shop for the first time. Your Regrowth Pendant should keep you alive through the laning phase.
Also, if you've gotten all six items and the game continues, you can sell your Philosopher's Stone to make room for another Legendary Item since you'll have more than enough Health Regen and you'll have enough total Mana that between shopping and just running around between fights, natural mana regen will be enough to let you unleash everything when needed.
Skillshot stun / Initiation ability. Very useful for starting team fights by grabbing an enemy and instantly using your ult. Try to aim for champions that can disable your ult mid-cast since the stun should stop them from being able to.
Anti-health ability. Drains almost 3% of all nearby enemy champions' health per second. Activate just before or after starting a Bandage Toss team fight to get the most use out of it. Also very useful for running away, keep it on if possible and the enemy champions won't usually realize their health decreasing as they chase you.
Close-combat AOE ability. Passively also increases your defenses. This is your minion-farming ability. Also useful during team fights, use it as many times as possible after using your ult to do more damage. The cooldown is decreased by one second very time Amumu is hit, so if you can get their team to all attack you it'll basically be a continuous spam.
Curse of the Sad Mummy
Team-Killer ability. With proper team coordination, this ability could potentially lead to your team scoring an ace. Does a good amount of damage as well as stunning all champions near Amumu for 2 seconds. Coupled with a Fiddlesticks or Katarina ult and they should be running away crying.
Play defensively! Unless you have a great pusher as your lane partner, expect to be sitting under your own turret quite a bit. Tough out the dullness of standing around behind your minions and get that Philosopher's Stone asap to help you generate money since you won't be getting much from minion kills.
If you have a good mid that has taken one of their turrets, get them to help push your lane's turret and/or set up an ambush. You should have your ult by now, so try to catch both enemy champions in it and gank them to further push. If there's an opportunity to use your ult on only one enemy but doing so would possibly mean one of their turrets down soon after, go for it. Also use it to trap enemies near your turrets if they're making moves on it.
You should have so much damage mitigation that, depending on their team makeup and your items, you may be able to survive a 5v1 and let your teammates escape if needed. Try to be the instigator for team fights since your ult is what should definitely win it for your team.