This is the first guide I've ever written, so why not write it about one of my favorite champions to play, Amumu. I would like to point out I am not top tier, I do not have IPL championships under my belt, but I have some real first hand experience. The reason I always come to this website was when I buy a new champion, I have no idea what to build, what their mastery is, and how they should play. This guide is not going to consist of number crunching or how to maximize your dps to the thousandth degree. It is solely here for the summoner who just bought Amumu, has no clue what to do and wants to jump right into the game.
The masteries I set up are really for tanking in general. It can be swapped around a little so you do more damage to creeps/minions if you wanted to level faster for example. The reason I do not set it up like that is because I have no problems killing creeps/minions and late game I feel like it helps more to be able to move around faster and take less damage.
The runes are not the ideal setup. With that being said I am writing this guide as I play right now. I do use everything I put in this guide (the items the mastery the runes.) If you had the money for it or already do have the runes maybe go with some armor magic resist, the standard runes page, but I like the attack damage and attack speed.
I start off at blue. Hopefully you can get a good leash, if you do then you won't even have to use your pot until after the wolves heading over to wraiths. Get the blue buff and head over to wolves. Kill the wolves and walk over to the wraiths. Please for the sake of mana conservation turn off your despair when you aren't using it. Be wary though, if a monster is about to die from one more "tick" of despair damage, and you turn it off, the monster wont die. You'll probably get the hang of it after the wolves or wraiths. After the wraiths head over to the golems. Kill them off and if you're low on hp don't hesitate to recall back and heal up. I do not believe in dying in the jungle just to get back faster. After golems get the red buff. The way the timing works out is at level 1 you will have despair. After getting the blue buff you will be level 2 and have tantrum for the wolves. Keep leveling up tantrum first as it does more damage since the cooldown is reduced the more you're hit. I experimented leveling up despair to five(5) first rather then tantrum but found tantrum is the better route. Only get bandage toss at level 5! This is very important because almost useless in the jungle and I don't really gank until level 6 when I have my ult up. With that being said bandage toss is a nice stun and odds are if your enemy jungler tries to counter jungle you, you're more then likely going to win because amumu is a monster tank as you will soon learn. If you are in the middle of jungle and happen to have enough golf for the heart of gold item, I personally go back for it and continue the jungle.
The order of items has its purpose. To start the regrowth pendant is used to regain hp during and between jungle creep fights. The time it takes you to run from wolves to wraiths on the first jungle clear, it will give you about 1-2 bars of hp back. The boots help you run around the jungle faster and of course if and when you start to gank help you keep up with the enemy. The heart of gold serves two purposes, health pool increase and the gold/10. It takes a while for the heart of gold or any gold/10 item to pay for itself but by the time you do its when you'll need that spot for another major item and it will have paid itself back and then some. The sunfire cape is just another despair but better cause its free. It gives a nice boost to your health pool as well as the constant damage. The mercury treads are for the magic resist as well as the tenacity. The next few items are kind of situational. As you can see they all give massive amounts of armor and that's good and all but if the enemy team is a hybrid (AD as well as AP) then get a mix of magic resist as well as armor. The example build I gave is for AD heavy teams. I've seen some Amumu's go AP heavy and it seems to work some times, but I like the tanky route because it is more reliable just like AD Yi (don't hate if you like AP) but AP anything like that is just more for fun/trolling in my opinion.
Pros / Cons
He is very tanky
He can farm well
He is underused IMO and you won't generally be mirror matched with another Amumu jungler
Insane 1v1 dominance
You can chain bandage chase someone
His ganks are really dependent on landing his Q(bandage shot) it's called a skill-shot for a reason
No life steal whatsoever, unless you build for life steal which is just silly, you will have 0 life steal ability/items
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This is a screenshot of a game I played with this very setup, the same runes, the same mastery, the same items. I hope this guide helps, it isn't super fancy with links to all kinds of websites and pictures left and right, its just something to read if you just bought him and are unsure of how to play him. Best of luck.