Sick of tanking as Amumu? Want to do something different for once? This is the solution for you: AP Amumu. Due to Amumu's Bandage Toss and Ultimate, you are able to charge and stun the opposing team without having to worry about your health bar. Why play tank when you won't die anyway? The purpose of this build is to provide you with enough ability power to kill the opposing champions (with the use of your ultimate) before they have a chance to fight back. This build order also has maxed out CDR, so that you should have your ultimate for every team fight.

One important thing you should do when using this build is to tell your team beforehand what you are doing. A lot of people won't approve of you making an AP Amumu, but most are sensible enough to accept it as long as they are properly informed ahead of time. This will also help the team know to choose their builds and champions accordingly. They may pick an extra tank, instead of a mage. It may also work better to actually build an extra DPS though, instead of another tank. This way, you can kill the other team even faster in team fights, since they'll be stunned most of the time anyway (assuming you get a good ult in). After all, initiators are really more important than "tanks" imo. An AP Amumu can most certainly initiate team fights, just about as well as a tank Amumu can.

I'm not exactly an expert on runes. My suggested runes are about the best advice I have, but do whatever you want really. You may want to tweak them a bit also, depending on whether or not you're jungling.

I prefer the utility tree on Amumu. The speed buff, cooldowns, and all that can be really helpful. The mana regeneration masteries are probably the most important, since Amumu tends to run low on mana, especially early on. Obviously, you will want the utility masteries if you are jungling, so the gollum buff lasts longer. Regardless of where you spend most of your mastery points, you should definitely go at least 9 on the attack tree, to get you to Archaic Knowledge.

Rod of Ages - Awesome first item. Health/Mana/AP are all things you need early, making this an ideal start, especially with its passive.

Boots of Lucidity - Great for the cooldown reduction, it saves you having to buy a Morello's or something to max your CDR. These can be replaced with Merc's Treads for the tenacity. You can also replace them with Boots of Swiftness or Boots of Mobility, so you can charge in and ult faster and easier.

Shurelya's Reverie - This is great on Amumu. The health, regen, and CDR are all great. The best part of it though is the active. It's a great way to help chase down retreating opponents, which is great when they are all weak from your ult.

Rylai's Crystal Scepter - The health and ability power are obviously good. The passive also gives you a slow for your W and E.

Lich Bane - Movement speed, magic resistance, ability power, and a passive that helps kill faster (goes great with bandage toss). This is a great item in every way.

Rabadon's Deathcap - This is the clincher. Get this and you'll start killing so fast that it'll just be funny. You can also solo baron after getting this, but it's always better to bring along at least one other mate to be safe.

My recommended tankier AP build - Rod of Ages - Boots of Lucidity - Sunfire Cape - Abyssal Scepter - Rylai's - Lich Bane. Use this if you need extra survivability in order to effectively initiate.

The skill sequence I chose is for jungling. You will probably want to do the same one or a similar one if you are solo laning. If you are 2v2 laning, you might want to start Bandage Toss or at least get it earlier.

If you're jungling, you normally won't do any ganking until you have bandage toss, so try to get bandage toss around the time you'll want to start with your ganking. Sometimes, you can wait until level 5 before bandage toss and even go up to level 6 before you start ganking, so you have your ultimate. You will definitely still want bandage toss before your ultimate though, no matter what you're doing and how the game is going.

If you're in a lane, bandage toss is just as important for ganks as it is when you're jungling. You will want to have bandage toss in time for the gank, to help make sure no one gets away alive.

I would advise getting Ghost, Flash, Exhaust, you can choose which two. These are probably best, because you aren't tanking, so you might need help to get away so you don't die. Ghost and Flash also both help with initiating and getting to team fights faster. Of course, you should get Smite if you're jungling though. Another possible choice would be to get Ignite, Teleport, or Fortify. Along with Exhaust, Ignite could help you win a lot of fights earlier in the game. Teleport can just help get you where you need to be for team fights. Fortify can be an amazing spell if you get the perfect situation for it: using your ultimate on a team attack your turret.

I hope people at least try this, without hating it too hard. The reason it's more viable than going AP or AD on other tanks is because Amumu has a stun and he can damage the entire opposing team simultaneously. Unlike other tanks, Amumu can still play like a tank with this build. He can still initiate and stun. The difference is, with this build you weaken everyone uber fast so that your allies can run in and get the kills.