Olympus Coliseum

There are 2 Blue Trinity marks here that you can activate immediately for some treasures. (the one to the right has Puppies 22-24) Also, try putting out the flames on all the torches with your Blizzard magic (if you have enough MP) to reveal a chest with an Orichalcum. Check behind the pillars too for a Mega-Potion. Proceed to the Lobby afterwards.

Talk to Phil and he'll ask you to move the huge block. Push it until the phrase "it weighs a ton" appears then talk to Phil again. There'll be a cut-scene where Phil refuses to believe that Sora is a hero, while Donald and Goofy tries to convince him otherwise. Eventually, Phil decides to train you. You'll have to break all the barrels within the given time limit. 20 barrels within 30 seconds in the first round and 25 barrels within 1 minute in the second round. You'll get the Thunder magic after that.

He still won't let you enter the games so go outside. You'll see a short cut-scene with Hades and he'll give you the Entry Pass. Go back inside and show it to Phil. You can now enter the preliminaries. Make sure your characters are stocked with Potions and Ethers before you enter the tournament since you won't be given the chance to restock on items.

There'll be 7 matches to fight here. After the 5th battle you'll see a cut-scene between Cloud and Hades. You'll then fight your 6th battle followed by Cloud.

Boss Battle - Cloud

He's really fast and his attacks are powerful, therefore this is going to be a tough battle. But thankfully winning isn't necessary. Now Dodge Roll to avoid his dash attacks while trying get close to him at the same time. Once you're close enough keep attacking him from the side or better yet head on. Doing this will make him concentrate on blocking your blows rather than attack you. Be sure to move around a lot though to dodge his plunge attack. You'll win eventually.

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After the battle, Cerberus will show up and so will Hercules. When you get back to the Lobby, restock on items and SAVE your game then go back to the arena. It is also a good idea to customize your partners so that they don't do physical attacks often. (This will prevent them from trying to smack Cerberus silly and end up dying.)

Boss Battle - Cerberus

Never ever attempt to attack him head on; I'd bet you're suicidal if you did. Instead stay on the side and lock on to any head then attack. You can also try to jump on his back to get on top of him and start smashing. Avoid getting bitten at all cost as you definitely wouldn't want to see how much damage it can do. Watch out for his fire balls as well. When he starts releasing black stuffs after you, just Dodge Roll or jump to avoid it. Be sure to keep your partners alive if you can so that they can help with healing when you need it.

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After the battle, there will be a cut-scene and you'll receive the Hero License. Outside talk to Cloud for another cut-scene. You'll receive the Sonic Blade ability afterwards.