I recently managed to finish Pokemon Pokemon X and Y (I was playing the Y version) and was disappointed to not a lot of stuff in the post-game. So while searching around I managed to run in Moltres who ran away before the battle started. After doing some research it seems that players will be able to catch Moltres, Articuno or Zapdos after defeating the Elite Four depending on the starter they chose at the start.
The bird you’re able to catch is of the opposite type to your starter; so choosing Froakie gets you Moltres, Chespin gets you Articuno and Fennekin will net you Zapdos. Naturally I chose Froakie so I ran into Moltres.
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So catching your playthrough’s Legendary Bird is to Roam around Kalos after finishing the game (beating the Elite Four) and hope to encounter them in battle. It won’t be a proper battle as per say since the bird will escape immediately after you send your first Pokemon in. Now after the first encounter you’re gonna have to go to the Pokedex and track the bird’s location, now you have to go to the location and try to find them. You can use Super Repels (not Max Repels since Super Repels are the most cost effective) and it won’t stop the encounter. You’ll have to encounter the bird around 10 times before they permanently move to Sea Spirit’s Den.
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To get to Sea Spirit’s Den you need to Coumarine City and take the monorail to the seaside. Now you need to head west to Router 12 and Surf across the water and keep going north until you reach an area called Azure Bay. Now we’re gonna go west a bit till the north area opens up, then surf northwards again while avoiding the rocks along the way. Eventually you’ll reach the cave and here is where you should save your game.
After ending the cave either Moltres, Articuno or Zapdos will sweep down and start the battle. All three birds are level 70 so make sure you’re properly prepared. Moltres’ move set includes Safeguard, Sunny Day, Air Slash and Heat Wave; Articuno’s move set includes – Reflect, Hail, Tailwind and Ice Beam while finally Zapdos’ move set includes Agility, Light Screen, Rain Dance and Discharge. I’d recommend using Dusk and Timer balls in this battle but the decision is up to you. Good luck!