All iPads Removing games question

Depends on the game. For a game that uses GameCenter syncing, it'll restore your progress as soon as you log in. (for instance, Infinity Blade II and something like Hundreds both do this, and in fact, when I installed them new on my iPad, they both updated and started right where I currently was on my iPhone (with all levels and such the same).

However, other apps don't do this, and store the data locally. If you just reinstall, your progress will likely be gone (though some seem to keep your save game data on the device when you delete it or restore from a backup). If you restore your phone from a backup with the game, all your data will still be there.

 

I just want to add that it's not just GameCenter syncing, but rather iCloud syncing. To check if the game supports iCloud syncing, go to Settings>iCloud>Storage & Backup>Manage Storage>Documents & Data to see if your game is listed there.