You seem to have some fundamental misunderstandings of iOS programming paradigms. You have segues attached to your buttons, but you also then call performSegue in code. When you have a segue attached to a control, you don't need any code (and shouldn't have any) to cause the segue to execute. You also shouldn't go back to a previous view controller with a segue, other than an unwind segue; you're not really going back, you're creating a new instance of the controller you think you're going back to. This will cause a build up of controllers (as none will be deallocated) until your app runs out of memory.
So, you should delete the function, letsPlayButton: from MainScreen, and also get rid of it in the storyboard (the segue attached to that button is all you need).
Delete the segue you have going "back" from GameScreen to MainScreen, and change the code in backToMainScreen to this,
@IBAction func backToMainScreen(sender: AnyObject) {
self.dismissViewControllerAnimated(true, completion: nil)
}
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