Get 3 swans on any level to unlock Time Attack. Get the best time (#1 on leaderboard) on Time Attack on Bromley Stage 5 to win the free swan. Just post a screenshot of your leaderboard on Facebook.
Find the leaderboard for the screenshot by clicking on the golden cup image in the top right corner of the game home screen. In the Leaderboard's menu scroll to "Bromley Stage 5" and click on it. If the #1 spot is occupied by "Me", take a screenshot and send it in.
See example images below:
Google images look slightly different as shown below:
You must have at least one friend's name in your screenshot with "Me" (so that we can tell its your unique screenshot) so be sure to sign in to the game center on your device.