Use the menu to navigate. Click the pull button to spin the slots. Try as many times as you want. If it says to wait for them to finish spinning and they have, just click pull again and it should work. If you get a matching pair, you get that egg! If you get a match of all three, you can choose any egg! Go to the nest and click the egg to hatch it. The nest can only hold one egg at a time, so if you don't hatch an egg before getting a new one, you'll lose it! All by me (@EvelynJ).
