I started by purchasing and taking apart a Magic 8 Ball.
Many people on the Internet will tell you that this isn't that difficult. "Run a knife around the seam and then gently pry apart," they'll say. "Use a flathead screwdriver and plexiglass knife to gently separate the two halves," they'll say.
The Internet is full of lies.
I spent two days using various knives and razor blades on this thing. Finally, I broke out my Dremel tool. Or I should actually say I broke my Dremel tool. Only four of the ceramic cutting wheels, mind you, but I broke four ceramic cutting wheels getting this thing apart. Mattel does not want you getting inside.
Check out the image: there's half a centimeter of plastic to cut through here. Anyone who does this with a razor knife is either more patient than I am or a stupendous badass with arms of steel.
Once I cut through it enough, the halves separated cleanly.