What the best way to remove gum from the carpet?

Use an ice cube at a time and press the cube into the gum. Tackle each corner at a time and as the ice melts and becomes less effective rush back to the freezer to fetch a fresh one. Then continue on the next little corner. Use an eating fork (which is much stiffer) to scratch out the little bits of gum that have now become brittle from the ice. Wipe them away with a damp sponge.