Sketch the mascot onto all sides of the Styrofoam block with a pencil or marker. Give yourself an idea of the general shape.
Make the first and largest cuts with a hack saw. The largest cuts will most likely involve the corners of the block.
Take smaller, medium-sized chunks off with a floral knife. Move from side to side after one or two cuts, rotating around the block. Take less than you think you need and return to that side if need be.
Cut into the block with an art knife or box cutter to make details. This is where you'll begin to see things such as a nose or a mouth. Replace the blade every eight cuts or so when it gets dull.
Smooth the foam with sandpaper. Start with coarse sandpaper and move to a fine-grit paper to give it a polished look.
Paint the mascot with acrylic paint. Acrylic paint dries as a plastic, dries quickly and will be permanent. Paint the figure's details, including a logo somewhere on the body.