Drain olives and place them on a paper towel to dry.
In a small bowl, combine butter and shredded sharp cheddar with a wooden spoon. Add flour, paprika, salt, and pepper. Mix as well as you can with the wooden spoon, and then finish by combining with your hands. It will turn into a moist, dense dough that you can easily form into a ball.
Divide the large ball of dough into 18 smaller portions. I like to use a small 2-teaspoon scoop, tightly packed. Roll each portion into a ball and place it in the palm of one hand. With the thumb of your other hand, press the dough ball to flatten it. Place an olive in the center of the flattened dough and mold the dough up around the olive. Give it a gentle roll between your palms to finish. Place olive cheese balls two inches apart on an ungreased baking sheet. Cover tightly and refrigerate for 1 to 2 hours.
Preheat oven to 400°F. Place chilled baking sheet in oven and bake for about 15 minutes, or until lightly browned. Let cool on the baking sheet for 5 to 10 minutes before serving. These are best when eaten while still warm. Serve with homemade buttermilk ranch if you like to dip!
