How many clips do I need to hang a rug?

It depends upon the weight and size of the rug.
A 12×18 feet rug with a thickness of 5mm generally weighs around 60 pounds. A rug of this size needs 4 Standard Rug Clips to hang on an arm. To calculate the weight of a 5mm thick rug, a simple formula is every square inch weighs 0.04 Ounce.

For standard sized rugs, a general formula is, for every 50 square feet, you need one heavy duty clip. 
For shag rugs that are pretty thick and heavy weight, you need extra heavy duty clips that have wide opening to engage the thick rugs and designed to bear a heavy load. If you are using extra heavy duty clips the 50 square feet to one clips formula stands well. Extra heavy duty clips are designed differently, and they are more resistant to distort. If you are trying to hang shag rugs with the standard rug clips (if you can get it engage in its jaws), the formula is 40 square feet to one clip. The more the number of clips the less is the chance of their deformity. 

If you are going to hang the clips on a string the better choice is wire-based rug clips. They are strong enough and stands well for the formula of one clip for 50 square feet.