Your mention of modular arithmetic makes me draw a connection between this story and the way I think about ceiling fan pull chains. Suppose the fan is clearly

Your mention of modular arithmetic makes me draw a connection between this story and the way I think about ceiling fan pull chains. Suppose the fan is clearly starting on high and you want to turn it off, but you don't know if the order of settings is "high-low-off", or "high-medium-low-off" (three states or four). It's often difficult to tell if the fan is coasting off or just spinning down to low. If you get it wrong and pull one time too many, it goes back to high and you have to start over. Therefore I always pull fan chains eleven (11) times to turn them off, so that I don't have to know if it's a three or four state fan. (Also works for on-off pull chains.) Pulling the chain 59 times would extend this technique to cover five-state fans, but I've never encountered one.