This question already has an answer here:
When seeing groups and such for the first time, the abstraction often seems pointless and unnecessary to students. (Most students at my school leave their introductory abstract algebra class thinking it was pointless and unnecessary, too.)
What are some nice questions that can be used during the class to motivate the development of groups (or rings), that groups solve nicely? I'd rather Galois theory wasn't used as an example - it takes a bit too much work and wouldn't fit into the type of introductory course I'm thinking of.