In how many different ways can 12 dimes be divided into three piles with an odd number of dimes in each pile?
To divide 12 dimes into three piles with an odd number of dimes in each pile is to express 12 as a sum of three odd numbers. This is impossible to do, since $12=2\cdot6$ is even and the sum of three odd integers is odd: $(2l+1)+(2m+1)+(2n+1)=2l+2m+2n+3=2(l+m+n+1)+1$. Therefore, there are $\boxed{0}$ ways to divide the dimes as described.