TITLE: Minimum possible value of change
QUESTION [7 upvotes]: My physics professor loves giving little logic puzzles at the end of our homework...
This weeks was “Sheila is a cashier at a convenience store and has run out of one-dollar bills, but she has plenty of coins - pennies, nickels, dimes, and quarters. She always gives the change with the fewest possible coins. The change she just gave a customer contained 20 coins. What is the minimum possible value of the change?” 
I’m asking this question here because a friend and I are disagreeing as to what the answer should be. Our general strategy is the same: we start by saying she gave 20 pennies since this is the least possible value of 20 coins. Then we note that this is impossible because Sheila wouldn’t give more than 4 pennies because she could just give a nickel instead (condition of least coins given). So now we have Sheila giving 4 pennies and 16 nickels. But this can’t be either because she would never give more than a nickel since she could just replace 2 nickels with a dime. And so on...but we’re getting a bit confused and have dissenting opinions when it comes to the dimes. 
Could someone help us out here? 
Thanks!

REPLY [9 votes]: [Glossary for those who need it: a nickel is 5 cents, a dime is 10 cents and a quarter is 25 cents.]
We can't have 5 cents in the solution (because we could replace them with a nickel), so we have at most 4 cents. So we must actually have exactly 4 cents, because otherwise we could replace a larger coin by a cent and get a better solution.
If there is a nickel, there is exactly one nickel because 2 nickels can be replaced by a dime. In this case, there is also at most one and hence exactly one dime (because 2 dimes and a nickel can be replaced by a quarter, while if there are no dimes, a quarter can be replaced by a dime).
If there is no nickel, then there can be at most 2 dimes (because 3 dimes can be replaced by a nickel and a quarter).
So we have EITHER:

4 cents, 1 nickel, 1 dime and 14 quarters, OR 
4 cents, 2 dimes and 14 quarters. 

So option 1 wins, with the solution as $3.69. (And Ned's guess was right.)
Many thanks to Jens for pointing out an error in an earlier version of this answer.