Least wasteful use of stamps to achieve a given postage 
You have sheets of $42$-cent stamps and
  $29$-cent stamps, but you need at least
  $\$3.20$ to mail a package.  What is the
  least amount you can make with the $42$-
  and $29$-cent stamps that is sufficient
  to mail the package?

A contest problem such as this is probably most easily solved by tabulating the possible combinations, using $0$ through ceiling(total/greater value) of the greater-value stamp and computing the necessary number of the smaller stamp and the total postage involved.  The particular example above would be solved with a $9$-row table, showing the minimum to be $\$3.23$, made with seven $42$-cent stamps and one $29$-cent stamp.
Is there a better algorithm for solving this kind of problem?  What if you have more than two values of stamps?

A similar problem is known in mathematics.  It is called the "Postage-Stamp" problem, and usually asks which postal values can be realized and which cannot.  Dynamic programming is a common but not polynomial-time solution for this.  A polynomial time solution using continued fractions in the case of two stamp denominations exists, and more complex algorithms exist for three or more stamp denominations.  Look up "Frobenius Number" for more technical information.
For the given problem, I would roughly estimate the Frobenius number, and based on the estimate make an easy guess as to a near-solution and then use a dynamic programming/tabular solution, depending on the amount of time I had to solve the problem.
