"A problem Seminar" by Donald J Newman has many excellent problems which require math undergrad/grad knowledge. Below is one of the more accessible problems, solveable by a non-math person.
Each coin is either 10 or 9 grams. Given 4 such coins, find the weight of each using a scale (not a balance, a scale which gives the true weight) no more than 3 times.
Scroll down for a solution.
Suppose the coins are A,B,C,D.
Weigh A+B. It must be 19, otherwise it is trivial. Now weigh A+C. It must be 19, otherwise it is trivial.
Now we must have B=C. Now weigh B+C+D = 2B+D. This has the same parity (odd or even) as D. Thus we know D, which gives us B=C and then A.
