One coin among $N$ identical-looking coins is a fake and is slightly heavier than the others, which all have the same weight. To compare two groups of coins you are allowed to use a set of scales with two pans which balance exactly when the weight in each pan is the same. What is the largest value of $N$ for which the fake coin can be identified using a maximum of two such comparisons?
If you liked this problem, here is an NRICH task
that challenges you to use similar mathematical ideas.
This problem is taken from the UKMT Mathematical Challenges.