Three bags with different named labels. A t-shirt or jumper. Of course, you can dress this problem up in many different ways so if you have got a t-shirt or jumper to hand you could have three pencil cases with blue and yellow pencils, for example.
En route to China our luggage got mixed up. Each of our bags look the same, but the name tags have all been mixed up such that we know that no bag has the correct label (a key point to reiterate). We are allowed to put our hand in a bag of our choice and pull out an object at random (we can look in the bag). Can we identify which bag belongs to each person, from this single piece of information?
You should ensure that the audience understands that we are looking for a definite strategy. We do not need to consider probabilities at all.
To be honest, I cannot think of one. This question is pretty much self contained. However, the idea behind this question, i.e. logical deduction, is one of the crucial weapons in the mathematical arsenal. If you have any suggestions for extensions please write them in the comments below.