I don't think this fits the specific criteria you've set out, but I believe it is in the right spirit. It involves using a mathematical concept all undergrads learn to give an elegant solution to a very down to earth puzzle that anyone can understand.

**Background**

The puzzle is about the game of SET. If you know how the game is played, don't bother reading this paragraph or the next. In the game of SET, you have a number of cards, and on each card there's a simple image. The image consists of 1, 2 or 3 identical shapes. These shapes can be one of three shapes: diamonds, squiggles, and ovals (on a given card, all the shapes are the same, but different cards may have different shapes). On a given card, all the shapes are coloured the same, in one of three possible colours: red, green, and purple. And finally, on a given card, all the shapes are shaded in the same manner, in one of three different possible manners: outlined, filled in, or hatched.

The game involves multiple players, and the goal for each player is to collect the greatest number of sets, where a set is a collection of three cards such that for each category (shape, number, colour, shading), all three cards are either the same or all different. So for example if you have three cards, all three of which are made up of 2 squiggles, but are three different colours and shaded in the three different ways, then this is a set. At the beginning of the game, the cards are all laid out, and then players form sets as quickly as the can until all the cards are gone, and the winner is the one who has made the most sets.

**The Puzzle**

Suppose you're playing Set, and as you're approaching the end of the game, you notice there are only 11 cards remaining. There ought to be a multiple of three remaining, so you figure one card must have gone missing from your deck. Looking only at the 11 remaining cards, can you figure out which card is missing?

Alternatively, draw up an example of some 11 cards, and ask the student to solve the same problem. When presented with a specific example, the student may not think that there is a general solution, and so making the problem more concrete actually adds an interesting twist to the problem, it makes it a nice test of analytical thinking.

**The Solution**

The set of all cards can be regarded as a 4-dimensional vector space over $\mathbb{F}_3$ in an obvious way. Why give it this vector space structure? Because a collection of three cards is a set iff, regarded as vectors in this vector space, their sum is the zero vector. Also, since you're supposed to be able to make sets until you've eliminated all the cards, you know that the sum of all the cards is 0. So:

0 = sum of cards put into sets so far + sum of remaining 11 cards + missing card

Clearly the sum of the cards put into sets is 0, and so:

missing card = -(sum of remaining 11)

And that's about it!

oneexample of such a phenomenon would be useful to people trying to answer this question. $\endgroup$1more comment