My Favorite Math Problems

Here are some great math problems for high school students. To make this list, a problem must be accessible and delightful, conducive to further exploration, but non-standard. Hints, solutions, and extensions are provided for each of them. However, no peeking at the hint until you've made an honest effort to solve the problem. Have fun with the mathematics!

...
Note: the phrase no three collinear means that any line in the plane will pass through at most two of the points at a time. We need this condition, for otherwise we could place all 101 points in a row, with point V on one end, to obtain a counterexample to the statement.
...

Note: a unit disc consists of all points inside or on a circle with radius one. Also, each segment must have positive length; no points are allowed. As usual, a tiling implies that none of the segments intersect and that every point is covered by exactly one segment.
...
Note: the three expressions appearing here are commonly called binomial coefficients. The first one on the left is usually read “n choose 3” and is often written as C(n,3) in text. It represents the number of ways to choose 3 objects from among a collection of n different objects. For example, C(4,3)=4. (Just count the number of ways there are to select three of the four fingers on your left hand.) The same sort of interpretation applies for C(n,4) and C(n+1,4). The goal here is to show that the two sides of the above equation are equal based solely on what they mean, without using any algebra.
......

I will gradually be expanding this list. The most recent addition was posted on 11/24/05. Feel free to Contact Us if you would like to suggest your own favorite math problem, but be forewarned that the editor is somewhat choosy about which questions to include.