Creative Thinking Questions?
Math is often intimidating to the average man due to its complex appearance. To show that math requires creative thinking, not just memorization, I was wondering if anyone had any math problems that required some out-of-the-box thinking.
I am looking for something that can be solved with just basic math skills. Please limit problems that can be solved with high school math. I do not want a problem with a complex solution, just one that requires a lot of thinking to solve.
Do there exist infinitely many primes?
I think the fly and two trains problem is a good example of a creative solution accessible to those with basic math skills.
What I think of as the "standard examples" in this area:
The simple reason why you can't tile a chessboard with opposite corners removed with dominoes (what Wikipedia calls the mutilated chessboard problem).
The issues connected to the bridges of Konigsberg problem and its generalizations.
Since I don't like the Wikipedia entry for the second thing as much: the basic problem is: given a (connected) graph, to determine whether or not there is a path in the graph that goes over each edge exactly once (what is now called an Eulerian path). In a small graph, if such a path exists, one can easily be found by trial and error (even very young children can do it). But when one cannot find such a path by trial and error, at first glance, it is very difficult to understand (let alone prove) that there aren't any.
Euler's insight was that the answer depends only on the degrees of the vertices in the graph: if a graph has more than two vertices of odd degree, it cannot have such a path. And if you think about it a while, this becomes "obvious", although it is far from obvious if your only experience with the problem is using pen and paper to look for Eulerian paths.
(I should say: it's significantly harder to show that that if a connected graph has two or fewer vertices of odd degree, then it must have an Eulerian path. But with any graph that is small enough to draw in a short amount of time, the truth of this statement can at least be checked by hand.)