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. 
 A: I think the fly and two trains problem is a good example of a creative solution accessible to those with basic math skills.
A: 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.)
A: This question is much too vague and too vast to admit a meaningful list of answers.
(What are basic math skills? What is a complex solution?)
I suggest that you look at the site
http://www.artofproblemsolving.com/Forum/index.php
where you can find nice problems of different difficulties that only use high school mathematics.
A: How about calculating the sum of the first n integers, and (the legend of) Gauss' solution?
A: I think the proof of Cantor's Theorem is simple enough to show the creativity found in mathematical arguments without being too deep.
A: 
Say I have a square. I pick 3 random points on the perimeter of the square, each on a different side. What are the odds that the resulting triangle contains the centre of the square?
Can be solved (possibly sacrificing a little rigor) without a single line of calculation.
A: Project Euler has a number of both interesting and challenging problems. The discussion board of solutions that opens up after you solve a problem is also insightful and full of creative solutions: http://projecteuler.net/
A: 
Do there exist infinitely many primes?

A: Try 
The Penguin Book of Curious and Interesting Numbers: Revised Edition (Penguin Press Science) [Paperback]
David Wells (Author) 
and other books by him. 
June 7: Here is a puzzle with a solution by Dirac. 
5 sailors are shipwrecked on a South Sea Island, and before they go to sleep they collect a pile of bananas to share out in the morning. One of then wakes up, does not trust the others, so divides the pile into 5 equal parts, with one left over, which he gives to the monkey, and then takes his own part for himself. This process repeats with the other sailors. In the morning, they find the pile divides into 5 equal parts, with one left for the monkey. 
Problem: How many bananas were there in the initial pile? 
Easy solution by Dirac: Start with $-4$ bananas. With one for the monkey, that makes $-5$ which divides into $5$ equal parts;  take away $-1$ for the sailor leaves $-4$, as before. So the process repeats. To get a positive solution, add $5^5$. (No wonder the problem was difficult to solve directly!) 
As an exercise for students, you can ask them to generalise the problem, since that is what mathematicians do. 
A: The social Golfer Problem.
Kirkmans Fifteen School Girl problem.
The motorcycle Problem.
The desert exploration problem.
Should Phil chop his big toe off problem ?
A: This might look advanced but trust me, simple reasoning and logic gives you a beautiful answer.
A permutation of first $N$ numbers is said to be good if $x$ and $x+1$ do not occur consecutively.
For $N=3$, they are
$(1,3,2)$, $(2,1,3)$ and $(3,2,1)$, and the number of such permutations are $3$.
Find the number of ways for any $n$.
A: There is no greatest natural number (or $\mathbb{N}$ does not have an upper bound in $\mathbb{R}$).
It seems daft but a lot of non-mathematicians have argued with me about this.
A: $$ \lim \sqrt[n] n = 1$$
Requires fairly creative thinking to apply the Squeeze Theorem. $\sqrt[n] n = 1 + t_n$ where $t_n \ge 0$ for each $n \in \Bbb N$. Then $n = (1 + t_n)^n = 1 + nt_n + \frac 1 2n(n - 1)t_n^2 + ... + t_n^n \implies n \gt \frac 1 2n(n - 1)t_n^2$ $\implies 0 \le t_n \lt \sqrt {\frac 2 {n - 1}} \implies \lim t_n = 0 \implies \lim(1 + t_n) = \lim \sqrt[n] n = 1$
