Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I am interested in theorems with unexpected conclusions. I don't mean an unintuitive result (like the existence of a space-filling curve), but rather a result whose conclusion seems disconnected from the hypotheses. My favorite is the following. Let $f(n)$ be the number of ways to write the nonnegative integer $n$ as a sum of powers of 2, if no power of 2 can be used more than twice. For instance, $f(6)=3$ since we can write 6 as $4+2=4+1+1=2+2+1+1$. We have $(f(0),f(1),\dots) = $ $(1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,\dots)$. The conclusion is that the numbers $f(n)/f(n+1)$ run through all the reduced positive rational numbers exactly once each. See A002487 in the On-Line Encyclopedia of Integer Sequences for more information. What are other nice examples of "unexpected conclusions"?

share|improve this question
Very interesting question! But since it has no right answer, and you are asking for a big list, I think it should be community wiki. –  Grétar Amazeen Mar 13 '10 at 21:32
I'll send you a shirt: thenerdiestshirts.com/blog/math-shirt-cw/#more-116 –  Douglas Zare Mar 13 '10 at 21:48
Thanks for great shirt! I guess it's difficult to make precise the difference between "unintuitive" and "unexpected conclusion." Some examples are at math.dartmouth.edu/~pw/solutions.pdf. Another nice example is the cake icing problem (demonstrations.wolfram.com/TheCakeIcingPuzzle). –  Richard Stanley Mar 13 '10 at 22:47
Stolen from Kevin Buzzard's comment at mathoverflow.net/questions/15050/linear-algebra-problems: If A and B are real n x n matrices, A^2+B^2=AB, and AB-BA is invertible, then n is a multiple of 3. –  Jonas Meyer Mar 13 '10 at 23:14
add comment

closed as no longer relevant by Felipe Voloch, Bill Johnson, Andres Caicedo, Mark Sapir, Ryan Budney Jan 5 '12 at 21:50

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

35 Answers

Polya's Theorem: Simple random walk on $\mathbb{Z}^d$ is recurrent for $d\leq2$ and transient for $d>2$.

There is also a nice connection between this theorem and infinite networks of resistors. It turns out that the resistance of the whole network $\mathbb{Z}^d$ (one puts a unit source in one point and takes away sinks to $\infty$) is finite iff corresppnding random walk is transient :)

share|improve this answer
add comment

The classical differential geometry results should definitely be mentioned here. Although it may seem not surprising for us, Gauss found his Theorema Egregium to be truly remarkable and unexpected. My favorite example is Gauss-Bonnet Theorem.

share|improve this answer
add comment

In the strict spirit of your question, the hypothesis that 1 = -1 has as conclusion that the moon is made of green cheese.

share|improve this answer
show 2 more comments

The first time I ever saw Cayley's Fundamental Theorem of Group Theory - i.e. every group is isomorphic to a group of permutations on a nonempty set - I was floored and I knew anything that contained a statement that bizarre that's true was something I wanted to do in life.

share|improve this answer
add comment

Not the answer you're looking for? Browse other questions tagged or ask your own question.