TITLE: What are some counter-intuitive results in mathematics that involve only finite objects?
QUESTION [231 upvotes]: There are many counter-intuitive results in mathematics, some of which are listed here. However, most of these theorems involve infinite objects and one can argue that the reason these results seem counter-intuitive is our intuition not working properly for infinite objects.
I am looking for examples of counter-intuitive theorems which involve only finite objects. Let me be clear about what I mean by "involving finite objects". The objects involved in the proposed examples should not contain an infinite amount of information. For example, a singleton consisting of a real number is a finite object, however, a real number simply encodes a sequence of natural numbers and hence contains an infinite amount of information. Thus the proposed examples should not mention any real numbers.
I would prefer to have statements which do not mention infinite sets at all. An example of such a counter-intuitive theorem would be the existence of non-transitive dice. On the other hand, allowing examples of the form $\forall n\ P(n)$ or $\exists n\ P(n)$ where $n$ ranges over some countable set and $P$ does not mention infinite sets would provide more flexibility to get nice answers.
What are some examples of such counter-intuitive theorems?

REPLY [3 votes]: $()()$ is not a palindrome but $())($ is.
Intuition tells us that if we can put a mirror in the centre and it reflects, then it's a palindrome.  But because the mirror exchanges left brackets for right ones, our intuition deceives us in this particular instance.