What are some counter-intuitive results in mathematics that involve only finite objects? 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?

Not sure whether this is the kind of thing you were expecting, but here goes:
Some statements about constructive mathematics can seem very counter-intuitive (at first, this is probably because one is misinterpreting what they mean), e.g.:


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*the induction principle holds, but on the other hand: that every non-empty (or inhabited) set of naturals has a smallest element is in general false

*given a set $A$, consider the statements: (i) "there is a finite set $B$ and an injection $A\to B$", (ii) "there is a finite set $B$ and a surjection $B\to A$". None of the statements imply each other or that $A$ is finite

