Simple & Intuitive Statements that are Difficult to Prove

Question

Looking through the webcomic, I came across one of my favorite comics:

^{(from Saturday Morning Breakfast Cereal)}

It seems that people have an ongoing interest in results in mathematics that are true, but highly unintuitive, like the Banach-Tarski Paradox. However, what results are there that are seemingly obvious and intuitive, but difficult to prove (or perhaps have non-obvious intricacies)?

I feel that such examples are important for helping people understand the necessity of rigor or that seemingly obvious results are not at all obvious from a mathematical perspective.

(Personally, I don't feel that 'simple' number theory conjectures/results like the ABC conjecture fall under this category, because while simple to state, they are often very removed from reality.)

Answer 1

The Jordan curve theorem asserts that every a non-self-intersecting continuous loop divides the plane into an "interior" region bounded by the curve and an "exterior" region.

Another nice example is the P vs NP problem, which basically says verifying is easier than finding solutions. But it is still unsolved , one of the Clay Millennium problems.

Answer 2

Prove the reflexive property, or that $x = x$.

http://www.tondering.dk/claus/sur16.pdf

Crazy stuff.