My favorite mathematical embarassment: what is the chromatic number of the plane? That is, what is the minimum number of colors needed to color each point in the plane such that no two points that are a distance of 1 apart are the same color? It is known that the answer is 4, 5, or 6.
As far as I know, there is a "proof" of the 4 color map theorem, but it requires the use of a computer to analyze about a 1,000 different scenarios, so its hard to check, unintuitive, and "not elegant."
This is not the problem described by the post you are responding to. Search for "chromatic number of the plane" or "Hadwiger-Nelson problem" to find clarification.
Why should the Goldbach Conjecture be included? Just because a problem's formation is easy for many people to understand doesn't mean it is an easy problem that should have been solved. The Goldbach Conjecture would be better classified as what that blog calls a 'mathematical disease', a legitimately difficult problem that infects the attentions of many mathematicians. I'm not sure the Goldbach would even qualify for this because I feel like it's famous more for attracting the attention of pseudomatheticians than genuine ones.
