Assume you have a collection of perfect, identical, solid cubes. Choose one of these cubes to be in the center. Now your task is to arrange the other cubes in such a way that as many as possible are touching the center cube. "Touching" here is defined as sharing some finite amount of face to face contact; contact at edges or corners does not count.
Over the years that I gave this problem to my math students, many of them found a solution that I call the "very good" solution. (I received some lovely alphabet block models of the "very good" solution.) But, it turns out, the "very good" solution is not the best solution. Only one of my students, sixth grader Andrew L., ever found the optimal arrangement of cubes, and he found it several months after the problem was assigned! He won the prize for most persistent puzzle solver ever.