10.64 On the plane, what is the minimum number of points needed so that the distances between each pair of points can take on each of the values $1,2,4,8,16,32,64$?
[Solution] Take 8 points $A_{1}, A_{2}, \cdots, A_{8}$, such that $A_{1} A_{2}=1, A_{2} A_{3}=2, A_{3} A_{4}$ $=4, \cdots, A_{7} A_{8}=64$ to meet the requirement. It is clear that the minimum number of points required is no more than 8.

Take 7 line segments with lengths $1,2,4,8,16,32,64$ respectively. Clearly, no combination of these line segments can form a closed polyline. Therefore, when they are connected end-to-end as much as possible, the number of endpoints is always one more than the number of line segments, i.e., there are at least 8 different endpoints.
Thus, a minimum of 8 points are required to meet the requirements of the problem.