1. [3] How many different values can $\angle A B C$ take, where $A, B, C$ are distinct vertices of a cube?
Answer: 5 . In a unit cube, there are 3 types of triangles, with side lengths $(1,1, \sqrt{2}),(1, \sqrt{2}, \sqrt{3})$ and $(\sqrt{2}, \sqrt{2}, \sqrt{2})$. Together they generate 5 different angle values.