a) All vertices of the pyramid lie on the faces of the cube, but not on its edges, and at least one vertex lies on each face. What is the maximum number of vertices that the pyramid can have? 

b) All vertices of the pyramid lie in the planes of the faces of the cube, but not on the lines containing its edges, and at least one vertex lies in the plane of each face. What is the maximum number of vertices that the pyramid can have?
a) The section of the cube by the plane of the pyramid's base intersects all its faces and, therefore, is a convex hexagon. The vertices of the base lie on the sides of this hexagon, but not at its vertices. It is easy to see that if more than two vertices of the base lie on any one side, it is impossible to connect them with a non-self-intersecting broken line lying inside the hexagon. Therefore, the base has no more than 12 vertices, and the pyramid has no more than 13. The existence of a pyramid with 13 vertices is obvious. b) In this case, the vertices of the base can lie on the lines containing the sides of the hexagon, and their number can be arbitrarily large.

![](https://cdn.mathpix.com/cropped/2024_05_06_7dfbf12cfa0901bdf2bfg-49.jpg?height=915&width=1595&top_left_y=703&top_left_x=240)

## Answer

a) 13. b) Arbitrarily large.