29. Space Division. Through a fixed point in space, we draw planes to divide the space into as many parts as possible. One plane will divide the space into two parts, two intersecting planes will divide it into four parts, and three planes intersecting at a point and having no other common point will divide the space into eight parts. What is the maximum number of parts that can be obtained with four planes? And what about with $n$ planes?
29. Instead of the entire space, we will divide the tar, through the center of which we draw planes. On the surface of the sphere (bounded by its sphere), intersecting great circles will arise. We will take one of them as the әquator and project all other circles from the center of the sphere onto a plane tangent to the sphere at the pole. The projections of our circles (except for one, which is the әquator and does not project at all) will be straight lines. Therefore, we need to calculate the maximum number of regions on the plane divided by \( n-1 \) lines. By induction, it is easy to obtain that this number is \( 1+1+2+3+\ldots+(n-1) = 1 + \frac{1}{2} n(n-1) \), because with \( k-1 \) existing lines, the \( k \)-th line can increase the number of regions by at most \( k \). Since there are twice as many regions on the sphere as on its planar projection, the desired number will be twice the one we calculated above, i.e., it is \( n(n-1) + 2 \). In particular, for \( n=4 \), the desired number is 14.

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-081.jpg?height=472&width=1146&top_left_y=1534&top_left_x=478)

Fig. 38.