Dividing Spaces with Lines and Hyperplanes

Requirements Clarification & Assessment

When preparing to answer the interview question regarding dividing spaces with lines and hyperplanes, it's crucial to understand the requirements and assumptions inherent in the problem:

1. Understanding the Problem Context:

   • The interview question revolves around determining the maximum number of regions that can be formed by 4 lines in a 2D plane and by 4 hyperplanes in a 3D space.
   • It's important to clarify whether the lines/hyperplanes are assumed to be in general position, meaning no two lines are parallel, and no three lines intersect at a single point in 2D. Similarly, for 3D, no two hyperplanes are parallel, and no three intersect along the same line.

2. Mathematical Formulation:

   • Recognize that this is a combinatorial geometry problem, and it involves understanding the combinatorial relationships between lines and hyperplanes.

3. Assumptions Made:

   • No lines or hyperplanes are parallel.
   • In 2D, no three lines intersect at the same point.
   • In 3D, no three hyperplanes intersect along the same line.

4. Key Concepts Required:

   • Familiarity with combinatorial mathematics and binomial coefficients.
   • Visualization of geometric division in both 2D and 3D spaces.