I originally posted this in the pre-university geometry forum without much luck, and since it's a puzzle I'll repost it here and hope that someone will take up the challenge!
Basically, I wanted to see what the maximum number of areas I could divide a circle into with N number of straight cuts -- basically, drawing a circle on a sheet of paper and then using a ruler to draw straight lines through it. After that I worked out the same thing but for cutting up a sphere with N 2D planes -- or in other words cutting up a cake with a large, straight knife, where you may not move the pieces between cuts.
I managed to get these right and have confirmed it with other sources (see links below). These two work as great puzzles in their own right, if anyone wants to give them a try before looking at the answers.
For cutting circles, see: Circle Division by Lines -- from Wolfram MathWorld
And for cutting spheres (or in this case cubes), see: Cube Division by Planes -- from Wolfram MathWorld
Lastly (and the one I'm currently stuck on), I want to work out the maximum number of areas N three-dimensional planes can divide a 4D hypersphere into.
I have an answer, but it doesn't make much sense and I'm already out of my depth, so I was hoping someone could help me out. The answer I've come up with is as follows:
I hope you have fun with these puzzles and are able to help me out with the more difficult 4 dimensional question. Thanks!