From Pizza Cut
, we know the formula for maximum partitioning (pieces) of the circle, given n
straight lines (cuts).
- Determine the formula for the maximum number of pieces if you cut a crescent moon (think capital letter 'C') with n straight lines.
- Determine the formula for the maximum number of pieces of cheesecake (a cylinder) produced by n plane cuts.
(In reply to re: I think this is right.
My first three cuts happened to be all normal to each other Ė kind of like coordinate axes except planes (they don't have to be exactly that way, though). In other words, my first two cuts were like I was cutting the cheesecake like a "normal" person, and then my third was horizontal. Thatís how I had a total of 1, then 2, then 4, then 8 pieces of cheesecake after my first three cuts (if I didnít make any horizontal ones, it would have been like pizza problem: 1, then 2, then 4, then 7 pieces).
My fourth cut, though, couldnít cut through all 8 pieces again to make 16 (doubling the number of pieces). If the fourth cut went through the intersection point of the first three planes, I would be missing at least two pieces. If I move that cut a little so that itís not going through the intersection point of the first three planes, I get a little better: I now cut through one of the pieces I missed before, but Iíve abandoned the other one. So I was able to cut at most 7 pieces, not 8, making a total of 15 pieces (not 16).
Then I ran out of brain RAM and couldnít visualize the next cut so I hoped my pattern would hold for future cuts. So my formula could be wrong, but I donít think I was wrong that the number of pieces doesnít just keep doubling in the cheesecake problem.
Posted by nikki
on 2004-08-26 08:03:02