In how many maximum/minium parts can a round cake($\large\text{ cylinder}$) be divided with $n$ cuts when each cut is necessarily a straight line?
What would be the case if we have a circle,sphere,toroid,etc
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The minimum amount seems to be trivial. With $n$ cuts you get $n+1$ pieces. – Darth Geek Jul 27 '14 at 18:02
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@DarthGeek Max?? – Hashir Omer Jul 27 '14 at 18:05
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The phrasing of your questions makes it unclear whether this is a 2-d or 3-d problem. Are we cutting with planes or lines? – JosephSlote Jul 27 '14 at 19:43
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The sphere case will be the same as the cylinder case if so. – JosephSlote Jul 27 '14 at 19:48
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I am cutting with planes – Hashir Omer Jul 27 '14 at 19:52
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I guess you need to specify if the cake slices can be rearranged after each cut. – knedlsepp Jul 28 '14 at 16:50
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Do they have to be vertical planes, or are horizontal planes and slanting planes allowed? – bof Oct 27 '14 at 06:55
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This is known as the lazy caterer's sequence, the number is $\frac{n(n+1)+2}{2}$. The idea is to use Euler's Formula $E-V+F=2$ and the fact that the number of edges increases $1$ more than the number of vertices in each line.
Asinomás
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However I am sure that the problem can be solved using a similar technique. – Hashir Omer Jul 27 '14 at 19:04
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1@HashirOmer The wikipedia article link also mentions the so called "cake number", which is the 3D equivalent you are looking for. – knedlsepp Jul 28 '14 at 18:00
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When $n=3$ your formula gives $7$ pieces. However, if the centroid of the cake is at the origin, the three coordinate planes cut it into $8$ pieces. Are you assuming some restriction on the cutting planes that is not stated in the question? – bof Oct 27 '14 at 04:10
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yes thank you, what I gave is for 2d, the formula for 3-d is given in kneldsepp's comment, the cake number. – Asinomás Oct 28 '14 at 03:44
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I'd like to say $2^n$ in dimension $n$ for $n$ cut, because you can always find a hyperplane (I consider a cut being a hyperplane) cutting all the ones already drawn... At least, with one cut (line cut, of course) you can only double the number of parts, so $2^n$ is an easy upper bound, and the previous argument should work properly written.
For the plane case (usual cake without horizontal cut), I'd like to say $2n$, but you can do much more, cutting a bit away from the center : with 3 cuts you get not 6 but 7 parts.
With one cut you can at best double the number of parts, but of course you should try to keep the maximum number of part intersecting one half of the cake (cutting implies not cutting in one half plane !)...
Well, merely some thoughts.
Desiderius Severus
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In case the pieces are allowed to be moved this is certainly correct. – knedlsepp Jul 28 '14 at 18:02
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is it possible to get 2^n pieces without moving pieces/changing their orientations? In 3 cut it simple to visualize that 8 pieces can be made(2 vertical cut and 1horizontal) but how to get 16 pieces from these 8 pieces and just using 1 more cut. I not able to think of any hyper-plane for this.() – Deepankar Singh Jul 30 '16 at 16:37