Optimal Cuts

I was reading a problem in Brilliant in which you had to tell the maximum quantity of pieces you could divide an orange into with 3 straight cuts, and then compared with the slicing of objects with less spatial dimensions: a circle and a line segment.

(From now on, each time I refer to "cuts" I mean "straight cuts", and when I refer to "dimensions" I mean "spatial simensions", unless I explicitely state otherwise)

I realized the following:

  • With 3 cuts, I could divide the orange into 8 pieces by duplicating the quantity of pieces each time. However, that wouldn't work with 4 cuts: I am able to cut all the pieces of the sphere in half with each cut in the first three cuts, but it becomes impossible to do so at the 4th.

  • With 2 cuts, I could divide a circle into 4 pieces, but I couldn't divide the circle into 8 pieces with a third cut.

  • With 1 cut, I could divide a line segment into 2 pieces, but the second cut would only allow me to divide one of the pieces in two.

The first thing to notice is that the best cut you can ever do will duplicate the quantity of pieces the object was into. The second thing is that, in all three situations, the last cut which could duplicate the ammount of pieces was the nth, being n equal to the number of dimensions of the object.

Is this a coincidence?

(To not complicate things up, let's assume the tool used to cut the objects and the entity cutting them have more dimensions than the objects themselves)

Let's suppose we want to cut an n-dimensional hypersphere (also called (n-1)-sphere) into the maximum number of pieces possible with n cuts. How would we do it?

We could consider the centre of the hypersphere the centre of a n-dimensional referential, which would have n axis, each one perpendicular to all the others. By cutting through a (n-1)-dimensional object perpendicular to each axis and which intersects the hypersphere, one would be able to divide each of the previous pieces into two. The result would be 2^n pieces, given that the number of pieces duplicated with each cut, starting with one object, and n cuts were made.

I don't know how to find a proof for any dimensions quantity which tells me that the (n+1)th cut can never duplicate the number of pieces, though.

Feel free to contribute.

Note by Anonymous Person
6 years, 10 months ago

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Actually that was my problem.

Anuj Shikarkhane - 6 years, 10 months ago

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Do you know how to solve the problem:

What is the maximum number of regions that nn lines can divide a disk into?

I used "disk" to denote that it is a filled circle, as opposed to just the perimeter.

You can use similar ideas, to approach this higher dimensional version.

Calvin Lin Staff - 6 years, 10 months ago

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