Into how many pieces can you cut a four-dimensional cake with four slices? How about an N-dimensional cake with N slices?

kaylofgorons

If you take a two dimensional cake and use two slices, you get three pieces, and if you assume they are equal, you get thirds. 1/3. If you take a three dimensional cake and cut it three times, you get four pieces or fourths. So if you take it that way... a four-dimensional cake can be cut if fifths. And a N-dimensional cake is cut in peices 1/N+1. Other than that, I have no clue. Apr 01 05, 7:45 PM

kaylofgorons

But wait, there's more. If you take a three dimensional cake, cut twice from top to bottom, you get thirds, then once from one side to the other, you now have sixths--doubling the amount possible. And the same for the two dimensional one, it can be cut in quarters. O_o So from this way, an N cake seems to be cut to 1/ 2N... I'll be working on this one all night, because I can't really test it on another dimension! Apr 01 05, 7:48 PM

kaylofgorons

That's not doubling the amount possible, scratch that. It's getting two pieces for each dimension, or something like that... Apr 01 05, 7:50 PM

gmackematix

With each increase in dimension, you can slice at right angles to all the previous cuts and so slice each of the preceding pieces into two. So a one-dimensional "cake" is cut into two by a point. A 2D cake (such as a square one) can be cut into four by two lines. A 3D cake (such as a cubic one) can be cut into eight by three plane cuts. A 4D cake will be cut into 16 pieces by 4 3D cuts. An N-dimensional cake will be cut into 2 to the power of N (2^N) pieces by N cuts. Apr 01 05, 8:10 PM

kaylofgorons

Phew, now I can get some sleep! Apr 01 05, 10:15 PM

SOTHC

Surely it would take time to cut a four dimensional cake Apr 02 05, 3:10 PM

ogicu8abruok

That was an excellent explanation. Apr 03 05, 1:01 PM

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