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Into how many chunks can one cut a round piece of cheese with n straight cuts?

Consider the $3D$ version

My try:

f(x) = number of pieces and $'x'$ as number of cuts.

$f(1)=2$

$f(2)= 2 + f(1)$

and therefore

$f(n) = n + f(n-1) $ and after solving the recurrence relation I got

$f(n) = [n^{2}+n+2]/2 $

Is this correct?

max
  • 677
  • I don't understand what "straight" means in 3D. Do you means planar? – user202729 Apr 20 '16 at 07:25
  • Just Imagine this: If you are given a knife , how would cut a pizza or cheese? – max Apr 20 '16 at 07:26
  • Well, $f(3) = 8$, so this is not correct. Consider slice a $2\times2\times2$ cube into 8 pieces by 3 slices... – user202729 Apr 20 '16 at 07:26
  • No, there will be 7 pieces not 8 :/ – max Apr 20 '16 at 07:28
  • Why 7 pieces? Consider a cube $([-1..1],[-1..1],[-1..1])$. Slice it by 3 planes $xy, xz, yz$. You have 8 pieces correspond to 8 octants. You are considering 3D, don't you? – user202729 Apr 20 '16 at 07:31
  • I got it, I did it for 2D version. You are correct – max Apr 20 '16 at 07:33
  • How can I approach this for 3D? – max Apr 20 '16 at 07:33
  • Do the cuts have to pass through the center? If so, I count 6 pieces with three cuts... If not, 4, 5, 6 or 7 pieces are possible. Even 8 if you consider cuts in 3D, as @user202729 mentioned. – AugSB Apr 20 '16 at 07:35
  • There is nothing else mentioned in the question so we can assume that there is no condition – max Apr 20 '16 at 07:37

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