[Haskell-cafe] Harder Question - Chapter 4 - 4.12 - haskell the craft of functional programming - Second Edition
Anton Nikishaev
anton.nik at gmail.com
Tue Jul 9 14:06:26 CEST 2013
Manoel Menezes <manoel.menezes.jr at gmail.com> writes:
> Hi everybody!
>
> I am trying to solve the question for a long time:
>
> [4.12 Harder] Find out the maximum number of pieces we can get by
> making a given number of flat (that is planar) cuts through a solid
> block. It is not the same answer as we calculated for straight-line
> cuts of a flat piece of paper. I find out that this function has
> the following results:
>
> f 0 = 1
> f 1 = 2
> f 2 = 4
> f 3 = 8
>
> That is, from 0 to 3, the flat cuts all the pieces in two other
> pieces, so the number of pieces is doubled.
>
> But, starting from f 4, the flat can not cuts all the pieces, in case
> of f 4, the flat can cut 6 out of the 8 pieces, resulting in 12 pieces
> plus 2 pieces 2 = 14 pieces.
It can cut 7, so it is 2*7+1.
Forget about block borders and suppose you have n-1 planes and add a new
place. It will cut each n-1 planes. Now look at this new plane with
intersection lines (n-1 ones) in it. Each region in plane adds one new
3D piece, so
F(0) = 1;
F(n) = F(n-1) + P(n-1),
where P(n) is max number of pieces slicing plane with n cuts.
Spoiler: cake numbers
--
lelf
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