Manoel Menezes 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