What is the maximum number of the pieces that a watermelon can be cut into 10 knife cuts? (circle cutting problem extension) I saw this problem in a game "Brain out" level 85. In this game, the answer is 1024, simply 10th power of 2. But I learned circle cutting problem in school and by solving recurrence formula, I could get explicit formula for the problem.
I think this problem also require recurrence formula similarly and the answer is not just 1024. What is the true answer about this game stage?
The original watermelon is (presumably) a convex shape. Performing a straight cut with a convex shape results in wo convex shapes. If you can arrange $n$ convex shapes in a way such that the cutting plane intersects each part, this gives you $2n$ convex shapes. So, if you are allowed to rearrange the pieces before applying your next knife cut, you can produce $2^k$ pieces with $k$ cuts.

If you are not allowed to rearrange the pieces, the answer is strictly lower for $k&gt;3$ cuts of a convex watermelon in $\Bbb  R^3$.

In fact, if we count cuts through different pieces as different cuts (even when performed in the same movement), the answer is much simpler: After $k$ cuts, we have $k+1$ pieces.

If the expected answer is $1024$, it appears (though could perhaps not be inferred from the problem statement) that rearrangement of pieces is allowed.
i think that answer's actually right since it doesn't have any limitations like the watermelon cannot be rearranged. But if you weren't allowed to rearrange it and some force were holding the watermelon pieces together as you cut, the answer would be $176$. It's the same as the cake number problem and follows the formula $(n^3+5n+6)/6$