1
$\begingroup$

If we have a circular pizza then the maximum number of pieces we can get by making $7$ cuts in it?

The fact that I know the solution only got me the way to find it but it was like a kid trying to make all those cuts on a piece of a paper. is there some logical way of getting the answer?

$\endgroup$
  • 1
    $\begingroup$ Same as for the plane, math.stackexchange.com/q/209672 $\endgroup$ – user147263 Feb 15 '16 at 18:48
  • $\begingroup$ Hope this helps: I once went into a pizza shop and ordered a pie to go. When it finally came out, the counterman asked me, "Do you want it cut into six or eight pieces?" I thought about it a bit, then answered, "Better cut it into six pieces: I don't think I could eat eight." $\endgroup$ – Senex Ægypti Parvi Feb 15 '16 at 23:28
7
$\begingroup$

One line splits the plane in two parts. If we add a second line, we split it in four parts. With a third line that crosses both the previous line we split the plane in seven parts. Till now, the situation is the following: $$ \begin{array}{ccccc}\text{number of lines:} & 0 & 1 & 2 & 3 \\ \text{number of parts:} & 1 & 2 & 4 & 7\end{array}$$ Any further line may cross the previous $n$ lines, giving $(n+1)$ extra parts and no more.

It follows that the number of parts is a second-degree polynomial in the number of lines: since $\sum_{k=1}^{n}k = \frac{n(n+1)}{2}$, we have that $n$ lines split the plane in at most $$ \frac{n(n+1)}{2}+1 $$ parts, so seven lines may split the plane (or the circle) in $\color{red}{29}$ parts.

$\endgroup$
  • 1
    $\begingroup$ Each additional line (1) must not be parallel to any existing line and (2) must not pass through any existing intersection of two existing lines. $\endgroup$ – Senex Ægypti Parvi Feb 15 '16 at 23:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.