Life of Pie

We maximize the number of pieces by making sure:

1. Every pair of cuts intersect in the pie.
2. No two pairs of cuts have overlapping intersections.

Each new cut intersects each of the previous cuts. In between individual intersections (as well as between intersections and the edge of the pie) the cut moves through a single piece. The $n^{th}$ cut intersects the previous $n-1$ cuts. Therefore it moves through $n$ distinct pieces, adding $n$ new pieces. Therefore after n cuts we have

$\dfrac{n(n+1)}{2}+1$ total pieces. For n=6 this is 22.

You can do this by trial and error, but the sketch will get a bit messy, and it is not particularly insightful. Better to think up a rule. So, think about what is happening when you add straight cuts.

The first cut splits the pie into 2 pieces.

The second cut makes 2 more pieces, bringing the total to 4

The third cut makes 3 more pieces, bringing the total to 7.

It looks like each cut adds the total number of pieces by the number of the cut – and with a little more thought we can see why this is true.

So, the fourth cut will make as many as 4 new pieces, the fifth cut 5 and the sixth cut 6. The largest number of pieces therefore is 22.

Slice1: 2x1 = 2 Slice2: 2x2 = 4 Slice3: 4+3 = 7 Slice4: 7+4= 11 Slice5: 11+5= 16 Slice6: 16+6=22. slice7: 22+7 =29 and so on ?

Maximum number of regions in which 'n' straight lines can cut a plane is given by {n(n+1)}/{2} +1 here n=6, so the answer is 22