Different, motivated proofs: How do I cut a cake into as many pieces while making a fixed number of cuts? What are some different ways of motivating and deriving from first principles, such that a high schooler, and not a particularly good high schooler at math, can see that the formula for the number of ways of cutting a cake into as many pieces while making a fixed number of cuts is$${{x^2 + x + 2}\over2}?$$
 A: The strategy for getting the most number of pieces is to make sure each successive cut intersects every previous cut, but doesn't pass through the intersection of any two previous cuts.  To motivate that, imagine that a cut does pass through a previous intersection.  By moving the cut just a bit, a new piece is created.  Then imagine the new cut not intersecting a previous cut.  By moving it till it intersects, we get a new piece.  
Second, how many new pieces are created with the $n$th cut?  You create one by starting the cut.  Then you create one more each time you intersect a previous cut.  So by making the $n$th cut, you create $(n-1)+1 = n$ new pieces.  
Third, (this is really induction) if the formula works for $n-1$ cuts, then it works for $n$ cuts: The paragraph above says that "new number of pieces equals old number of pieces plus $n$."  If the formula works for $n-1$, then after $n-1$ cuts, there should be $\frac{(n-1)^2+(n-1)+2}{2}$ pieces.  So the new number of pieces is
$$\frac{(n-1)^2+(n-1)+2}{2} + n = \cdots \mbox{ some algebra } \cdots =\frac{n^2+n+2}{2}.$$
Since the formula works for $n=0$, we should be convinced.
