From Chengyu Wu comes a puzzle that cuts right to the chase:
You have a square piece of paper. You fold it in half along an axis parallel to two of its sides. You then fold it in half along another axis that is perpendicular to the original one, so that you again have a square shape that is one-quarter the size of the original square.
At this point, you make three cuts along three straight lines, all the way through the folded paper. As you unfold the paper, what is the greatest number of separate pieces you could have?
Extra credit: Instead of three cuts, suppose you make N straight cuts. Now what is the greatest number of pieces you could have?
The third cut will split two pieces specific to each quadrant, and two pieces shared by two quadrants.
$C(3) = C(2) + 4*2 + 2$
The fourth cut will split three pieces specific to each quadrant, and two pieces shared by two quadrants.
$C(4) = C(3) + 4*3 + 2$
This pattern continues. $C(N) = C(N-1) + 4(N-1) + 2$. The table below summarizes.
| Cuts | Maximum Pieces |
|---|---|
| $1$ | $5$ |
| $2$ | $12$ |
| $3$ | $22$ |
| $4$ | $36$ |
| ... | ... |
| $N$ | $2N^2 + 4$ |
