Stick Areas
Suppose you have 12 sticks to use to enclose a region with horizontal and vertical sides. These three possible ways for doing that enclose areas of size 3, 5, and 9.
THE CHALLENGE
Find all the possible areas that you can create with 12 sticks.
EXPLORATION
What happens when you change the number of sticks? Are there some numbers of sticks that don’t work at all?
Notes
THE CHALLENGE
Here are some examples for creating areas from 3 to 9.
EXPLORATION
The sticks can always be paired up, so only even numbers of sticks will work. For the even numbers, two kinds of things can happen.
Multiples of 4: These numbers will behave just as 12 did. For a number of the form 4n, the smallest area will be produced by n 1 by 1 squares, and the largest by an n by n square.
Multiples of 4 plus 2: These numbers will be similar to a multiple of four, only they will have an extra bump. For a number of the form 4n + 2, the smallest area will be produced by (n – 1) 1 by 1 squares plus one 2 by 1 rectangle. The largest area will be an n by (n + 1) rectangle.