Doodles

What's a doodle? Is it what you do while talking on the phone? Do you daydream and doodle while the boring math professor mumbles on and on about the marvels of the quadratic formula? Of course you do! Or is a doodle more than that? Engineers who design your cell phone and cable connections call a doodle by its more technical name: network. Every doodle is made up of points, lines, and areas. Mathematicians, who prefer Latin and Greek rather than English, call these vertices, edges, and regions.

This doodle has 5 points, 8 lines, and 4 areas.
A point is where two or more lines intersect.
An area is enclosed by one or more lines.

This doodle has 5 points, 10 lines, and 6 areas.
Lines may be either crooked or straight.
A line connecting two points is a segment.
A line connecting a point to itself is a loop.

Oodles of Doodles...

Observe the doodles illustrated below. Notice that all the doodles consist of connected lines. There are no eyeballs staring out at you, nor disconnected dots floating around.

In any doodle, it is harder to count the lines than to count the points and areas. It would be helpful if we had a formula that predicted the number of lines if we knew the number of points and areas.

FOR 5 POINTS:
Write a formula that relates the number of lines, L, in a doodle to its number of points, P, and areas, A. Your formula should work for any connected doodle. You may use the table below to help you count the points, lines, and areas in the 38 doodles above. You do not need to complete the table. Complete only as many rows as it takes to see the pattern. Write this pattern as your formula. Hint: Describe the pattern first in words. This will help you write the formula easier.

Try out your formula by creating doodles of your own in the doodle box at the top of the page. Do all your points, lines, and areas relate as your formula predicts they will? If you have forgotten what formulas look like, here are some famous examples using P, L, and A as variables:

FAMOUS FORMULAS