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Hawthorne Traffic Equation

I refer to this formula as the Hawthorne Traffic Equation, named after a friend Casey Hawthorne, who scored an 800 on the math portion of the SAT. Some time ago while Casey was still an undergrad (he's 10 years younger than me), I asked him to help me figure out the number of possible routes in a network. It is a simple factoral equation.

If you start from one corner of a grid at Point A, and want to go to the opposite corner at Point B, then where x = number of east/west blocks, and y = number of north/south blocks, the number of possible routes = (x+y)! / x!y!. Note, this formula works if you can only move in the direction of A to B (no doubling back). If you did allow doubling back, the number of possible routes would be even higher.

Thus if you have a grid of two blocks by two blocks, with Point A at the southwest corner of the grid, and you want to get to point B at the northeast corner of the grid, the number of possible routes = (2+2)! / 2! x 2! = 4 x 3 x 2 x 1 / 2 x 1 x 2 x 1 = 24 / 4 = 6 possible routes. If your grid is three blocks by two blocks, then the number of possible routes = (3+2)! / 3! x 2! = 5 x 4 x 3 x 2 x 1 / 3 x 2 x 1 x 2 x 1 = 120 / 12 = 10 possible routes.

If your grid is 10 blocks by 10 blocks, then using the formula you will find that there are 184,756 possible routes. The reason this is so important is that it allows the streets to be small and on more of a human scale. When I give a presentation, I always follow this little demonstration with a quote from Jesus who said, according to Matthew 7:13-14 "Enter through the narrow gate. For wide is the gate and broad is the road that leads to destruction, and many enter through it. But small is the gate and narrow the road that leads to life. And only a few find it." I have found this quote to be very effective when arguing with D.O.T. officials.

Incidentally, if you've ever wondered how computer chips become increasingly powerful, this is part of the reasoning. Computer engineers are able to fit ever more circuits on a single chip.

Victor Dover put this presenation into a nice power point presentation.

Vince Graham


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Last Updated: April 18, 2005