Q: You are walking north down the sidewalk on the west side of the street towards an intersection with a stoplight. You need to cross to the northeast corner of the intersection. The street you are walking along is quiet and you could cross diagonally at any time.
What will be the fastest way, on average, for you to get to the northeast corner?
A: Do NOT cross the street diagonally. Wait until you reach the intersection and then cross each way when you have the green light.
Why? When you reach the intersection, you will be able to cross in one of the two directions. You will have, on average, half the time of the light in one direction to get across. The light will then change and you can cross in the perpendicular direction.
If you cross the street diagonally, you will reach the intersection a few seconds later. You only need to cross in the north direction - but if the light just changed, you may need to wait for a full cycle before you can cross that way. If you had waited until you reached the intersection, you would have had the light to continue walking north, and then the light would change and you could cross to the northesat corner. Instead you are stuck on the southeast corner waiting for the light to turn green again.
So, the best case scenario is that you save a couple of seconds from the diagonal, and the light is green when you get to the intersection. But the worst case scenario is that you have to wait a full extra half cycle to get across.
Of course, if you can see the light up ahead and you know the length of the cycle, you calculate which direction would be green when you reach the intersection and then decide whether to cross diagonally based on that calculation. I have tried to do this calculation, but it's very difficult to time everything correctly.
Posted by Stephen Bronstein at September 25, 2006 11:09 PM