Competition Introduction and Overview
Part I: When are Traffic Lights Feasible?
Part II: What Happens when a Light Turns Green?
Part III: Find the Best Light Timings
Details on The Mathematical Model Used
Run the Computer Simulation
Switch to text-only version (no graphics)
Access printed version in PostScript format (requires PostScript printer)
Go to University of Toronto Mathematics Network
Welcome to a competition that was held in 1996 by the
UNIVERSITY OF TORONTO MATHEMATICS NETWORK.
Although that competition is over, you are still invited to try your
hand at exploring mathematics in action in this real life
Keep The Traffic Moving!
Everybody knows how frustrating it is to sit at a red light. Worse than
that, a light which is red too long can cause cars to back up endlessly,
leading to gridlock and other chaos. It is the job of the traffic engineer
to time things so that the traffic has the least amount of waiting to do.
In this competition, you are the traffic engineer. You control the
timings of the lights at the six intersections shown in the picture
below. Each road has two lanes, one in each direction. The lanes are
each 4 metres wide (so each road is 8 metres wide), and the spacing
between the roads is as shown. For convenience we've numbered the
lanes from 1 to 10, indicated their directions, and labelled the
lights A through F.
You can control these lights yourself through a
computer simulation available on our web site.
What You (the Traffic Engineer) Already Know
You have measured the traffic densities d, . . . , d: the
number of cars per minute trying to use each lane.
You have also found that the traffic behaviour (the way a car speeds
up or slows down when faced with an obstacle in front, such as a red
light or another car) follows a certain mathematical model governed by
three numbers c, l, and L (the model is described in the section
Details on the Mathematical Model
What You Need to Figure Out
This competition has three parts; we encourage you to test your
mathematical skills on any or all of them! Parts I and II require you
to do some serious mathematical thinking, but in Part III all you have
to do is time the lights and see if you can beat the current "high
scores" on the computer. So, if you're not in a serious mathematical
mood right now, you can skip to part III.
To get the full details on each part you
can use the "I", "II", "III" icons at the top and
bottom of each page. Here's a brief description of them:
- How can you use the information you have (the
thirteen numbers d, . . . , d, c, l, and L)
to decide whether traffic lights
are even feasible, or whether a more drastic solution is required, such
as widening the roads or building overpasses?
In the detailed discussion of this part, we'll
give you hints on how basic algebra can be used to come up with a
simple set of inequalities that tells whether or not traffic lights
would be capable of handling the traffic flow. See if you can figure
out this formula. You can test your answer on the computer
- What happens when a long light of cars is waiting at
a red light, and the line turns green? If you've ever been near the
back of such a line, you've probably sat in frustration wishing the
cars in front would get a move on! But is it only their slowness in
In the detailed discussion of this part, we'll explain
how ideas of calculus and " differential equations" can be used
to answer the question, and challenge you
to find out how long it takes for the line to
get moving even if everybody's reaction time is instantaneous and
there's no limit on the cars' acceleration power. You may be surprised at
the answer, and less inclined to blame the cars in front of you next
time you're kept waiting! Note: the mathematics in this part
is quite challenging and requires a knowledge of calculus. But you can
still watch what happens on the computer
simulation even if you can't solve the problem
- Find the best light timings for three specific
situations. In the detailed discussion of this
part, we give you three different situations (three different sets of
numbers for d, . . . , d, c, l, and L). Your job: time
the traffic lights in a way that causes the least amount of delay.
This part of the competition is a free-for-all: you can use any combination
of mathematical reasoning or pure experimentation. The computer
lets you enter timing values, watch what happens, and get a
score for how well your timings did. It keeps track of the high score,
so see if you can beat it and get a new high score!
Use the links above or below
to get more information on each of these three competition
questions and on the mathematical model used, or to access
the computer simulation.
This page last updated: April 12, 1997
Original Web Site Creator / Mathematical Content Developer:
Current Network Coordinator and Contact Person:
Joel Chan - firstname.lastname@example.org