Stop Press: 2006 Competion announced. See my challenge here.
In 2004 and 2005 I was the Faculty Advisor for a team in the Mathematical Contest in modelling (MCM). It was very interesting especially for the students who got to have their own office in Burnside for a long weekend. I will happily continue to supervise teams in the future.
See the MCM homepage at the COMAP website for more details on the contest.
Make a model to help you determine the optimal number of tollbooths to deploy in a barrier-toll plaza. Explicitly consider the scenario where there is exactly one tollbooth per incoming travel lane. Under what conditions is this more or less effective than the current practice? Note that the definition of "optimal" is up to you to determine.
Team 244: Mathieu Guay-Paquet, Marc-Andre Rousseau, Haining Wang
Department of Mathematics and Statistics, McGill University
To model the behaviour of the cars as they approach the toll booths, we compute their acceleration based on the distance to the car in front of them, using modified spring equations. The general idea is that cars slow down when they are close together, and speed up when there is enough room between them. We also take into account the fact that cars do not move backwards in general, and have to wait a certain time at the toll booth itself. The variables we considered are the average acceleration and speed of a vehicle, the number of lanes in the toll station, the time spend at the toll booths, and the length of the toll station. We assume the cars come into the toll station randomly.
Based on this theory, we used Matlab to make simulations of cars going through a toll booth. Our goal was to display graphs of density vs position and time to show how the traffic would flow through the gates depending on different values for the number of gates. Using these graphs, we establish how many toll booths are needed to let traffic flow through relatively undisturbed.
Our model should describe the details of car behaviour when approaching a toll booth and show how a traffic jam could occur. It does not take into account the specific distribution of the cars coming into the toll station (e.g., peak hours), and it does not provide us with a simple relationship between the number of lanes, the distribution of cars coming in, and the building up of traffic.
This solution was obtained between 8pm on Thursday 3th February and 8pm on Monday 7th February. It earned an "Honorable Mention".
Team 421: Alexandre Gadbois, Jesse McKeown, Colin McNally
Department of Mathematics and Statistics, McGill University
We propose a model estimating the probability of a coincidence of thumb prints occurring among the cumulative world population. Our approach attempts to maintain a maximum of abstraction beyond the biomechanical details giving rise to thumb prints in individuals. We construct a hypothetical space of thumb prints, regarding specifics of minutiae ("ridge topology") and medium-scale ridge directionality ("curve pattern") as independent aspects of a thumb print. Using this decomposition, we calculate probabilities of each ridge topology, and each curve pattern , arriving at the aforementioned probability estimates. Model parameters primarily include feasibly measurable quantities, and we find that our model is stable with respect to variations in these inputs.
Our model assumes that an entire, minimally distorted thumb print is available for identification, and so does not give results useful to most forensic circumstances. However, under such properly controlled, idealized circumstances, we conclude that the probability of a measured thumbprint coincidence occurring among a population of one hundred billion people is on the order of one in 1016. Furthermore, we find that given an observed coincidence, there is a probability of 3 in 4 that the repeated print is of arch type and has no minutiae. Unfortunately, owing to the equivocal nature of published opinions on the reliability of DNA-based identification, we find it difficult to compare this result with the probability of misidentification based on DNA evidence.
This solution was obtained by Colin, Alexandre and Jesse between 8pm on Thursday 5th February and 8pm on Monday 9th February. It earned an "Honorable Mention".