Batch 3 - Class 155 - Traffic Jams

• Arrange six people in a pattern so that each of them can see exactly 4 other people. Note that a person in middle can block the view for what is beyond them.
• Answer:

• (MartinShCol - 6.8) Design a cork so that it fits a square, a circular and a triangular hole snugly.
• Answer: Take a cylindrical cork standing on a face, and cut a triangle as you look at it. The base provides the circular fit, the cylinder's cross section provides square, and the triangle is the third.

• What causes traffic jams? Why do traffic jams take time to clear up? If cars were all moving at the same speed at all times, would jams happen? Why do we see traffic flowing smoothly and suddenly having to stop?
• Let us try to model this out. Imagine a road which has "slots" for cars. Initially, cars are all moving at the same speed. To give them space to move, let us assume that at any instance, the next slot is available for a car to move into. Draw what the movement pattern of cars looks like from one time instance to another.

• Now let us assume a chicken crosses a road. That is, it blocks an empty slot for one time instance. What will happen?
• Let us see the effect of this. For this let us assume some rules. If a car sees its next slot as empty, it will move there in the next time instance, else it will remain where it is. Draw what happens to the pattern of traffic for next 7-8 time instances. What do you observe?

• Note that the stoppage percolates through the traffic. Where is it moving? Why is it not clearing off, even though the chicken is long gone? How can it go away?
• In reality, if a car is stopped, human reactions take time to restart. What if we had the following rules?
• If a car is moving (black color), and next slot is empty, it moves forward and continues to move
• If a car is moving (black color), and next slot is occupied, it brakes (doesn't move forward and turns red)
• If a car is stopped (red color), and next slot is occupied, it remains stopped (doesn't move forward and remains red)
• If a car is stopped (red color), and next slot is empty, it gears up (doesn't move and turns orange)
• If a car is geared up (orange color), and next slot is empty, it moves forward and turns black
• If a car is geared up (orange color), and next slot is occupied, it stops again (doesn't move and turns red)
• Can you model a few time steps with the above rules. What happens?

• What are your observations? What's happening to the length of the jam? What is happening in front of the jam?
• What happens in real life, how does the jam clear?
• What happens on intersections? Does traffic move smoothly, or does it tend to clog up? Can you reason why? Why are high traffic intersections converted to flyovers?
• What happens when we change lanes?
• Imagine a circular track where cars are moving at a uniform pace. What will happen if a car slows down momentarily.
• What should we do to avoid traffic jams?
• Why do ants not get stuck in traffic, even though they move in a line?

• If we were driving cars on a highway, how should we position ourselves so as to minimize the jams? When should different people accelerate? Can we think about a system to make it happen? Why don't we see jams on metro lanes?

• Suppose we want to go from START to END, and there is a certain network of roads to get there. Let us assume that we upgrade the network to improve one of the roads. Will the time to get from START to END improve? Can it actually worsen?
• Lets take a real example as under. You can go from START to END via Town A or Town B. START to Twon A and Town B to END are highways and it takes a car 105 minutes regardless of number of cars. START to Town B and Town A to END are slower rates, and if there are N cars on the road, it takes all of them N minutes each. There is a old dirt road from Town A to Town B which takes 100 minutes. What route will cars take, and how much time will it take them from START TO END.

• Now lets say we upgrade the road from Town A to Town B so that it only takes 2 minutes. What happens?

• This is called Braess's paradox. An even more counter-intuitive example is here https://www.youtube.com/watch?v=ekd2MeDBV8s
• This was actually used in South Korea to tear down a motorway to restore a river, which actually led to improvement in traffic. Such initiatives are being considered in other cities.  https://www.theguardian.com/environment/2006/nov/01/society.travelsenvironmentalimpact

• Alice and Bob are trapped by an evil logician. In their cells, Alice can see 12 trees and Bob can see 8 trees. They are told that together they see all the trees and no tree is seen by both people. 

• Answer: Yes https://mindyourdecisions.com/blog/2016/03/20/the-seemingly-impossible-escape-sunday-puzzle/