### Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

### Overarch 2

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

### Cushion Ball

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

# Epidemic Modelling

##### Stage: 4 and 5 Challenge Level:

This is an entirely open problem to which there are no 'correct' answers. Indeed the answers are not even known. You are doing genuine research. You can choose the characteristics of the disease you want to model and study.

The environment is designed so that the researcher can use a mathematical model to explore the possibilities of the spread of the disease before it actually hits a real population. Doctors can estimate the probability of death and the duration of the infectious period for different diseases. Based on this knowledge the public health officials may decide on advising the public whether sick individuals should stay at home to limit the risk of spreading the infection, or even advise that all infected individuals are put into isolation to curb the spread of the disease.

These projects can be carried out with only a basic knowledge of probability and statistics or with a higher level of knowledge. The more mathematics you use to analyse your results the more potential value there might be in your findings.

• Decide how many individuals there are in your population.
• Decide how many individuals are infected with the disease at the start.
• Decide on how many individuals (if any) will be vaccinated in advance.
• Decide on the probability $p$ of catching the disease on contact with it.
• Decide on the probability $q$ of dying and $1-q$ of recovering from the disease.
• Decide how many days the disease lasts.
• Decide on the incubation period during which time the sick person is in circulation and passes on the disease to others.
• Choose one of these three options: After the incubation period, and while the illness lasts, the sick individual can either (1) circulate in the population as normal or (2) stay at home (modelled by remaining static) or (3) be put in isolation and have no contact with others.
• Decide whether recovered individuals can be re-infected or not. If not they will develop an immunity in which case they become dark green on the display.
• Model a day as a trial during which every healthy individual either stays in the same place or moves one square in one of 8 directions.
• Note that paths touching the edge of the board continue along the 'reflected' direction back into the village. To avoid collisions both individuals stop in their tracks.
• If a healthy individual, who is not immune, lands on a square that touches the space of a sick individual, either corner to corner or edge to edge, then the healthy individual is immediately infected.