### Rule of Three

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

### Crossing the Atlantic

Every day at noon a boat leaves Le Havre for New York while another boat leaves New York for Le Havre. The ocean crossing takes seven days. How many boats will each boat cross during their journey?

### On Time

On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands?

# Buses

### Why do this problem?

This challenge requires a logical, systematic approach. The solution might be unexpected to some, and might cause disagreement amonst students. Whilst the problem might seem to be algebraic, it is more an exercise in visualisation, time and motion.

### Possible approach

Ask students to tell their neighbours their initial guesses as to the number of buses that will be passed. Give students time to work on paper, checking out their ideas and developing convincing arguments. Ask for volunteers to explain their ideas to the group. There is likely to be disagreement which (ideally) should prompt clarity of explanation.
It could be useful to model the process with the students as follows:

Split some volunteers into two queues of buses, one on each side of the room.
Set up three intermediate bus stops. Every ten seconds ask each bus to move to the next stop.
They must count each time they walk past someone. Continue until at least 6 buses have crossed in each direction.
How many crossings did each person make?
Is there anything about the journey that is more obvious now that it has been acted out?

Students could be asked to write a full clear solution. Some of these could be shared with the group.
How convincing and clear did the group find these explanations?

This may be an opportunity to introduce a distance-time graph to represent the situation. It demonstrates the number of solutions very elegantly, but students often find these graphs confusing. The teacher could put a completed graph on the board, and ask the students to make sense of it. It would be useful to discuss the meaning of:
a horizontal strip - the view from a particular point on the route,
a vertical strip - a summary of all bus positions at a specific moment in time,
a diagonal strip - a description of the journey for a particular bus.

Emphasise that this is not an artistic picture of events, and certainly not a road map, but it summarises a lot of information, to give an overview of a complicated situation.

### Key questions

• Is there a difference between the first bus of the day and a bus which sets off later on in the day?
• Does each bus make the same number of crossings?
• Is there a clear way of recording the motion of the buses?

### Possible extension

Investigate the number of crossings if the journey time and/or gap between buses is altered.
Does it matter if the gap time isn't a factor of the journey time?
Let the bus journey be up (or down) a long hill - so that the speed in one direction is different to the speed on the return journey.
Can students summarise/generalise their solutions to the variations they have tried?

### Possible support

Simplify the numbers - eg 30 minute journey time.
Provide counters to allow students to model the motion of the buses, establishing bus stops at 10 minute intervals to make the movement easier.