# Mathematical Modelling - November 2007, All Stages

## Problems

### How Many Pieces?

##### Age 5 to 7 Challenge Level:

How many loops of string have been used to make these patterns?

### How Many Pieces This Time?

##### Age 5 to 7 Challenge Level:

How many pieces of string have been used in these patterns? Can you describe how you know?

### Train Routes

##### Age 5 to 7 Challenge Level:

This train line has two tracks which cross at different points. Can you find all the routes that end at Cheston?

### Simple Train Journeys

##### Age 5 to 11 Challenge Level:

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

### Reef and Granny

##### Age 7 to 11 Challenge Level:

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?

### Turning Granny

##### Age 7 to 11 Challenge Level:

A brief video looking at how you can sometimes use symmetry to distinguish knots. Can you use this idea to investigate the differences between the granny knot and the reef knot?

### Twisting and Turning

##### Age 11 to 14 Challenge Level:

Take a look at the video and try to find a sequence of moves that will take you back to zero.

### More Twisting and Turning

##### Age 11 to 14 Challenge Level:

It would be nice to have a strategy for disentangling any tangled ropes...

### Turning Tangles

##### Age 11 to 14 Challenge Level:

Look carefully at the video of a tangle and explain what's happening.

### All Tangled Up

##### Age 11 to 14 Challenge Level:

Can you tangle yourself up and reach any fraction?

### Cola Can

##### Age 11 to 14 Challenge Level:

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

### Under the Ribbon

##### Age 14 to 16 Challenge Level:

A ribbon is nailed down with a small amount of slack. What is the largest cube that can pass under the ribbon ?

### The Legacy

##### Age 14 to 16 Challenge Level:

Your school has been left a million pounds in the will of an ex- pupil. What model of investment and spending would you use in order to ensure the best return on the money?

### The Fire-fighter's Car Keys

##### Age 14 to 16 Challenge Level:

A fire-fighter needs to fill a bucket of water from the river and take it to a fire. What is the best point on the river bank for the fire-fighter to fill the bucket ?.

### The Invertible Trefoil

##### Age 14 to 16 Challenge Level:

When is a knot invertible ?

### Funnel

##### Age 14 to 16 Challenge Level:

A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?

### Bridge Builder

##### Age 16 to 18 Challenge Level:

In this short problem we investigate the tensions and compressions in a framework made from springs and ropes.

### Overarch 1

##### Age 16 to 18 Short Challenge Level:

This short question asks if you can work out the most precarious way to balance four tiles.

### Odd One Out

##### Age 16 to 18 Short Challenge Level:

In this short problem, can you deduce the likely location of the odd ones out in six sets of random numbers?

### Making More Tracks

##### Age 16 to 18 Challenge Level:

Given the equation for the path followed by the back wheel of a bike, can you solve to find the equation followed by the front wheel?

### More Bridge Building

##### Age 16 to 18 Challenge Level:

Which parts of these framework bridges are in tension and which parts are in compression?