# Number Operations - October 2005, All Stages

## Problems

### Same Length Trains

##### Stage: 1 Challenge Level:

How many trains can you make which are the same length as Matt's, using rods that are identical?

### Making Trains

##### Stage: 1 Challenge Level:

Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?

### Fractional Wall

##### Stage: 2 Challenge Level:

Using the picture of the fraction wall, can you find equivalent fractions?

### Fractions Made Faster

##### Stage: 2 Challenge Level:

Use the fraction wall to compare the size of these fractions - you'll be amazed how it helps!

### Combining Cuisenaire

##### Stage: 2 Challenge Level:

Can you find all the different ways of lining up these Cuisenaire rods?

### Fault-free Rectangles

##### Stage: 2 Challenge Level:

Find out what a "fault-free" rectangle is and try to make some of your own.

### Rod Fractions

##### Stage: 3 Challenge Level:

Pick two rods of different colours. Given an unlimited supply of rods of each of the two colours, how can we work out what fraction the shorter rod is of the longer one?

### Colour Building

##### Stage: 3 Challenge Level:

Using only the red and white rods, how many different ways are there to make up the other colours of rod?

### Different by One

##### Stage: 4 Challenge Level:

Make a line of green and a line of yellow rods so that the lines differ in length by one (a white rod)

### Stretching Fractions

##### Stage: 4 Challenge Level:

Imagine a strip with a mark somewhere along it. Fold it in the middle so that the bottom reaches back to the top. Stetch it out to match the original length. Now where's the mark?

### Harmonic Triangle

##### Stage: 4 Challenge Level:

Can you see how to build a harmonic triangle? Can you work out the next two rows?

### Data Chunks

##### Stage: 4 Challenge Level:

Data is sent in chunks of two different sizes - a yellow chunk has 5 characters and a blue chunk has 9 characters. A data slot of size 31 cannot be exactly filled with a combination of yellow and. . . .

### Pythagorean Fibs

##### Stage: 5 Challenge Level:

What have Fibonacci numbers got to do with Pythagorean triples?

### Fibonacci Fashion

##### Stage: 5 Challenge Level:

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

### Plus or Minus

##### Stage: 5 Challenge Level:

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.