# November 2009, Stage 4&5

This month many of our problems focus on properties of number and calculating. 'Playing' with the mathematics of a situation is often a good way to get into a problem and several resources this month lend themselves to that approach.

## Problems

### What's Possible?

##### Stage: 4 Challenge Level:

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

### A Question of Scale

##### Stage: 4 Challenge Level:

Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?

### Plus Minus

##### Stage: 4 Challenge Level:

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

### Farey Approximation

##### Stage: 5 Short Challenge Level:

Use Farey sequences to obtain rational approximations to irrational numbers.

### Farey Fibonacci

##### Stage: 5 Short Challenge Level:

Investigate Farey sequences of ratios of Fibonacci numbers.

### Implicitly

##### Stage: 5 Challenge Level:

Can you find the maximum value of the curve defined by this expression?

### Aim High

##### Stage: 5 Challenge Level:

How do you choose your planting levels to minimise the total loss at harvest time?

### Ford Circles

##### Stage: 5 Challenge Level:

Small circles nestle under touching parent circles when they sit on the axis at neighbouring points in a Farey sequence.

### Cobalt Decay

##### Stage: 5 Challenge Level:

Investigate the effects of the half-lifes of the isotopes of cobalt on the mass of a mystery lump of the element.