# Your Solutions - Stage 4 & 5

### Take Three from Five

##### Age 11 to 16Challenge Level

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

### Number Rules - OK

##### Age 14 to 16Challenge Level

Can you produce convincing arguments that a selection of statements about numbers are true?

### Always Perfect

##### Age 14 to 18Challenge Level

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

### Common Divisor

##### Age 16 to 18Challenge Level

Can you find out what numbers divide these expressions? Can you prove that they are always divisors?

### Polite Numbers

##### Age 16 to 18Challenge Level

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?

### Digital Equation

##### Age 16 to 18Challenge Level

Can you find a three digit number which is equal to the sum of the hundreds digit, the square of the tens digit and the cube of the units digit?

### The Better Choice

##### Age 14 to 16Challenge Level

Here are two games you can play. Which offers the better chance of winning?

##### Age 14 to 16Challenge Level

Can you match each graph to one of the statements?

### Curve Hunter

##### Age 14 to 18Challenge Level

This problem challenges you to sketch curves with different properties.

### Curve Fitter

##### Age 14 to 18Challenge Level

This problem challenges you to find cubic equations which satisfy different conditions.

### Exploring Cubic Functions

##### Age 14 to 18Challenge Level

Quadratic graphs are very familiar, but what patterns can you explore with cubics?

### Curvy Equation

##### Age 16 to 18Challenge Level

This problem asks you to use your curve sketching knowledge to find all the solutions to an equation.