### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

### Natural Sum

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural numbers.

### Prime AP

Show that if three prime numbers, all greater than 3, form an arithmetic progression then the common difference is divisible by 6. What if one of the terms is 3?

# Charlie's Delightful Machine

##### Stage: 3 and 4 Challenge Level:

Note: if you have an iPod Touch, iPhone, or iPad, or have Safari on your phone, there's a webapp called 'Light Delights' that you can use instead of the machine below.

Charlie has a delightful machine with four coloured lights on it.
You can enter numbers into Charlie's machine. Each light is controlled by a rule, and if your number satisfies the rule, the light will go on. Some numbers may turn on more than one light!

Type in some numbers and see which lights you can switch on.
To start again with a new set of rules, click the Restart button.

Can you develop a strategy to work out the rules controlling each light?

What is special about a rule where all the 'light on' numbers
• are odd?
• are even?
• are a mixture of odd and even?
Now explore how to turn on several lights at once in the problem A Little Light Thinking.
You may also like to try Shifting Times Tables.