### 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.

### Series Sums

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

# Charlie's Delightful Machine

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

You may wish to look at Shifting Times Tables before trying this problem.

Charlie's Delightful Machine has four coloured lights. Each light is controlled by a rule.
If you choose a number that satisfies the rule, the light will go on.
Some numbers may turn on more than one light!

Start by exploring Level 1. Type in some numbers and see which lights you can switch on.
To start again with a new set of rules, click the Level 1 button.

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

Once you have a strategy, challenge yourself to find some four-digit numbers that turn on each light.

Once you are confident that you can work out the rules for Level 1 lights, have a go at A Little Light Thinking, where you can explore how to turn on several lights at once.

You may also wish to explore the Level 2 and 3 lights (which use a different type of sequence) in the same way.