Braille uses a matrix of six dots to represent letters, numbers and other language features.
The rectangles are two dots wide by three dots high - some examples are shown below:
The black dots represent "bumps" which are raised so that they can be read using a finger.
We want to know how many possible patterns of bumps and flat spaces you can make.
The first pattern above has no bumps, and is not used in Braille, but it is still a valid pattern.

 Question 1 It is quite tricky to work out how many patterns there are for six dots, so we'll start with one dot. There are two possible patterns in this case, because the single dot can be on or off. Enter the two possible patterns in the grid below and click the "Check It!" button after each one. The computer will tell you when you have found all the patterns.
 Question 2 Now we want to see how many different patterns you can make with two dots. Remember each dot can be on or off. Enter each pattern in the grid and click the "Check It!" button after each one. The computer will tell you when you have found all the patterns.
 Question 3 Now we will look for all the patterns you can make with three dots. The computer will tell you when you have found all the patterns.

Question 4

We want to predict how many patterns you can make with four dots.
 How many patterns could you make with one dot? How many patterns could you make with two dots? How many patterns could you make with three dots? How many patterns do you think you will be able to make with four dots?
 Question 5 Now check that you can find all the patterns which use four dots.

Question 6
Now we should be able to predict how many patterns you can make with six dots.
Fill in the table below, including your predictions for how many patterns you could make with 5 and 6 dots.
 Number of dots in the grid 1 2 3 4 5 6 Maximum number of patterns

Question 7

Can you explain what the pattern is in the sequence of numbers above?
Write your answer in words, but keep it as simple as possible.

Question 8

This question is quite hard - you may need to ask a teacher for help!
If n is the number of dots in a grid, and p is the maximum
number of different patterns you can make with that grid,
what is the formula for p in terms of n?