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Break it Up!

Stage: 1 and 2 Challenge Level: Challenge Level:1

You have a stick of $7$ interlocking cubes. You cannot change the order of the cubes.

stick of seven differently coloured cubes

You break off a bit of it leaving it in two pieces.

Here are $3$ of the ways in which you can do it:

2 interlocking cubes and 5 interlocking cubes

3 interlocking cubes and 4 interlocking cubes

5 interlocking cubes and 2 interlocking cubes

In how many different ways can it be done?

Now try with a stick of $8$ cubes and a stick of $6$ cubes:

eight interlocking cubes of different colours six interlocking cubes of different colours

Make a table of your results like this:

Number of cubes

Number of ways

$6$ cubes

?

$7$ cubes

?

$8$ cubes

?

Now predict how many ways there will be with $5$ cubes.

Were you right?

How many ways with $20$ cubes? $50$ cubes? $100$ cubes?

ANY number of cubes?

* * * * * * * * * * * * * * * * * * * *

If all the cubes are the same colour, a split of $4$ and $2$ will look the same as a split of $2$ and $4$.

How many ways are there of splitting $6$ cubes now?

Can you predict how may ways there will be with any number of cubes?