### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### It Figures

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

### Bracelets

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

# Counting Cogs

##### Stage: 2 and 3 Challenge Level:

This problem has been designed to be worked on in a group. For more details about how you might go about doing this, please read the Teachers' Notes.

Here are nine different cogs:

Take a pair of cogs. Mark a tooth on the first cog with a black dot. As the two cogs move around each other, note which gaps on second cog the marked tooth goes in to.

Here are some examples, where the first cog in the pair is one with six teeth.

When the second cog also has six teeth, the marked tooth only ever meets one of the six gaps on the second cog (the one also marked with a black dot):

When the second cog has seven teeth, the marked tooth meets each of the different coloured gaps on the second cog:

When the second cog has nine teeth, the marked tooth only goes in to the cogs marked with black or yellow dots:

Which pairs of cogs let the coloured tooth go into every 'gap' on the other cog?

Which pairs do not let this happen? Why?

Can you explain how to determine which pairs will work, and why?

You could cut out the cogs from these sheets or you could use the cogs interactive environment to try out your ideas.